B
[Bab85] László Babai. Trading Group Theory for Randomness. In STOC'85, pages 421-429. ACM Press, May 1985.
@inproceedings{stoc1985-Bab,
  author =              {Babai, L{\'a}szl{\'o}},
  title =               {Trading Group Theory for Randomness},
  booktitle =           {{P}roceedings of the 17th {A}nnual {ACM} {S}ymposium
                         on the {T}heory of {C}omputing ({STOC}'85)},
  acronym =             {{STOC}'85},
  publisher =           {ACM Press},
  pages =               {421-429},
  year =                {1985},
  month =               may,
}
[BAS02] Armin Biere, Cyrille Artho, and Viktor Schuppan. Liveness Checking as Safety Checking. In FMICS'02, Electronic Notes in Theoretical Computer Science 66(2). Elsevier, July 2002.
@inproceedings{fmics2002-BAS,
  author =              {Biere, Armin and Artho, Cyrille and Schuppan,
                         Viktor},
  title =               {Liveness Checking as Safety Checking},
  editor =              {Cleaveland, Rance and Garavel, Hubert},
  booktitle =           {{P}roceedings of the 7th {I}nternational {ERCIM}
                         {W}orkshop in {F}ormal {M}ethods for {I}ndustrial
                         {C}ritical {S}ystems ({FMICS}'02)},
  acronym =             {{FMICS}'02},
  publisher =           {Elsevier},
  series =              {Electronic Notes in Theoretical Computer Science},
  volume =              {66},
  number =              {2},
  year =                {2002},
  month =               jul,
}
[BB91] Jos C. M. Baeten and Jan A. Bergstra. Real Time Process Algebra. Formal Aspects of Computing 3(2):142-188. Springer-Verlag, 1991.
@article{fac3(2)-BB,
  author =              {Baeten, Jos C. M. and Bergstra, Jan A.},
  title =               {Real Time Process Algebra},
  publisher =           {Springer-Verlag},
  journal =             {Formal Aspects of Computing},
  volume =              {3},
  number =              {2},
  pages =               {142-188},
  year =                {1991},
}
[BB02] Dietmar Berwanger and Achim Blumensath. The Monadic Theory of Tree-like Structures. In Erich Grädel, Wolfgang Thomas, and Thomas Wilke (eds.), Automata, Logics, and Infinite Games, Lecture Notes in Computer Science 2500, pages 285-302. Springer-Verlag, 2002.
@incollection{lncs2500-BB,
  author =              {Berwanger, Dietmar and Blumensath, Achim},
  title =               {The Monadic Theory of Tree-like Structures},
  editor =              {Gr{\"a}del, Erich and Thomas, Wolfgang and Wilke,
                         Thomas},
  booktitle =           {Automata, Logics, and Infinite Games},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2500},
  pages =               {285-302},
  year =                {2002},
}
[BB04] Laura Brandán Briones and Ed Brinksma. A Test Generation Framework for Quiescent Real-Time Systems. In FATES'04, Lecture Notes in Computer Science 3395, pages 64-78. Springer-Verlag, September 2004.
@inproceedings{BB-fates2004,
  author =              {Briones, Laura Brand{\'a}n and Brinksma, Ed},
  title =               {A~Test Generation Framework for Quiescent Real-Time
                         Systems},
  editor =              {Grabowski, Jens and Nielsen, Brian},
  booktitle =           {{P}roceedings of the 4th {I}nternational {W}orkshop
                         on {F}ormal {A}pproaches to {S}oftware {T}esting
                         ({FATES}'04)},
  acronym =             {{FATES}'04},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3395},
  pages =               {64-78},
  year =                {2004},
  month =               sep,
  doi =                 {10.1007/978-3-540-31848-4_5},
}
[BBB+09] Christel Baier, Nathalie Bertrand, Patricia Bouyer, and Thomas Brihaye. When are timed automata determinizable?. In ICALP'09, Lecture Notes in Computer Science 5556, pages 43-54. Springer-Verlag, July 2009.
@inproceedings{icalp2009-BBBB,
  author =              {Baier, {\relax Ch}ristel and Bertrand, Nathalie and
                         Bouyer, Patricia and Brihaye, {\relax Th}omas},
  title =               {When are timed automata determinizable?},
  editor =              {Albers, Susanne and Marchetti{-}Spaccamela Alberto
                         and Matias, Yossi and Nikoletseas, Sotiris and
                         Thomas, Wolfgang},
  booktitle =           {{P}roceedings of the 36th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'09)~-- Part~{II}},
  acronym =             {{ICALP}'09},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {5556},
  pages =               {43-54},
  year =                {2009},
  month =               jul,
  doi =                 {10.1007/978-3-642-02930-1_4},
}
[BBB+07] Christel Baier, Nathalie Bertrand, Patricia Bouyer, Thomas Brihaye, and Marcus Größer. Probabilistic and Topological Semantics for Timed Automata. In FSTTCS'07, Lecture Notes in Computer Science 4855, pages 179-191. Springer-Verlag, December 2007.
@inproceedings{fsttcs2007-BBBBG,
  author =              {Baier, {\relax Ch}ristel and Bertrand, Nathalie and
                         Bouyer, Patricia and Brihaye, {\relax Th}omas and
                         Gr{\"o}{\ss}er, Marcus},
  title =               {Probabilistic and Topological Semantics for Timed
                         Automata},
  editor =              {Arvind, Vikraman and Prasad, Sanjiva},
  booktitle =           {{P}roceedings of the 27th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'07)},
  acronym =             {{FSTTCS}'07},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4855},
  pages =               {179-191},
  year =                {2007},
  month =               dec,
}
[BBB+08] Christel Baier, Nathalie Bertrand, Patricia Bouyer, Thomas Brihaye, and Marcus Größer. Almost-sure Model Checking of Infinite Paths in One-clock Timed Automata. In LICS'08, pages 217-226. IEEE Comp. Soc. Press, June 2008.
@inproceedings{lics2008-BBBBG,
  author =              {Baier, {\relax Ch}ristel and Bertrand, Nathalie and
                         Bouyer, Patricia and Brihaye, {\relax Th}omas and
                         Gr{\"o}{\ss}er, Marcus},
  title =               {Almost-sure Model Checking of Infinite Paths in
                         One-clock Timed Automata},
  booktitle =           {{P}roceedings of the 23rd {A}nnual {S}ymposium on
                         {L}ogic in {C}omputer {S}cience ({LICS}'08)},
  acronym =             {{LICS}'08},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {217-226},
  year =                {2008},
  month =               jun,
  doi =                 {10.1109/LICS.2008.25},
}
[BBB+07] Éric Badouel, Marek Bednarczyk, Andrzej Borzyszkowski, Benoît Caillaud, and Philippe Darondeau. Concurrent secrets. Discrete Event Dynamic Systems 17(4):425-446. Kluwer Academic, December 2007.
@article{deds17(4)-BBBCD,
  author =              {Badouel, {\'E}ric and Bednarczyk, Marek and
                         Borzyszkowski, Andrzej and Caillaud, Beno{\^\i}t and
                         Darondeau, Philippe},
  title =               {Concurrent secrets},
  publisher =           {Kluwer Academic},
  journal =             {Discrete Event Dynamic Systems},
  volume =              {17},
  number =              {4},
  pages =               {425-446},
  year =                {2007},
  month =               dec,
  doi =                 {10.1007/s10626-007-0020-5},
}
[BBB+08] Nathalie Bertrand, Patricia Bouyer, Thomas Brihaye, and Nicolas Markey. Quantitative Model-Checking of One-Clock Timed Automata under Probabilistic Semantics. In QEST'08, pages 55-64. IEEE Comp. Soc. Press, September 2008.
Abstract

In [Baier et al., Probabilistic and Topological Semantics for Timed Automata, FSTTCS'07] a probabilistic semantics for timed automata has been defined in order to rule out unlikely (sequences of) events. The qualitative model-checking problem for LTL properties has been investigated, where the aim is to check whether a given LTL property holds with probability 1 in a timed automaton, and solved for the class of single-clock timed automata.

In this paper, we consider the quantitative model-checking problem for ω-regular properties: we aim at computing the exact probability that a given timed automaton satisfies an ω-regular property. We develop a framework in which we can compute a closed-form expression for this probability; we furthermore give an approximation algorithm, and finally prove that we can decide the threshold problem in that framework.

@inproceedings{qest2008-BBBM,
  author =              {Bertrand, Nathalie and Bouyer, Patricia and Brihaye,
                         {\relax Th}omas and Markey, Nicolas},
  title =               {Quantitative Model-Checking of One-Clock Timed
                         Automata under Probabilistic Semantics},
  booktitle =           {{P}roceedings of the 5th {I}nternational
                         {C}onference on {Q}uantitative {E}valuation of
                         {S}ystems ({QEST}'08)},
  acronym =             {{QEST}'08},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {55-64},
  year =                {2008},
  month =               sep,
  doi =                 {10.1109/QEST.2008.19},
  abstract =            {In [Baier \emph{et~al.}, \textit{Probabilistic and
                         Topological Semantics for Timed Automata},
                         FSTTCS'07] a probabilistic semantics for timed
                         automata has been defined in order to rule out
                         unlikely (sequences of) events. The qualitative
                         model-checking problem for LTL properties has been
                         investigated, where the aim is to check whether a
                         given LTL property holds with probability~\(1\) in a
                         timed automaton, and solved for the class of
                         single-clock timed automata.\par In this paper, we
                         consider the quantitative model-checking problem for
                         \(\omega\)-regular properties: we aim at computing
                         the exact probability that a given timed automaton
                         satisfies an \(\omega\)-regular property. We develop
                         a framework in which we can compute a closed-form
                         expression for this probability; we furthermore give
                         an approximation algorithm, and finally prove that
                         we can decide the threshold problem in that
                         framework.},
}
[BBB+14] Nathalie Bertrand, Patricia Bouyer, Thomas Brihaye, Quentin Menet, Christel Baier, Marcus Größer, and Marcin Jurdziński. Stochastic Timed Automata. Logical Methods in Computer Science 10(4). December 2014.
@article{lmcs10(4)-BBBMBGJ,
  author =              {Bertrand, Nathalie and Bouyer, Patricia and Brihaye,
                         {\relax Th}omas and Menet, Quentin and Baier,
                         {\relax Ch}ristel and Gr{\"o}{\ss}er, Marcus and
                         Jurdzi{\'n}ski, Marcin},
  title =               {Stochastic Timed Automata},
  journal =             {Logical Methods in Computer Science},
  volume =              {10},
  number =              {4},
  year =                {2014},
  month =               dec,
  doi =                 {10.2168/LMCS-10(4:6)2014},
}
[BBB+07] Patricia Bouyer, Thomas Brihaye, Véronique Bruyère, and Jean-François Raskin. On the Optimal Reachability Problem of Weighted Timed Automata. Formal Methods in System Design 31(2):135-175. Springer-Verlag, October 2007.
@article{fmsd31(2)-BBBR,
  author =              {Bouyer, Patricia and Brihaye, {\relax Th}omas and
                         Bruy{\`e}re, V{\'e}ronique and Raskin, Jean-Fran{\c
                         c}ois},
  title =               {On the Optimal Reachability Problem of Weighted
                         Timed Automata},
  publisher =           {Springer-Verlag},
  journal =             {Formal Methods in System Design},
  volume =              {31},
  number =              {2},
  pages =               {135-175},
  year =                {2007},
  month =               oct,
  doi =                 {10.1007/s10703-007-0035-4},
}
[BBC10] Albert Benveniste, Anne Bouillard, and Paul Caspi. A unifying view of loosely time-triggered architectures. In EMSOFT'10, pages 189-198. ACM Press, October 2010.
@inproceedings{emsoft2010-BBC,
  author =              {Benveniste, Albert and Bouillard, Anne and Caspi,
                         Paul},
  title =               {A~unifying view of loosely time-triggered
                         architectures},
  editor =              {Carloni, Luca P. and Tripakis, Stavros},
  booktitle =           {{P}roceedings of the 10th {I}nternational
                         {C}onference on {E}mbedded {S}oftware ({EMSOFT}'10)},
  acronym =             {{EMSOFT}'10},
  publisher =           {ACM Press},
  pages =               {189-198},
  year =                {2010},
  month =               oct,
  doi =                 {10.1145/1879021.1879047},
}
[BBD14] Thomas Brihaye, Véronique Bruyère, and Julie De Pril. On Equilibria in Quantitative Games with Reachability/Safety Objectives. Theory of Computing Systems 54(2):150-189. Springer-Verlag, February 2014.
@article{tcsyst54(2)-BBD,
  author =              {Brihaye, {\relax Th}omas and Bruy{\`e}re,
                         V{\'e}ronique and De{~}Pril, Julie},
  title =               {On Equilibria in Quantitative Games with
                         Reachability{{\slash}}Safety Objectives},
  publisher =           {Springer-Verlag},
  journal =             {Theory of Computing Systems},
  volume =              {54},
  number =              {2},
  pages =               {150-189},
  year =                {2014},
  month =               feb,
}
[BBD+11] Thomas Brihaye, Véronique Bruyère, Laurent Doyen, Marc Ducobu, and Jean-François Raskin. Antichain-based QBF Solving. In ATVA'11, Lecture Notes in Computer Science 6996, pages 183-197. Springer-Verlag, October 2011.
@inproceedings{atva2011-BBDDR,
  author =              {Brihaye, {\relax Th}omas and Bruy{\`e}re,
                         V{\'e}ronique and Doyen, Laurent and Ducobu, Marc
                         and Raskin, Jean-Fran{\c c}ois},
  title =               {Antichain-based {QBF} Solving},
  editor =              {Bultan, Tevfik and Hsiung, Pao-Ann},
  booktitle =           {{P}roceedings of the 9th {I}nternational {S}ymposium
                         on {A}utomated {T}echnology for {V}erification and
                         {A}nalysis ({ATVA}'11)},
  acronym =             {{ATVA}'11},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6996},
  pages =               {183-197},
  year =                {2011},
  month =               oct,
}
[BBD+12] Thomas Brihaye, Véronique Bruyère, Julie De Pril, and Hugo Gimbert. Subgame Perfection for Equilibria in Quantitative Reachability Games. In FoSSaCS'12, Lecture Notes in Computer Science 7213, pages 286-300. Springer-Verlag, March 2012.
@inproceedings{fossacs2012-BBDG,
  author =              {Brihaye, {\relax Th}omas and Bruy{\`e}re,
                         V{\'e}ronique and De{~}Pril, Julie and Gimbert,
                         Hugo},
  title =               {Subgame Perfection for Equilibria in Quantitative
                         Reachability Games},
  editor =              {Birkedal, Lars},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'12)},
  acronym =             {{FoSSaCS}'12},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {7213},
  pages =               {286-300},
  year =                {2012},
  month =               mar,
}
[BBD+02] Gerd Behrmann, Johan Bengtsson, Alexandre David, Kim Guldstrand Larsen, Paul Pettersson, and Wang Yi. UPPAAL Implementation Secrets. In FTRTFT'02, Lecture Notes in Computer Science 2469, pages 3-22. Springer-Verlag, September 2002.
@inproceedings{ftrtft2002-BBDLPY,
  author =              {Behrmann, Gerd and Bengtsson, Johan and David,
                         Alexandre and Larsen, Kim Guldstrand and Pettersson,
                         Paul and Yi, Wang},
  title =               {UPPAAL Implementation Secrets},
  editor =              {Damm, Werner and Olderog, Ernst R{\"u}diger},
  booktitle =           {{P}roceedings of the 7th {F}ormal {T}echniques in
                         {R}eal-Time and {F}ault-Tolerant {S}ystems
                         ({FTRTFT}'02)},
  acronym =             {{FTRTFT}'02},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2469},
  pages =               {3-22},
  year =                {2002},
  month =               sep,
  doi =                 {10.1007/3-540-45739-9_1},
}
[BBF+06] Tomáš Brázdil, Václav Brožek, Vojtěch Forejt, and Antonín Kučera. Stochastic Games with Branching-Time Winning Objectives. In LICS'06, pages 349-358. IEEE Comp. Soc. Press, July 2006.
@inproceedings{lics2006-BBFK,
  author =              {Br{\'a}zdil, Tom{\'a}{ s} and V{\'a}clav
                         Bro{{z}}ek and Forejt, Vojt{{e}}ch and Ku{
                         c}era, Anton{\'\i}n},
  title =               {Stochastic Games with Branching-Time Winning
                         Objectives},
  booktitle =           {{P}roceedings of the 21st {A}nnual {S}ymposium on
                         {L}ogic in {C}omputer {S}cience ({LICS}'06)},
  acronym =             {{LICS}'06},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {349-358},
  year =                {2006},
  month =               jul,
}
[BBF+03] Gerd Behrmann, Patricia Bouyer, Emmanuel Fleury, and Kim Guldstrand Larsen. Static Guard Analysis in Timed Automata Verification. In TACAS'03, Lecture Notes in Computer Science 2619, pages 254-270. Springer-Verlag, April 2003.
@inproceedings{tacas2003-BBFL,
  author =              {Behrmann, Gerd and Bouyer, Patricia and Fleury,
                         Emmanuel and Larsen, Kim Guldstrand},
  title =               {Static Guard Analysis in Timed Automata
                         Verification},
  editor =              {Garavel, Hubert and Hatcliff, John},
  booktitle =           {{P}roceedings of the 9th {I}nternational
                         {C}onference on {T}ools and {A}lgorithms for
                         {C}onstruction and {A}nalysis of {S}ystems
                         ({TACAS}'03)},
  acronym =             {{TACAS}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2619},
  pages =               {254-270},
  year =                {2003},
  month =               apr,
}
[BBF+18] Giovanni Bacci, Patricia Bouyer, Uli Fahrenberg, Kim Guldstrand Larsen, Nicolas Markey, and Pierre-Alain Reynier. Optimal and Robust Controller Synthesis Using Energy Timed Automata with Uncertainty. In FM'18, Lecture Notes in Computer Science 10951, pages 203-221. Springer-Verlag, July 2018.
Abstract

In this paper, we propose a novel framework for the synthesis of robust and optimal energy-aware controllers. The framework is based on energy timed automata, allowing for easy expression of timing-constraints and variable energy-rates. We prove decidability of the energy-constrained infinite-run problem in settings with both certainty and uncertainty of the energy-rates. We also consider the optimization problem of identifying the minimal upper bound that will permit existence of energy-constrained infinite runs. Our algorithms are based on quantifier elimination for linear real arithmetic. Using Mathematica and Mjollnir, we illustrate our framework through a real industrial example of a hydraulic oil pump. Compared with previous approaches our method is completely automated and provides improved results.

@inproceedings{fm2018-BBFLMR,
  author =              {Bacci, Giovanni and Bouyer, Patricia and Fahrenberg,
                         Uli and Larsen, Kim Guldstrand and Markey, Nicolas
                         and Reynier, Pierre-Alain},
  title =               {Optimal and Robust Controller Synthesis Using Energy
                         Timed Automata with Uncertainty},
  editor =              {Havelund, Klaus and Peleska, Jan and Roscoe, Bill W.
                         and de Vink, Erik},
  booktitle =           {{P}roceedings of the 22nd {I}nternational
                         {S}ymposium on {F}ormal {M}ethods ({FM}'18)},
  acronym =             {{FM}'18},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {10951},
  pages =               {203-221},
  year =                {2018},
  month =               jul,
  doi =                 {10.1007/978-3-319-95582-7_12},
  abstract =            {In this paper, we propose a novel framework for the
                         synthesis of robust and optimal energy-aware
                         controllers. The framework is based on energy timed
                         automata, allowing for easy expression of
                         timing-constraints and variable energy-rates. We
                         prove decidability of the energy-constrained
                         infinite-run problem in settings with both certainty
                         and uncertainty of the energy-rates. We also
                         consider the optimization problem of identifying the
                         minimal upper bound that will permit existence of
                         energy-constrained infinite runs. Our algorithms are
                         based on quantifier elimination for linear real
                         arithmetic. Using Mathematica and Mjollnir, we
                         illustrate our framework through a real industrial
                         example of a hydraulic oil pump. Compared with
                         previous approaches our method is completely
                         automated and provides improved results.},
}
[BBF+01] Béatrice Bérard, Michel Bidoit, Alain Finkel, François Laroussinie, Antoine Petit, Laure Petrucci, Philippe Schnoebelen, and Pierre McKenzie. Systems and Software Verification: Model-Checking Techniques and Tools. Springer-Verlag, 2001.
@book{SSV2001-BBFLPPS,
  author =              {B{\'e}rard, B{\'e}atrice and Bidoit, Michel and
                         Finkel, Alain and Laroussinie, Fran{\c c}ois and
                         Petit, Antoine and Petrucci, Laure and Schnoebelen,
                         {\relax Ph}ilippe and McKenzie, Pierre},
  title =               {Systems and Software Verification: Model-Checking
                         Techniques and Tools},
  publisher =           {Springer-Verlag},
  year =                {2001},
}
[BBF+13] Aaron Bohy, Véronique Bruyère, Emmanuel Filiot, and Jean-François Raskin. Synthesis from LTL Specifications with Mean-Payoff Objectives. In TACAS'13, Lecture Notes in Computer Science 7795, pages 169-184. Springer-Verlag, March 2013.
@inproceedings{tacas2013-BBFR,
  author =              {Bohy, Aaron and Bruy{\`e}re, V{\'e}ronique and
                         Filiot, Emmanuel and Raskin, Jean-Fran{\c c}ois},
  title =               {Synthesis from {LTL} Specifications with Mean-Payoff
                         Objectives},
  editor =              {Piterman, Nir and Smolka, Scott A.},
  booktitle =           {{P}roceedings of the 19th {I}nternational
                         {C}onference on {T}ools and {A}lgorithms for
                         {C}onstruction and {A}nalysis of {S}ystems
                         ({TACAS}'13)},
  acronym =             {{TACAS}'13},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {7795},
  pages =               {169-184},
  year =                {2013},
  month =               mar,
}
[BBG+07] Christel Baier, Tomáš Brázdil, Marcus Größer, and Antonín Kučera. Stochastic Game Logic. In QEST'07, pages 227-236. IEEE Comp. Soc. Press, September 2007.
@inproceedings{qest2007-BBGK,
  author =              {Baier, {\relax Ch}ristel and Br{\'a}zdil,
                         Tom{\'a}{ s} and Gr{\"o}{\ss}er, Marcus and Ku{
                         c}era, Anton{\'\i}n},
  title =               {Stochastic Game Logic},
  booktitle =           {{P}roceedings of the 4th {I}nternational
                         {C}onference on {Q}uantitative {E}valuation of
                         {S}ystems ({QEST}'07)},
  acronym =             {{QEST}'07},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {227-236},
  year =                {2007},
  month =               sep,
}
[BBJ+14] Dimitri Bohlender, Harold Bruintjes, Sebastian Junges, Jens Katelaan, Viet Yen Nguyen, and Thomas Noll. A Review of Statistical Model Checking Pitfalls on Real-Time Stochastic Models. In ISoLA'14, Lecture Notes in Computer Science 8803, pages 177-192. Springer-Verlag, October 2014.
@inproceedings{BBJKNN-isola2014,
  author =              {Bohlender, Dimitri and Bruintjes, Harold and Junges,
                         Sebastian and Katelaan, Jens and Nguyen, Viet Yen
                         and Noll, Thomas},
  title =               {A~Review of Statistical Model Checking Pitfalls on
                         Real-Time Stochastic Models},
  editor =              {Margaria, Tiziana and Steffen, Bernhard},
  booktitle =           {{P}roceedings of the 6th {I}nternational {S}ymposium
                         on {L}everaging {A}pplications of {F}ormal
                         {M}ethods, {V}erification and {V}alidation
                         ({IS}o{LA}'14)~-- {P}art~{II}: {S}pecialized
                         {T}echniques and {A}pplications},
  acronym =             {{IS}o{LA}'14},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8803},
  pages =               {177-192},
  year =                {2014},
  month =               oct,
  doi =                 {10.1007/978-3-662-45231-8_13},
}
[BBL04] Patricia Bouyer, Ed Brinksma, and Kim Guldstrand Larsen. Staying Alive as Cheaply as Possible. In HSCC'04, Lecture Notes in Computer Science 2993, pages 203-218. Springer-Verlag, March 2004.
@inproceedings{hscc2004-BBL,
  author =              {Bouyer, Patricia and Brinksma, Ed and Larsen, Kim
                         Guldstrand},
  title =               {Staying Alive as Cheaply as Possible},
  editor =              {Alur, Rajeev and Pappas, George J.},
  booktitle =           {{P}roceedings of the 7th {I}nternational {W}orkshop
                         on {H}ybrid {S}ystems: {C}omputation and {C}ontrol
                         ({HSCC}'04)},
  acronym =             {{HSCC}'04},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2993},
  pages =               {203-218},
  year =                {2004},
  month =               mar,
}
[BBL08] Patricia Bouyer, Ed Brinksma, and Kim Guldstrand Larsen. Optimal infinite scheduling for multi-priced timed automata. Formal Methods in System Design 32(1):2-23. Springer-Verlag, February 2008.
@article{fmsd32(1)-BBL,
  author =              {Bouyer, Patricia and Brinksma, Ed and Larsen, Kim
                         Guldstrand},
  title =               {Optimal infinite scheduling for multi-priced timed
                         automata},
  publisher =           {Springer-Verlag},
  journal =             {Formal Methods in System Design},
  volume =              {32},
  number =              {1},
  pages =               {2-23},
  year =                {2008},
  month =               feb,
}
[BBL+13] Giorgio Bacci, Giovanni Bacci, Kim Guldstrand Larsen, and Radu Mardare. Computing Behavioral Distances, Compositionally. In MFCS'13, Lecture Notes in Computer Science 8087, pages 74-85. Springer-Verlag, August 2013.
@inproceedings{mfcs2013-BBLM,
  author =              {Bacci, Giorgio and Bacci, Giovanni and Larsen, Kim
                         Guldstrand and Mardare, Radu},
  title =               {Computing Behavioral Distances, Compositionally},
  editor =              {Chatterjee, Krishnendu and Sgall, Ji{ r}{\'\i}},
  booktitle =           {{P}roceedings of the 38th {I}nternational
                         {S}ymposium on {M}athematical {F}oundations of
                         {C}omputer {S}cience ({MFCS}'13)},
  acronym =             {{MFCS}'13},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8087},
  pages =               {74-85},
  year =                {2013},
  month =               aug,
  doi =                 {10.1007/978-3-642-40313-2_9},
}
[BBL+13] Giorgio Bacci, Giovanni Bacci, Kim Guldstrand Larsen, and Radu Mardare. The BisimDist Library: Efficient Computation of Bisimilarity Distances for Markovian Models. In QEST'13, pages 278-281. IEEE Comp. Soc. Press, August 2013.
@inproceedings{qest2013-BBLM,
  author =              {Bacci, Giorgio and Bacci, Giovanni and Larsen, Kim
                         Guldstrand and Mardare, Radu},
  title =               {The BisimDist Library: Efficient Computation of
                         Bisimilarity Distances for Markovian Models},
  booktitle =           {{P}roceedings of the 10th {I}nternational
                         {C}onference on {Q}uantitative {E}valuation of
                         {S}ystems ({QEST}'13)},
  acronym =             {{QEST}'13},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {278-281},
  year =                {2013},
  month =               aug,
  doi =                 {10.1007/978-3-642-40196-1_23},
}
[BBL+04] Gerd Behrmann, Patricia Bouyer, Kim Guldstrand Larsen, and Radek Pelánek. Lower and Upper Bounds in Zone-Based Abstractions of Timed Automata. In TACAS'04, Lecture Notes in Computer Science 2988, pages 312-326. Springer-Verlag, March 2004.
@inproceedings{tacas2004-BBLP,
  author =              {Behrmann, Gerd and Bouyer, Patricia and Larsen, Kim
                         Guldstrand and Pel{\'a}nek, Radek},
  title =               {Lower and Upper Bounds in Zone-Based Abstractions of
                         Timed Automata},
  editor =              {Jensen, Kurt and Podelski, Andreas},
  booktitle =           {{P}roceedings of the 10th {I}nternational
                         {C}onference on {T}ools and {A}lgorithms for
                         {C}onstruction and {A}nalysis of {S}ystems
                         ({TACAS}'04)},
  acronym =             {{TACAS}'04},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2988},
  pages =               {312-326},
  year =                {2004},
  month =               mar,
  doi =                 {10.1007/978-3-540-24730-2_25},
}
[BBL+06] Gerd Behrmann, Patricia Bouyer, Kim Guldstrand Larsen, and Radek Pelánek. Lower and Upper Bounds in Zone-Based Abstractions of Timed Automata. International Journal on Software Tools for Technology Transfer 8(3):204-215. Springer-Verlag, June 2006.
@article{sttt8(3)-BBLP,
  author =              {Behrmann, Gerd and Bouyer, Patricia and Larsen, Kim
                         Guldstrand and Pel{\'a}nek, Radek},
  title =               {Lower and Upper Bounds in Zone-Based Abstractions of
                         Timed Automata},
  publisher =           {Springer-Verlag},
  journal =             {International Journal on Software Tools for
                         Technology Transfer},
  volume =              {8},
  number =              {3},
  pages =               {204-215},
  year =                {2006},
  month =               jun,
  doi =                 {10.1007/s10009-005-0190-0},
}
[BBL+15] Nikola Beneš, Petr Bezdek, Kim Guldstrand Larsen, and Jiří Srba. Language Emptiness of Continuous-Time Parametric Timed Automata. In ICALP'15, Lecture Notes in Computer Science 9135, pages 69-81. Springer-Verlag, July 2015.
@inproceedings{icalp2015-BBLS,
  author =              {Bene{{s}}, Nikola and Bezdek, Petr and Larsen, Kim
                         Guldstrand and Srba, Ji{ r}{\'\i}},
  title =               {Language Emptiness of Continuous-Time Parametric
                         Timed Automata},
  editor =              {Halld{\'o}rsson, Magn{\'u}s M. and Iwana, Kazuo and
                         Kobayashi, Naoki and Speckmann, Bettina},
  booktitle =           {{P}roceedings of the 42nd {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'15)~-- Part~{II}},
  acronym =             {{ICALP}'15},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {9135},
  pages =               {69-81},
  year =                {2015},
  month =               jul,
  doi =                 {10.1007/978-3-662-47666-6_6},
}
[BBM06] Ramzi Ben Salah, Marius Bozga, and Oded Maler. On Interleaving in Timed Automata. In CONCUR'06, Lecture Notes in Computer Science 4137, pages 465-476. Springer-Verlag, August 2006.
@inproceedings{concur2006-BBM,
  author =              {Ben{~}Salah, Ramzi and Bozga, Marius and Maler,
                         Oded},
  title =               {On Interleaving in Timed Automata},
  editor =              {Baier, {\relax Ch}ristel and Hermanns, Holger},
  booktitle =           {{P}roceedings of the 17th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'06)},
  acronym =             {{CONCUR}'06},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4137},
  pages =               {465-476},
  year =                {2006},
  month =               aug,
}
[BBM06] Patricia Bouyer, Thomas Brihaye, and Nicolas Markey. Improved Undecidability Results on Weighted Timed Automata. Information Processing Letters 98(5):188-194. Elsevier, June 2006.
Abstract

In this paper, we improve two recent undecidability results of Brihaye, Bruyère and Raskin about weighted timed automata, an extension of timed automata with a cost variable. Our results rely on a new encoding of the two counters of a Minsky machine that only require three clocks and one stopwatch cost, while previous reductions required five clocks and one stopwatch cost.

@article{ipl98(5)-BBM,
  author =              {Bouyer, Patricia and Brihaye, {\relax Th}omas and
                         Markey, Nicolas},
  title =               {Improved Undecidability Results on Weighted Timed
                         Automata},
  publisher =           {Elsevier},
  journal =             {Information Processing Letters},
  volume =              {98},
  number =              {5},
  pages =               {188-194},
  year =                {2006},
  month =               jun,
  doi =                 {10.1016/j.ipl.2006.01.012},
  abstract =            {In this paper, we improve two recent undecidability
                         results of Brihaye, Bruy{\`e}re and Raskin about
                         weighted timed automata, an extension of timed
                         automata with a cost variable. Our results rely on a
                         new encoding of the two counters of a Minsky machine
                         that only require three clocks and one stopwatch
                         cost, while previous reductions required five clocks
                         and one stopwatch cost.},
}
[BBM10] Patricia Bouyer, Romain Brenguier, and Nicolas Markey. Nash Equilibria for Reachability Objectives in Multi-player Timed Games. In CONCUR'10, Lecture Notes in Computer Science 6269, pages 192-206. Springer-Verlag, September 2010.
Abstract

We propose a procedure for computing Nash equilibria in multi-player timed games with reachability objectives. Our procedure is based on the construction of a finite concurrent game, and on a generic characterization of Nash equilibria in (possibly infinite) concurrent games. Along the way, we use our characterization to compute Nash equilibria in finite concurrent games.

@inproceedings{concur2010-BBM,
  author =              {Bouyer, Patricia and Brenguier, Romain and Markey,
                         Nicolas},
  title =               {{N}ash Equilibria for Reachability Objectives in
                         Multi-player Timed Games},
  editor =              {Gastin, Paul and Laroussinie, Fran{\c c}ois},
  booktitle =           {{P}roceedings of the 21st {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'10)},
  acronym =             {{CONCUR}'10},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6269},
  pages =               {192-206},
  year =                {2010},
  month =               sep,
  doi =                 {10.1007/978-3-642-15375-4_14},
  abstract =            {We propose a procedure for computing Nash equilibria
                         in multi-player timed games with reachability
                         objectives. Our procedure is based on the
                         construction of a finite concurrent game, and on a
                         generic characterization of Nash equilibria in
                         (possibly infinite) concurrent games. Along the way,
                         we~use our characterization to compute Nash
                         equilibria in finite concurrent games.},
}
[BBM10] Patricia Bouyer, Romain Brenguier, and Nicolas Markey. Computing Equilibria in Two-Player Timed Games via Turn-Based Finite Games. In FORMATS'10, Lecture Notes in Computer Science 6246, pages 62-76. Springer-Verlag, September 2010.
Abstract

We study two-player timed games where the objectives of the two players are not opposite. We focus on the standard notion of Nash equilibrium and propose a series of transformations that builds two finite turn-based games out of a timed game, with a precise correspondence between Nash equilibria in the original and in final games. This provides us with an algorithm to compute Nash equilibria in two-player timed games for large classes of properties.

@inproceedings{formats2010-BBM,
  author =              {Bouyer, Patricia and Brenguier, Romain and Markey,
                         Nicolas},
  title =               {Computing Equilibria in Two-Player Timed Games
                         {\textit{via}}~Turn-Based Finite Games},
  editor =              {Chatterjee, Krishnendu and Henzinger, Thomas A.},
  booktitle =           {{P}roceedings of the 8th {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'10)},
  acronym =             {{FORMATS}'10},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6246},
  pages =               {62-76},
  year =                {2010},
  month =               sep,
  doi =                 {10.1007/978-3-642-15297-9_7},
  abstract =            {We study two-player timed games where the objectives
                         of the two players are not opposite. We focus on the
                         standard notion of Nash equilibrium and propose a
                         series of transformations that builds two finite
                         turn-based games out of a timed game, with a precise
                         correspondence between Nash equilibria in the
                         original and in final games. This provides us with
                         an algorithm to compute Nash equilibria in
                         two-player timed games for large classes of
                         properties.},
}
[BBM18] A. R. Balasubramanian, Nathalie Bertrand, and Nicolas Markey. Parameterized verification of synchronization in constrained reconfigurable broadcast networks. In TACAS'18 (Part II), Lecture Notes in Computer Science 10806, pages 38-54. Springer-Verlag, April 2018.
Abstract

Reconfigurable broadcast networks provide a convenient formalism for modelling and reasoning about networks of mobile agents broadcasting messages to other agents following some (evolving) communication topology. The parameterized verification of such models aims at checking whether a given property holds irrespective of the initial configuration (number of agents, initial states and initial communication topology). We focus here on the synchronization property, asking whether all agents converge to a set of target states after some execution. This problem is known to be decidable in polynomial time when no constraints are imposed on the evolution of the communication topology (while it is undecidable for static broadcast networks).

In this paper we investigate how various constraints on reconfigurations affect the decidability and complexity of the synchronization problem. In particular, we show that when bounding the number of reconfigured links between two communications steps by a constant, synchronization becomes undecidable; on the other hand, synchronization remains decidable in PTIME when the bound grows with the number of agents.

@inproceedings{tacas2018-2-BBM,
  author =              {Balasubramanian, A. R. and Bertrand, Nathalie and
                         Markey, Nicolas},
  title =               {Parameterized verification of synchronization in
                         constrained reconfigurable broadcast networks},
  editor =              {Beyer, Dirk and Huisman, Marieke},
  booktitle =           {{P}roceedings of the 24th {I}nternational
                         {C}onference on {T}ools and {A}lgorithms for
                         {C}onstruction and {A}nalysis of {S}ystems
                         ({TACAS}'18)~-- {P}art~{II}},
  acronym =             {{TACAS}'18 (Part~{II})},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {10806},
  pages =               {38-54},
  year =                {2018},
  month =               apr,
  doi =                 {10.1007/978-3-319-89963-3_3},
  abstract =            {Reconfigurable broadcast networks provide a
                         convenient formalism for modelling and reasoning
                         about networks of mobile agents broadcasting
                         messages to other agents following some (evolving)
                         communication topology. The parameterized
                         verification of such models aims at checking whether
                         a given property holds irrespective of the initial
                         configuration (number of agents, initial states and
                         initial communication topology). We~focus here on
                         the \emph{synchronization property}, asking whether
                         all agents converge to a set of target states after
                         some execution. This~problem is known to be
                         decidable in polynomial time when no constraints are
                         imposed on the evolution of the communication
                         topology (while~it~is undecidable for static
                         broadcast networks).\par In this paper we
                         investigate how various constraints on
                         reconfigurations affect the decidability and
                         complexity of the synchronization problem.
                         In~particular, we show that when bounding the number
                         of reconfigured links between two communications
                         steps by a constant, synchronization becomes
                         undecidable; on~the other hand, synchronization
                         remains decidable in PTIME when the bound grows with
                         the number of agents.},
}
[BBM+11] Patricia Bouyer, Romain Brenguier, Nicolas Markey, and Michael Ummels. Nash Equilibria in Concurrent Games with Büchi Objectives. In FSTTCS'11, Leibniz International Proceedings in Informatics 13, pages 375-386. Leibniz-Zentrum für Informatik, December 2011.
Abstract

We study the problem of the existence (and computation) of Nash equilibria in multi-player concurrent games with Büchi-definable objectives. First, when the objectives are Büchi conditions on the game, we prove that the existence problem can be solved in polynomial time. In a second part, we extend our technique to objectives defined by deterministic Büchi automata, and prove that the problem then becomes EXPTIME-complete. We prove PSPACE-completeness for the case where the Büchi automata are 1-weak.

@inproceedings{fsttcs2011-BBMU,
  author =              {Bouyer, Patricia and Brenguier, Romain and Markey,
                         Nicolas and Ummels, Michael},
  title =               {{N}ash Equilibria in Concurrent Games with
                         {B}{\"u}chi Objectives},
  editor =              {Chakraborty, Supratik and Kumar, Amit},
  booktitle =           {{P}roceedings of the 31st {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'11)},
  acronym =             {{FSTTCS}'11},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {13},
  pages =               {375-386},
  year =                {2011},
  month =               dec,
  doi =                 {10.4230/LIPIcs.FSTTCS.2011.375},
  abstract =            {We study the problem of the existence (and
                         computation) of Nash equilibria in multi-player
                         concurrent games with B{\"u}chi-definable
                         objectives. First, when the objectives are B{\"u}chi
                         conditions on the game, we prove that the existence
                         problem can be solved in polynomial time. In a
                         second part, we extend our technique to objectives
                         defined by deterministic B{\"u}chi automata, and
                         prove that the problem then becomes
                         EXPTIME-complete. We prove PSPACE-completeness for
                         the case where the B{\"u}chi automata are 1-weak.},
}
[BBM+12] Patricia Bouyer, Romain Brenguier, Nicolas Markey, and Michael Ummels. Concurrent games with ordered objectives. In FoSSaCS'12, Lecture Notes in Computer Science 7213, pages 301-315. Springer-Verlag, March 2012.
Abstract

We consider concurrent games played on graphs, in which each player has several qualitative (e.g. reachability or Büchi) objectives, and a preorder on these objectives (for instance the counting order, where the aim is to maximise the number of objectives that are fulfilled).

We study two fundamental problems in that setting: (1) the value problem, which aims at deciding the existence of a strategy that ensures a given payoff; (2) the Nash equilibrium problem, where we want to decide the existence of a Nash equilibrium (possibly with a condition on the payoffs). We characterise the exact complexities of these problems for several relevant preorders, and several kinds of objectives.

@inproceedings{fossacs2012-BBMU,
  author =              {Bouyer, Patricia and Brenguier, Romain and Markey,
                         Nicolas and Ummels, Michael},
  title =               {Concurrent games with ordered objectives},
  editor =              {Birkedal, Lars},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'12)},
  acronym =             {{FoSSaCS}'12},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {7213},
  pages =               {301-315},
  year =                {2012},
  month =               mar,
  doi =                 {10.1007/978-3-642-28729-9_20},
  abstract =            {We consider concurrent games played on graphs, in
                         which each player has several qualitative (e.g.
                         reachability or B{\"u}chi) objectives, and a
                         preorder on these objectives (for instance the
                         counting order, where the aim is to maximise the
                         number of objectives that are fulfilled).\par We
                         study two fundamental problems in that setting:
                         (1)~the \emph{value problem}, which aims at deciding
                         the existence of a strategy that ensures a given
                         payoff; (2)~the \emph{Nash equilibrium problem},
                         where we want to decide the existence of a Nash
                         equilibrium (possibly with a condition on the
                         payoffs). We characterise the exact complexities of
                         these problems for several relevant preorders, and
                         several kinds of objectives.},
}
[BBM+15] Patricia Bouyer, Romain Brenguier, Nicolas Markey, and Michael Ummels. Pure Nash Equilibria in Concurrent Games. Logical Methods in Computer Science 11(2:9). June 2015.
Abstract

We study pure-strategy Nash equilibria in multiplayer concurrent games, for a variety of omega-regular objectives. For simple objectives (e.g. reachability, Büchi objectives), we transform the problem of deciding the existence of a Nash equilibrium in a given concurrent game into that of deciding the existence of a winning strategy in a turn-based two-player game (with a refined objective). We use that transformation to design algorithms for computing Nash equilibria, which in most cases have optimal worst-case complexity. For automata-defined objectives, we extend the above algorithms using a simulation relation which allows us to consider the product of the game with the automata defining the objectives. Building on previous algorithms for simple qualitative objectives, we define and study a semi-quantitative framework, where all players have several boolean objectives equipped with a preorder; a player may for instance want to satisfy all her objectives, or to maximise the number of objectives that she achieves. In most cases, we prove that the algorithms we obtain match the complexity of the problem they address.

@article{lmcs11(2)-BBMU,
  author =              {Bouyer, Patricia and Brenguier, Romain and Markey,
                         Nicolas and Ummels, Michael},
  title =               {Pure {N}ash Equilibria in Concurrent Games},
  journal =             {Logical Methods in Computer Science},
  volume =              {11},
  number =              {2:9},
  year =                {2015},
  month =               jun,
  doi =                 {10.2168/LMCS-11(2:9)2015},
  abstract =            {We study pure-strategy Nash equilibria in
                         multiplayer concurrent games, for a variety of
                         omega-regular objectives. For simple objectives
                         (e.g. reachability, B{\"u}chi objectives), we
                         transform the problem of deciding the existence of a
                         Nash equilibrium in a given concurrent game into
                         that of deciding the existence of a winning strategy
                         in a turn-based two-player game (with a refined
                         objective). We use that transformation to design
                         algorithms for computing Nash equilibria, which in
                         most cases have optimal worst-case complexity. For
                         automata-defined objectives, we extend the above
                         algorithms using a simulation relation which allows
                         us to consider the product of the game with the
                         automata defining the objectives. Building on
                         previous algorithms for simple qualitative
                         objectives, we define and study a semi-quantitative
                         framework, where all players have several boolean
                         objectives equipped with a preorder; a player may
                         for instance want to satisfy all her objectives, or
                         to maximise the number of objectives that she
                         achieves. In most cases, we prove that the
                         algorithms we obtain match the complexity of the
                         problem they address.},
}
[BBR04] Thomas Brihaye, Véronique Bruyère, and Jean-François Raskin. Model-checking for weighted timed automata. In FORMATS-FTRTFT'04, Lecture Notes in Computer Science 3253, pages 277-292. Springer-Verlag, September 2004.
@inproceedings{formats2004-BBR,
  author =              {Brihaye, {\relax Th}omas and Bruy{\`e}re,
                         V{\'e}ronique and Raskin, Jean-Fran{\c c}ois},
  title =               {Model-checking for weighted timed automata},
  editor =              {Lakhnech, Yassine and Yovine, Sergio},
  booktitle =           {{P}roceedings of the {J}oint {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'04) and {F}ormal
                         {T}echniques in {R}eal-Time and {F}ault-Tolerant
                         {S}ystems ({FTRTFT}'04)},
  acronym =             {{FORMATS-FTRTFT}'04},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3253},
  pages =               {277-292},
  year =                {2004},
  month =               sep,
}
[BBR05] Thomas Brihaye, Véronique Bruyère, and Jean-François Raskin. On Optimal Timed Strategies. In FORMATS'05, Lecture Notes in Computer Science 3829, pages 49-64. Springer-Verlag, September 2005.
@inproceedings{formats2005-BBR,
  author =              {Brihaye, {\relax Th}omas and Bruy{\`e}re,
                         V{\'e}ronique and Raskin, Jean-Fran{\c c}ois},
  title =               {On Optimal Timed Strategies},
  editor =              {Pettersson, Paul and Yi, Wang},
  booktitle =           {{P}roceedings of the 3rd {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'05)},
  acronym =             {{FORMATS}'05},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3829},
  pages =               {49-64},
  year =                {2005},
  month =               sep,
  doi =                 {10.1007/11603009_5},
}
[BBS01] Jiří Barnat, Luboš Brim, and Jitka Stříbrná. Distributed LTL Model-Checking in SPIN. In SPIN'01, Lecture Notes in Computer Science 2057, pages 200-216. Springer-Verlag, May 2001.
@inproceedings{spin2001-BBS,
  author =              {Barnat, Ji{ r}{\'\i} and Brim, Lubo{ s} and
                         St{ r}{\'\i}brn{\'a}, Jitka},
  title =               {Distributed {LTL} Model-Checking in {SPIN}},
  editor =              {Dwyer, Matthew B.},
  booktitle =           {{P}roceedings of the 8th {I}nternational {SPIN}
                         {W}orkshop ({SPIN}'01)},
  acronym =             {{SPIN}'01},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2057},
  pages =               {200-216},
  year =                {2001},
  month =               may,
}
[BC03] Marco Benedetti and Alessandro Cimatti. Bounded Model Checking for Past LTL. In TACAS'03, Lecture Notes in Computer Science 2619, pages 18-33. Springer-Verlag, April 2003.
@inproceedings{tacas2003-BC,
  author =              {Benedetti, Marco and Cimatti, Alessandro},
  title =               {Bounded Model Checking for Past {LTL}},
  editor =              {Garavel, Hubert and Hatcliff, John},
  booktitle =           {{P}roceedings of the 9th {I}nternational
                         {C}onference on {T}ools and {A}lgorithms for
                         {C}onstruction and {A}nalysis of {S}ystems
                         ({TACAS}'03)},
  acronym =             {{TACAS}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2619},
  pages =               {18-33},
  year =                {2003},
  month =               apr,
}
[BC05] Patricia Bouyer and Fabrice Chevalier. On Conciseness of Extensions of Timed Automata. Journal of Automata, Languages and Combinatorics 10(4):393-405. 2005.
@article{jalc10(4)-BC,
  author =              {Bouyer, Patricia and Chevalier, Fabrice},
  title =               {On Conciseness of Extensions of Timed Automata},
  journal =             {Journal of Automata, Languages and Combinatorics},
  volume =              {10},
  number =              {4},
  pages =               {393-405},
  year =                {2005},
}
[BC06] Patricia Bouyer and Fabrice Chevalier. On the Control of Timed and Hybrid Systems. EATCS Bulletin 89:79-96. EATCS, June 2006.
@article{eatcs-bull89()-BC,
  author =              {Bouyer, Patricia and Chevalier, Fabrice},
  title =               {On the Control of Timed and Hybrid Systems},
  publisher =           {EATCS},
  journal =             {EATCS Bulletin},
  volume =              {89},
  pages =               {79-96},
  year =                {2006},
  month =               jun,
}
[BC10] Luboš Brim and Jakub Chaloupka. Using Strategy Improvement to Stay Alive. Technical Report FIMU-RS-2010-03, Faculty of Informatics, Masaryk University, Brno, Czech Republic, March 2010.
@techreport{FIMU-RS-2010-03-BC,
  author =              {Brim, Lubo{ s} and Chaloupka, Jakub},
  title =               {Using Strategy Improvement to Stay Alive},
  number =              {FIMU-RS-2010-03},
  year =                {2010},
  month =               mar,
  institution =         {Faculty of Informatics, Masaryk University, Brno,
                         Czech Republic},
}
[BCC98] Sergey Berezin, Sérgio Vale Aguiar Campos, and Edmund M. Clarke. Compositional Reasoning in Model Checking. In COMPOS'97, Lecture Notes in Computer Science 1536, pages 81-102. Springer-Verlag, 1998.
@inproceedings{compos1997-BCC,
  author =              {Berezin, Sergey and Campos, S{\'e}rgio Vale Aguiar
                         and Clarke, Edmund M.},
  title =               {Compositional Reasoning in Model Checking},
  editor =              {de Roever, Willem-Paul and Langmaack, Hans and
                         Pnueli, Amir},
  booktitle =           {{R}evised {L}ectures of the 1st {I}nternational
                         {S}ymposium on {C}ompositionality: {T}he
                         {S}ignificant {D}ifference ({COMPOS}'97)},
  acronym =             {{COMPOS}'97},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1536},
  pages =               {81-102},
  year =                {1998},
  confyear =            {1997},
  confmonth =           {9},
}
[BCC+03] Armin Biere, Alessandro Cimatti, Edmund M. Clarke, Ofer Strichman, and Yunshan Zhu. Bounded Model Checking. In Marvin Zelkowitz (eds.), Highly Dependable Software, Advances in Computers 58, pages 117-148. Academic Press, 2003.
@incollection{AC58-BCCSZ,
  author =              {Biere, Armin and Cimatti, Alessandro and Clarke,
                         Edmund M. and Strichman, Ofer and Zhu, Yunshan},
  title =               {Bounded Model Checking},
  editor =              {Zelkowitz, Marvin},
  booktitle =           {Highly Dependable Software},
  publisher =           {Academic Press},
  series =              {Advances in Computers},
  volume =              {58},
  pages =               {117-148},
  chapter =             {3},
  year =                {2003},
}
[BCC+99] Armin Biere, Alessandro Cimatti, Edmund M. Clarke, and Yunshan Zhu. Symbolic Model Checking without BDDs. In TACAS'99, Lecture Notes in Computer Science 1579, pages 193-207. Springer-Verlag, March 1999.
@inproceedings{tacas1999-BCCZ,
  author =              {Biere, Armin and Cimatti, Alessandro and Clarke,
                         Edmund M. and Zhu, Yunshan},
  title =               {Symbolic Model Checking without {BDD}s},
  editor =              {Cleaveland, Rance},
  booktitle =           {{P}roceedings of the 5th {I}nternational
                         {C}onference on {T}ools and {A}lgorithms for
                         {C}onstruction and {A}nalysis of {S}ystems
                         ({TACAS}'99)},
  acronym =             {{TACAS}'99},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1579},
  pages =               {193-207},
  year =                {1999},
  month =               mar,
}
[BCD05] Patricia Bouyer, Fabrice Chevalier, and Deepak D'Souza. Fault Diagnosis Using Timed Automata. In FoSSaCS'05, Lecture Notes in Computer Science 3441, pages 219-233. Springer-Verlag, April 2005.
@inproceedings{fossacs2005-BCD,
  author =              {Bouyer, Patricia and Chevalier, Fabrice and D'Souza,
                         Deepak},
  title =               {Fault Diagnosis Using Timed Automata},
  editor =              {Sassone, Vladimiro},
  booktitle =           {{P}roceedings of the 8th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'05)},
  acronym =             {{FoSSaCS}'05},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3441},
  pages =               {219-233},
  year =                {2005},
  month =               apr,
  doi =                 {10.1007/978-3-540-31982-5_14},
}
[BCD+07] Gerd Behrmann, Agnès Cougnard, Alexandre David, Emmanuel Fleury, Kim Guldstrand Larsen, and Didier Lime. UPPAAL-Tiga: Time for Playing Games!. In CAV'07, Lecture Notes in Computer Science 4590, pages 121-125. Springer-Verlag, July 2007.
@inproceedings{cav2007-BCDFLL,
  author =              {Behrmann, Gerd and Cougnard, Agn{\`e}s and David,
                         Alexandre and Fleury, Emmanuel and Larsen, Kim
                         Guldstrand and Lime, Didier},
  title =               {{UPPAAL-Tiga}: Time for Playing Games!},
  editor =              {Damm, Werner and Hermanns, Holger},
  booktitle =           {{P}roceedings of the 19th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'07)},
  acronym =             {{CAV}'07},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4590},
  pages =               {121-125},
  year =                {2007},
  month =               jul,
}
[BCD+11] Luboš Brim, Jakub Chaloupka, Laurent Doyen, Raffaella Gentilini, and Jean-François Raskin. Faster algorithms for mean-payoff games. Formal Methods in System Design 38(2):97-118. Springer-Verlag, April 2011.
@article{fmsd38(2)-BCDGR,
  author =              {Brim, Lubo{ s} and Chaloupka, Jakub and Doyen,
                         Laurent and Gentilini, Raffaella and Raskin,
                         Jean-Fran{\c c}ois},
  title =               {Faster algorithms for mean-payoff games},
  publisher =           {Springer-Verlag},
  journal =             {Formal Methods in System Design},
  volume =              {38},
  number =              {2},
  pages =               {97-118},
  year =                {2011},
  month =               apr,
  doi =                 {10.1007/s10703-010-0105-x},
}
[BCE+98] Jean-Jacques Borrelly, Ève Costemaniere, Bernard Espiau, Konstantinos Kapellos, Roger Pissard-Gibollet, Daniel Simon, and Nicolas Turro. The ORCCAD Architecture. International Journal of Robotics Research 17(4):338-359. April 1998.
@article{ijrr17(4)-BCEKPST,
  author =              {Borrelly, Jean-Jacques and Costemaniere, {\`E}ve and
                         Espiau, Bernard and Kapellos, Konstantinos and
                         Pissard{-}Gibollet, Roger and Simon, Daniel and
                         Turro, Nicolas},
  title =               {The {ORCCAD} Architecture},
  journal =             {International Journal of Robotics Research},
  volume =              {17},
  number =              {4},
  pages =               {338-359},
  year =                {1998},
  month =               apr,
}
[BCF03] Harry Buhrman, Richard Chang, and Lance Fortnow. One Bit of Advice. In STACS'03, Lecture Notes in Computer Science 2607, pages 547-558. Springer-Verlag, February 2003.
@inproceedings{stacs2003-BCF,
  author =              {Buhrman, Harry and Chang, Richard and Fortnow,
                         Lance},
  title =               {One Bit of Advice},
  editor =              {Alt, Helmut and Habib, Michel},
  booktitle =           {{P}roceedings of the 20th {S}ymposium on
                         {T}heoretical {A}spects of {C}omputer {S}cience
                         ({STACS}'03)},
  acronym =             {{STACS}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2607},
  pages =               {547-558},
  year =                {2003},
  month =               feb,
}
[BCF+04] Patricia Bouyer, Franck Cassez, Emmanuel Fleury, and Kim Guldstrand Larsen. Optimal Strategies in Priced Timed Game Automata. In FSTTCS'04, Lecture Notes in Computer Science 3328, pages 148-160. Springer-Verlag, December 2004.
@inproceedings{fsttcs2004-BCFL,
  author =              {Bouyer, Patricia and Cassez, Franck and Fleury,
                         Emmanuel and Larsen, Kim Guldstrand},
  title =               {Optimal Strategies in Priced Timed Game Automata},
  editor =              {Lodaya, Kamal and Mahajan, Meena},
  booktitle =           {{P}roceedings of the 24th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'04)},
  acronym =             {{FSTTCS}'04},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3328},
  pages =               {148-160},
  year =                {2004},
  month =               dec,
}
[BCF+04] Patricia Bouyer, Franck Cassez, Emmanuel Fleury, and Kim Guldstrand Larsen. Optimal Strategies in Priced Timed Games. Research Report RS-04-4, Basic Research in Computer Science, Aalborg University, Denmark, February 2004.
@techreport{TR-brics04-4,
  author =              {Bouyer, Patricia and Cassez, Franck and Fleury,
                         Emmanuel and Larsen, Kim Guldstrand},
  title =               {Optimal Strategies in Priced Timed Games},
  number =              {RS-04-4},
  year =                {2004},
  month =               feb,
  institution =         {Basic Research in Computer Science, Aalborg
                         University, Denmark},
  type =                {Research Report},
}
[BCG88] Michael C. Browne, Edmund M. Clarke, and Orna Grumberg. Characterizing Finite Kripke Structures in Propositional Temporal Logic. Theoretical Computer Science 59(1-2):115-131. Elsevier, July 1988.
@article{tcs59(1-2)-BCG,
  author =              {Browne, Michael C. and Clarke, Edmund M. and
                         Grumberg, Orna},
  title =               {Characterizing Finite {K}ripke Structures in
                         Propositional Temporal Logic},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {59},
  number =              {1-2},
  pages =               {115-131},
  year =                {1988},
  month =               jul,
}
[BCG+92] Samuel R. Buss, Stephen A. Cook, Vineet Gupta, and Vijaya Ramachandran. An Optimal Parallel Algorithm for Formula Evaluation. SIAM Journal on Computing 21(4):755-780. Society for Industrial and Applied Math., August 1992.
@article{siamcomp21(4)-BCGR,
  author =              {Buss, Samuel R. and Cook, Stephen A. and Gupta,
                         Vineet and Ramachandran, Vijaya},
  title =               {An Optimal Parallel Algorithm for Formula
                         Evaluation},
  publisher =           {Society for Industrial and Applied Math.},
  journal =             {SIAM Journal on Computing},
  volume =              {21},
  number =              {4},
  pages =               {755-780},
  year =                {1992},
  month =               aug,
}
[BCH+11] Udi Boker, Krishnendu Chatterjee, Thomas A. Henzinger, and Orna Kupferman. Temporal Specifications with Accumulative Values. In LICS'11, pages 43-52. IEEE Comp. Soc. Press, June 2011.
@inproceedings{lics2011-BCHK,
  author =              {Boker, Udi and Chatterjee, Krishnendu and Henzinger,
                         Thomas A. and Kupferman, Orna},
  title =               {Temporal Specifications with Accumulative Values},
  booktitle =           {{P}roceedings of the 26th {A}nnual {S}ymposium on
                         {L}ogic in {C}omputer {S}cience ({LICS}'11)},
  acronym =             {{LICS}'11},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {43-52},
  year =                {2011},
  month =               jun,
  doi =                 {10.1109/LICS.2011.33},
}
[BCH+14] Udi Boker, Krishnendu Chatterjee, Thomas A. Henzinger, and Orna Kupferman. Temporal Specifications with Accumulative Values. ACM Transactions on Computational Logic 15(4):27:1-27:25. ACM Press, August 2014.
@article{tocl15(4)-BCHK,
  author =              {Boker, Udi and Chatterjee, Krishnendu and Henzinger,
                         Thomas A. and Kupferman, Orna},
  title =               {Temporal Specifications with Accumulative Values},
  publisher =           {ACM Press},
  journal =             {ACM Transactions on Computational Logic},
  volume =              {15},
  number =              {4},
  pages =               {27:1-27:25},
  year =                {2014},
  month =               aug,
  doi =                 {10.1145/2629686},
}
[BCK+01] Luboš Brim, Ivana Černa, Pavel Krčál, and Radek Pelánek. Distributed LTL Model Checking Based on Negative Cycle Detection. In FSTTCS'01, Lecture Notes in Computer Science 2245, pages 96-107. Springer-Verlag, December 2001.
@inproceedings{fsttcs2001-BCKP,
  author =              {Brim, Lubo{ s} and {{C}}erna, Ivana and
                         Kr{{c}}{\'a}l, Pavel and Pel{\'a}nek, Radek},
  title =               {Distributed {LTL} Model Checking Based on Negative
                         Cycle Detection},
  editor =              {Hariharan, Ramesh and Mukund, Madhavan and Vinay,
                         V.},
  booktitle =           {{P}roceedings of the 21st {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'01)},
  acronym =             {{FSTTCS}'01},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2245},
  pages =               {96-107},
  year =                {2001},
  month =               dec,
}
[BCM05] Patricia Bouyer, Fabrice Chevalier, and Nicolas Markey. On the Expressiveness of TPTL and MTL. In FSTTCS'05, Lecture Notes in Computer Science 3821, pages 432-443. Springer-Verlag, December 2005.
Abstract

TPTL and MTL are two classical timed extensions of LTL. In this paper, we positively answer a 15-year-old conjecture that TPTL is strictly more expressive than MTL. But we show that, surprisingly, the TPTL formula proposed by Alur and Henzinger for witnessing this conjecture can be expressed in MTL. More generally, we show that TPTL formulae using only the F modality can be translated into MTL.

@inproceedings{fsttcs2005-BCM,
  author =              {Bouyer, Patricia and Chevalier, Fabrice and Markey,
                         Nicolas},
  title =               {On the Expressiveness of {TPTL} and {MTL}},
  editor =              {Ramanujam, R. and Sen, Sandeep},
  booktitle =           {{P}roceedings of the 25th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'05)},
  acronym =             {{FSTTCS}'05},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3821},
  pages =               {432-443},
  year =                {2005},
  month =               dec,
  doi =                 {10.1007/11590156_35},
  abstract =            {TPTL and MTL are two classical timed extensions of
                         LTL. In this paper, we positively answer a
                         15-year-old conjecture that TPTL is strictly more
                         expressive than MTL. But we show that, surprisingly,
                         the TPTL formula proposed by Alur and Henzinger for
                         witnessing this conjecture can be expressed in MTL.
                         More generally, we show that TPTL formulae using
                         only the \textbf{F} modality can be translated into
                         MTL.},
}
[BCM10] Patricia Bouyer, Fabrice Chevalier, and Nicolas Markey. On the Expressiveness of TPTL and MTL. Information and Computation 208(2):97-116. Elsevier, February 2010.
Abstract

TPTL and MTL are two classical timed extensions of LTL. In this paper, we prove the 20-year-old conjecture that TPTL is strictly more expressive than MTL. But we show that, surprisingly, the TPTL formula proposed by Alur and Henzinger for witnessing this conjecture can be expressed in MTL. More generally, we show that TPTL formulae using only modality F can be translated into MTL.

@article{icomp208(2)-BCM,
  author =              {Bouyer, Patricia and Chevalier, Fabrice and Markey,
                         Nicolas},
  title =               {On the Expressiveness of {TPTL} and {MTL}},
  publisher =           {Elsevier},
  journal =             {Information and Computation},
  volume =              {208},
  number =              {2},
  pages =               {97-116},
  year =                {2010},
  month =               feb,
  doi =                 {10.1016/j.ic.2009.10.004},
  abstract =            {TPTL and MTL are two classical timed extensions
                         of~LTL. In~this paper, we prove the 20-year-old
                         conjecture that TPTL is strictly more expressive
                         than~MTL. But we show that, surprisingly, the
                         TPTL~formula proposed by Alur and Henzinger for
                         witnessing this conjecture \emph{can} be expressed
                         in~MTL. More generally, we show that TPTL formulae
                         using only modality~F can be translated into~MTL.},
}
[BCM16] Patricia Bouyer, Maximilien Colange, and Nicolas Markey. Symbolic Optimal Reachability in Weighted Timed Automata. In CAV'16, Lecture Notes in Computer Science 9779, pages 513-530. Springer-Verlag, July 2016.
Abstract

Weighted timed automata have been defined in the early 2000's for modelling resource-consumption or -allocation problems in real-time systems. Optimal reachability is decidable in weighted timed automata, and a symbolic forward algorithm has been developed to solve that problem. This algorithm uses so-called priced zones, an extension of standard zones with cost functions. In order to ensure termination, the algorithm requires clocks to be bounded. For unpriced timed automata, much work has been done to develop sound abstractions adapted to the forward exploration of timed automata, ensuring termination of the model-checking algorithm without bounding the clocks. In this paper, we take advantage of recent developments on abstractions for timed automata, and propose an algorithm allowing for symbolic analysis of all weighted timed automata, without requiring bounded clocks.

@inproceedings{cav2016-BCM,
  author =              {Bouyer, Patricia and Colange, Maximilien and Markey,
                         Nicolas},
  title =               {Symbolic Optimal Reachability in Weighted Timed
                         Automata},
  editor =              {Chaudhuri, Swarat and Farzan, Azadeh},
  booktitle =           {{P}roceedings of the 28th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'16)},
  acronym =             {{CAV}'16},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {9779},
  pages =               {513-530},
  year =                {2016},
  month =               jul,
  doi =                 {10.1007/978-3-319-41528-4_28},
  url =                 {http://arxiv.org/abs/1602.00481},
  abstract =            {Weighted timed automata have been defined in the
                         early 2000's for modelling resource-consumption or
                         -allocation problems in real-time systems. Optimal
                         reachability is decidable in weighted timed
                         automata, and a symbolic forward algorithm has been
                         developed to solve that problem. This algorithm uses
                         so-called priced zones, an extension of standard
                         zones with cost functions. In order to ensure
                         termination, the algorithm requires clocks to be
                         bounded. For unpriced timed automata, much work has
                         been done to develop sound abstractions adapted to
                         the forward exploration of timed automata, ensuring
                         termination of the model-checking algorithm without
                         bounding the clocks. In this paper, we take
                         advantage of recent developments on abstractions for
                         timed automata, and propose an algorithm allowing
                         for symbolic analysis of all weighted timed
                         automata, without requiring bounded clocks.},
}
[BCM+92] Jerry R. Burch, Edmund M. Clarke, Kenneth L. McMillan, David L. Dill, and L. J. Hwang. Symbolic model checking: 1020 states and beyond. Information and Computation 98(2):142-170. Academic Press, June 1992.
@article{icomp98(2)-BCMDH,
  author =              {Burch, Jerry R. and Clarke, Edmund M. and McMillan,
                         Kenneth L. and Dill, David L. and Hwang, L. J.},
  title =               {Symbolic model checking: {\(10^{20}\)} states and
                         beyond},
  publisher =           {Academic Press},
  journal =             {Information and Computation},
  volume =              {98},
  number =              {2},
  pages =               {142-170},
  year =                {1992},
  month =               jun,
}
[BCO+92] François Baccelli, Guy Cohen, Geert Jan Olsder, and Jean-Pierre Quadrat. Shynchronization and Linearity. An Algebra For Discrete Event Systems. John Wiley & Sons, 1992.
@book{BCOQ-des2001,
  author =              {Baccelli, Fran{\c c}ois and Cohen, Guy and Olsder,
                         Geert Jan and Quadrat, Jean-Pierre},
  title =               {Shynchronization and Linearity. An Algebra For
                         Discrete Event Systems},
  publisher =           {John Wiley \& Sons},
  year =                {1992},
  url =                 {http://www-rocq.inria.fr/metalau/cohen/SED/book-online.html},
}
[BCZ99] Armin Biere, Edmund M. Clarke, and Yunshan Zhu. Combining Local and Global Model Checking. In SMC'99, Electronic Notes in Theoretical Computer Science 23(2). Elsevier, July 1999.
@inproceedings{smc1999-BCZ,
  author =              {Biere, Armin and Clarke, Edmund M. and Zhu, Yunshan},
  title =               {Combining Local and Global Model Checking},
  editor =              {Cimatti, Alessandro and Grumberg, Orna},
  booktitle =           {{P}roceedings of the 1st {I}nternational {W}orkshop
                         on {S}ymbolic {M}odel {C}hecking ({SMC}'99)},
  acronym =             {{SMC}'99},
  publisher =           {Elsevier},
  series =              {Electronic Notes in Theoretical Computer Science},
  volume =              {23},
  number =              {2},
  year =                {1999},
  month =               jul,
}
[BD00] Béatrice Bérard and Catherine Dufourd. Timed Automata and Additive Clock Constraints. Information Processing Letters 75(1-2):1-7. Elsevier, July 2000.
@article{ipl75(1-2)-BD,
  author =              {B{\'e}rard, B{\'e}atrice and Dufourd, Catherine},
  title =               {Timed Automata and Additive Clock Constraints},
  publisher =           {Elsevier},
  journal =             {Information Processing Letters},
  volume =              {75},
  number =              {1-2},
  pages =               {1-7},
  year =                {2000},
  month =               jul,
}
[BDF+04] Patricia Bouyer, Catherine Dufourd, Emmanuel Fleury, and Antoine Petit. Updatable timed Automata. Theoretical Computer Science 321(2-3):291-345. Elsevier, August 2004.
@article{tcs321(2-3)-BDFP,
  author =              {Bouyer, Patricia and Dufourd, Catherine and Fleury,
                         Emmanuel and Petit, Antoine},
  title =               {Updatable timed Automata},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {321},
  number =              {2-3},
  pages =               {291-345},
  year =                {2004},
  month =               aug,
  doi =                 {10.1016/j.tcs.2004.04.003},
}
[BDG+11] Thomas Brihaye, Laurent Doyen, Gilles Geeraerts, Joël Ouaknine, Jean-François Raskin, and James Worrell. On Reachability for Hybrid Automata over Bounded Time. In ICALP'11, Lecture Notes in Computer Science 6756, pages 416-427. Springer-Verlag, July 2011.
@inproceedings{icalp2011(2)-BDGORW,
  author =              {Brihaye, {\relax Th}omas and Doyen, Laurent and
                         Geeraerts, Gilles and Ouaknine, Jo{\"e}l and Raskin,
                         Jean-Fran{\c c}ois and Worrell, James},
  title =               {On Reachability for Hybrid Automata over Bounded
                         Time},
  editor =              {Aceto, Luca and Henzinger, Monika and Sgall, Ji{
                         r}{\'\i}},
  booktitle =           {{P}roceedings of the 38th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'11)~-- Part~{II}},
  acronym =             {{ICALP}'11},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6756},
  pages =               {416-427},
  year =                {2011},
  month =               jul,
}
[BDG+98] Béatrice Bérard, Volker Diekert, Paul Gastin, and Antoine Petit. Characterization of the Expressive Power of Silent Transitions in Timed Automata. Fundamenta Informaticae 36(2-3):145-182. IOS Press, November 1998.
@article{fundi36(2-3)-BDGP,
  author =              {B{\'e}rard, B{\'e}atrice and Diekert, Volker and
                         Gastin, Paul and Petit, Antoine},
  title =               {Characterization of the Expressive Power of Silent
                         Transitions in Timed Automata},
  publisher =           {IOS Press},
  journal =             {Fundamenta Informaticae},
  volume =              {36},
  number =              {2-3},
  pages =               {145-182},
  year =                {1998},
  month =               nov,
}
[BDH+06] Dietmar Berwanger, Anuj Dawar, Paul Hunter, and Stephan Kreutzer. DAG-Width and Parity Games. In STACS'06, Lecture Notes in Computer Science 3884, pages 524-536. Springer-Verlag, February 2006.
@inproceedings{stacs2006-BDHK,
  author =              {Berwanger, Dietmar and Dawar, Anuj and Hunter, Paul
                         and Kreutzer, Stephan},
  title =               {{DAG}-Width and Parity Games},
  editor =              {Durand, Bruno and Thomas, Wolfgang},
  booktitle =           {{P}roceedings of the 23rd {S}ymposium on
                         {T}heoretical {A}spects of {C}omputer {S}cience
                         ({STACS}'06)},
  acronym =             {{STACS}'06},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3884},
  pages =               {524-536},
  year =                {2006},
  month =               feb,
  doi =                 {10.1007/11672142_43},
}
[BDJ+16] Thomas Brihaye, Benoît Delahaye, Loïg Jezequel, Nicolas Markey, and Jiří Srba (eds.) Proceedings of the Cassting Workshop on Games for the Synthesis of Complex Systems (Cassting'16) and of the 3rd International Workshop on Synthesis of Complex Parameters (SynCoP'16). Electronic Proceedings in Theoretical Computer Science 220. July 2016.
@proceedings{cassting-syncop2016-DJMS,
  title =               {{P}roceedings of the {C}assting Workshop on {G}ames
                         for the {S}ynthesis of {C}omplex {S}ystems
                         ({C}assting'16) and of the 3rd {I}nternational
                         {W}orkshop on {S}ynthesis of {C}omplex {P}arameters
                         ({S}yn{C}o{P}'16)},
  editor =              {Brihaye, {\relax Th}omas and Delahaye, Beno{\^\i}t
                         and Jezequel, Lo{\"\i}g and Markey, Nicolas and
                         Srba, Ji{ r}{\'\i}},
  booktitle =           {{P}roceedings of the {C}assting Workshop on {G}ames
                         for the {S}ynthesis of {C}omplex {S}ystems
                         ({C}assting'16) and of the 3rd {I}nternational
                         {W}orkshop on {S}ynthesis of {C}omplex {P}arameters
                         ({S}yn{C}o{P}'16)},
  acronym =             {{C}assting{{\slash}}{S}yn{C}o{P}'16},
  series =              {Electronic Proceedings in Theoretical Computer
                         Science},
  volume =              {220},
  year =                {2016},
  month =               jul,
  confyear =            {2016},
  confmonth =           {4},
  doi =                 {10.4204/EPTCS.220},
  url =                 {http://eptcs.web.cse.unsw.edu.au/
                         content.cgi?CASSTINGSynCoP2016},
}
[BDL12] Benedikt Bollig, Normann Decker, and Martin Leucker. Frequency Linear-time Temporal Logic. In TASE'12, pages 85-92. IEEE Comp. Soc. Press, July 2012.
@inproceedings{tase2012-BDL,
  author =              {Bollig, Benedikt and Decker, Normann and Leucker,
                         Martin},
  title =               {Frequency Linear-time Temporal Logic},
  editor =              {Margaria, Tiziana and Qiu, Zongyang and Yang,
                         Hongli},
  booktitle =           {{P}roceedings of the 6th {I}nternational {S}ymposium
                         on {T}heoretical {A}spects of {S}oftware
                         {E}ngineering ({TASE}'12)},
  acronym =             {{TASE}'12},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {85-92},
  year =                {2012},
  month =               jul,
  doi =                 {10.1109/TASE.2012.43},
}
[BDL+06] Gerd Behrmann, Alexandre David, Kim Guldstrand Larsen, John Håkansson, Paul Pettersson, Wang Yi, and Martijn Hendriks. UPPAAL 4.0. In QEST'06, pages 125-126. IEEE Comp. Soc. Press, September 2006.
@inproceedings{qest2006-BDLHPYH,
  author =              {Behrmann, Gerd and David, Alexandre and Larsen, Kim
                         Guldstrand and H{\aa}kansson, John and Pettersson,
                         Paul and Yi, Wang and Hendriks, Martijn},
  title =               {UPPAAL~4.0},
  booktitle =           {{P}roceedings of the 3rd {I}nternational
                         {C}onference on {Q}uantitative {E}valuation of
                         {S}ystems ({QEST}'06)},
  acronym =             {{QEST}'06},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {125-126},
  year =                {2006},
  month =               sep,
  doi =                 {10.1109/QEST.2006.59},
}
[BDL+14] Peter Bulychev, Alexandre David, Kim Guldstrand Larsen, and Guangyuan Li. Efficient Controller Synthesis for a fragment of MTL0,&infin. Acta Informatica 51(3-4):165-192. Springer-Verlag, June 2014.
@article{acta51(3-4)-BDLL,
  author =              {Bulychev, Peter and David, Alexandre and Larsen, Kim
                         Guldstrand and Li, Guangyuan},
  title =               {Efficient Controller Synthesis for a fragment of
                         {{\(\textsf{MTL}_{0,\infty}\)}}},
  publisher =           {Springer-Verlag},
  journal =             {Acta Informatica},
  volume =              {51},
  number =              {3-4},
  pages =               {165-192},
  year =                {2014},
  month =               jun,
  doi =                 {10.1007/s00236-013-0189-z},
}
[BDL+12] Peter Bulychev, Alexandre David, Kim Guldstrand Larsen, Axel Legay, Guangyuan Li, Danny Bøgsted Poulsen, and Amélie Stainer. Monitor-based statistical model checking for weighted metric temporal logic. In LPAR'12, Lecture Notes in Computer Science 7180, pages 168-182. Springer-Verlag, March 2012.
@inproceedings{lpar2012-BDLLLPS,
  author =              {Bulychev, Peter and David, Alexandre and Larsen, Kim
                         Guldstrand and Legay, Axel and Li, Guangyuan and
                         Poulsen, Danny B{\o}gsted and Stainer, Am{\'e}lie},
  title =               {Monitor-based statistical model checking for
                         weighted metric temporal logic},
  editor =              {Bj{\o}rner, Nikolaj and Voronkov, Andrei},
  booktitle =           {{P}roceedings of the 18th {I}nternational
                         {C}onference {L}ogic {P}rogramming and {A}utomated
                         {R}easoning ({LPAR}'12)},
  acronym =             {{LPAR}'12},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {7180},
  pages =               {168-182},
  year =                {2012},
  month =               mar,
  doi =                 {10.1007/978-3-642-28717-6_15},
}
[BDL+12] Peter Bulychev, Alexandre David, Kim Guldstrand Larsen, Axel Legay, Marius Mikučionis, and Danny Bøgsted Poulsen. Checking and Distributing Statistical Model Checking. In NFM'12, Lecture Notes in Computer Science 7226, pages 449-463. Springer-Verlag, April 2012.
@inproceedings{nasaFM2012-BDLLMP,
  author =              {Bulychev, Peter and David, Alexandre and Larsen, Kim
                         Guldstrand and Legay, Axel and Miku{{c}}ionis,
                         Marius and Poulsen, Danny B{\o}gsted},
  title =               {Checking and Distributing Statistical Model
                         Checking},
  editor =              {Goodloe, Alwyn and Person, Suzette},
  booktitle =           {{P}roceedings of the 4th {NASA} {F}ormal {M}ethods
                         {S}ymposium ({NFM}'12)},
  acronym =             {{NFM}'12},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {7226},
  pages =               {449-463},
  year =                {2012},
  month =               apr,
}
[BDL+09] Thomas Brihaye, Arnaud Da Costa, François Laroussinie, and Nicolas Markey. ATL with Strategy Contexts and Bounded Memory. In LFCS'09, Lecture Notes in Computer Science 5407, pages 92-106. Springer-Verlag, January 2009.
Abstract

We extend the alternating-time temporal logics ATL and ATL* with strategy contexts and memory constraints: the first extension make strategy quantifiers to not "forget" the strategies being executed by the other players. The second extension allows strategy quantifiers to restrict to memoryless or bounded-memory strategies.

We first consider expressiveness issues. We show that our logics can express important properties such as equilibria, and we formally compare them with other similar formalisms (ATL, ATL*, Game Logic, Strategy Logic, ...). We then address the problem of model-checking for our logics, providing a PSPACE algoritm for the sublogics involving only memoryless strategies and an EXPSPACE algorithm for the bounded-memory case.

@inproceedings{lfcs2009-BDLM,
  author =              {Brihaye, {\relax Th}omas and Da{~}Costa, Arnaud and
                         Laroussinie, Fran{\c c}ois and Markey, Nicolas},
  title =               {{ATL} with Strategy Contexts and Bounded Memory},
  editor =              {Artemov, Sergei N. and Nerode, Anil},
  booktitle =           {{P}roceedings of the {I}nternational {S}ymposium
                         {L}ogical {F}oundations of {C}omputer {S}cience
                         ({LFCS}'09)},
  acronym =             {{LFCS}'09},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {5407},
  pages =               {92-106},
  year =                {2009},
  month =               jan,
  doi =                 {10.1007/978-3-540-92687-0_7},
  abstract =            {We extend the alternating-time temporal logics ATL
                         and ATL\textsuperscript{*} with \emph{strategy
                         contexts} and \emph{memory constraints}: the first
                         extension make strategy quantifiers to not
                         {"}forget{"} the strategies being executed by the
                         other players. The second extension allows strategy
                         quantifiers to restrict to memoryless or
                         bounded-memory strategies.\par We first consider
                         expressiveness issues. We show that our logics can
                         express important properties such as equilibria, and
                         we formally compare them with other similar
                         formalisms (ATL, ATL\textsuperscript{*}, Game Logic,
                         Strategy Logic,~...). We~then address the problem of
                         model-checking for our logics, providing a PSPACE
                         algoritm for the sublogics involving only memoryless
                         strategies and an EXPSPACE algorithm for the
                         bounded-memory case.},
}
[BDL+97] Joshua Berman, Arthur Drisko, François Lemieux, Cristopher Moore, and Denis Thérien. Circuit and Expressions with Non-Associative Gates. In CoCo'97, pages 193-203. IEEE Comp. Soc. Press, June 1997.
@inproceedings{coco1997-BDLMT,
  author =              {Berman, Joshua and Drisko, Arthur and Lemieux,
                         Fran{\c c}ois and Moore, Cristopher and Th{\'e}rien,
                         Denis},
  title =               {Circuit and Expressions with Non-Associative Gates},
  booktitle =           {{P}roceedings of the 12th {A}nnual {IEEE}
                         {C}onference on {C}omputational {C}omplexity
                         ({CoCo}'97)},
  acronym =             {{CoCo}'97},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {193-203},
  year =                {1997},
  month =               jun,
}
[BDL+11] Gerd Behrmann, Alexandre David, Kim Guldstrand Larsen, Paul Pettersson, and Wang Yi. Developing UPPAAL over 15 years. Software – Practice and Experience 41(2):133-142. IEEE Comp. Soc. Press, February 2011.
@article{spe41(2)-ADGPY,
  author =              {Behrmann, Gerd and David, Alexandre and Larsen, Kim
                         Guldstrand and Pettersson, Paul and Yi, Wang},
  title =               {Developing {UPPAAL} over 15~years},
  publisher =           {IEEE Comp. Soc. Press},
  journal =             {Software~-- Practice and Experience},
  volume =              {41},
  number =              {2},
  pages =               {133-142},
  year =                {2011},
  month =               feb,
}
[BDM+03] Patricia Bouyer, Deepak D'Souza, Parthasarathy Madhusudan, and Antoine Petit. Timed Control with Partial Observability. In CAV'03, Lecture Notes in Computer Science 2725, pages 180-192. Springer-Verlag, July 2003.
@inproceedings{cav2003-BDMP,
  author =              {Bouyer, Patricia and D'Souza, Deepak and Madhusudan,
                         Parthasarathy and Petit, Antoine},
  title =               {Timed Control with Partial Observability},
  editor =              {Hunt, Jr, Warren A. and Somenzi, Fabio},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'03)},
  acronym =             {{CAV}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2725},
  pages =               {180-192},
  year =                {2003},
  month =               jul,
}
[BDM+09] Patricia Bouyer, Marie Duflot, Nicolas Markey, and Gabriel Renault. Measuring Permissivity in Finite Games. In CONCUR'09, Lecture Notes in Computer Science 5710, pages 196-210. Springer-Verlag, September 2009.
Abstract

In this paper, we extend the classical notion of strategies in turn-based finite games by allowing several moves to be selected. We define and study a quantitative measure for permissivity of such strategies by assigning penalties when blocking transitions. We prove that for reachability objectives, most permissive strategies exist, can be chosen memoryless, and can be computed in polynomial time, while it is in NP∩coNP for discounted and mean penalties.

@inproceedings{concur2009-BDMR,
  author =              {Bouyer, Patricia and Duflot, Marie and Markey,
                         Nicolas and Renault, Gabriel},
  title =               {Measuring Permissivity in Finite Games},
  editor =              {Bravetti, Mario and Zavattaro, Gianluigi},
  booktitle =           {{P}roceedings of the 20th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'09)},
  acronym =             {{CONCUR}'09},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {5710},
  pages =               {196-210},
  year =                {2009},
  month =               sep,
  doi =                 {10.1007/978-3-642-04081-8_14},
  abstract =            {In this paper, we extend the classical notion of
                         strategies in turn-based finite games by allowing
                         several moves to be selected. We~define and study a
                         quantitative measure for permissivity of such
                         strategies by assigning penalties when blocking
                         transitions. We~prove that for reachability
                         objectives, most permissive strategies exist, can be
                         chosen memoryless, and can be computed in polynomial
                         time, while it is in
                         \(\textsf{NP}\cap\textsf{coNP}\) for discounted and
                         mean penalties.},
}
[BDR03] Véronique Bruyère, Emmanuel Dall'Olio, and Jean-François Raskin. Durations, Parametric Model Checking in Timed Automata with Presburger Arithmetic. In STACS'03, Lecture Notes in Computer Science 2607, pages 687-698. Springer-Verlag, February 2003.
@inproceedings{stacs2003-BDR,
  author =              {Bruy{\`e}re, V{\'e}ronique and Dall'Olio, Emmanuel
                         and Raskin, Jean-Fran{\c c}ois},
  title =               {Durations, Parametric Model Checking in Timed
                         Automata with {P}resburger Arithmetic},
  editor =              {Alt, Helmut and Habib, Michel},
  booktitle =           {{P}roceedings of the 20th {S}ymposium on
                         {T}heoretical {A}spects of {C}omputer {S}cience
                         ({STACS}'03)},
  acronym =             {{STACS}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2607},
  pages =               {687-698},
  year =                {2003},
  month =               feb,
}
[BDS+09] Hans Bherer, Jules Desharnais, and Richard St-Denis. Control of Parameterized Discrete Event Systems. Discrete Event Dynamic Systems 19(2):213-165. Kluwer Academic, June 2009.
@article{deds19(2)-BDSD,
  author =              {Bherer, Hans and Desharnais, Jules and St-Denis,
                         Richard},
  title =               {Control of Parameterized Discrete Event Systems},
  publisher =           {Kluwer Academic},
  journal =             {Discrete Event Dynamic Systems},
  volume =              {19},
  number =              {2},
  pages =               {213-165},
  year =                {2009},
  month =               jun,
  doi =                 {10.1007/s10626-008-0040-9},
}
[Bea03] Danièle Beauquier. On probabilistic timed automata. Theoretical Computer Science 292(1):65-84. Elsevier, January 2003.
@article{tcs292(1)-Bea,
  author =              {Beauquier, Dani{\`e}le},
  title =               {On~probabilistic timed automata},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {292},
  number =              {1},
  pages =               {65-84},
  year =                {2003},
  month =               jan,
  doi =                 {10.1016/S0304-3975(01)00215-8},
}
[Bed01] Nicolas Bedon. Logic over Words on Denumerable Ordinals. Journal of Computer and System Sciences 63(3):394-431. Academic Press, November 2001.
@article{jcss63(3)-Bed,
  author =              {Bedon, Nicolas},
  title =               {Logic over Words on Denumerable Ordinals},
  publisher =           {Academic Press},
  journal =             {Journal of Computer and System Sciences},
  volume =              {63},
  number =              {3},
  pages =               {394-431},
  year =                {2001},
  month =               nov,
}
[BEG+15] Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Kazuhisa Makino. Markov Decision Processes and Stochastic Games with Total Effective Payoff. In STACS'15, Leibniz International Proceedings in Informatics 30, pages 103-115. Leibniz-Zentrum für Informatik, March 2015.
@inproceedings{stacs2015-BEGM,
  author =              {Boros, Endre and Elbassioni, Khaled and Gurvich,
                         Vladimir and Makino, Kazuhisa},
  title =               {{M}arkov Decision Processes and Stochastic Games
                         with Total Effective Payoff},
  editor =              {Mayr, Ernst W. and Ollinger, Nicolas},
  booktitle =           {{P}roceedings of the 32nd {S}ymposium on
                         {T}heoretical {A}spects of {C}omputer {S}cience
                         ({STACS}'15)},
  acronym =             {{STACS}'15},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {30},
  pages =               {103-115},
  year =                {2015},
  month =               mar,
  doi =                 {10.4230/LIPIcs.STACS.2015.103},
}
[Bei91] Richard Beigel. Bounded Queries to SAT and the Boolean Hierarchy. Theoretical Computer Science 84(2):199-223. Elsevier, 1991.
@article{tcs84(2)-Bei,
  author =              {Beigel, Richard},
  title =               {Bounded Queries to {SAT} and the {B}oolean
                         Hierarchy},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {84},
  number =              {2},
  pages =               {199-223},
  year =                {1991},
}
[Bel57] Richard Bellman. Dynamic Programming. Princeton University, 1957.
@book{DP-Bel,
  author =              {Bellman, Richard},
  title =               {Dynamic Programming},
  publisher =           {Princeton University},
  year =                {1957},
}
[Bel58] Richard Bellman. On a Routing Problem. Quarterly of Applied Mathematics 16(1):87-90. 1958.
@article{qam16(1)-Bel,
  author =              {Bellman, Richard},
  title =               {On a Routing Problem},
  journal =             {Quarterly of Applied Mathematics},
  volume =              {16},
  number =              {1},
  pages =               {87-90},
  year =                {1958},
}
[BEM96] Julian C. Bradfield, Javier Esparza, and Angelika Mader. An effective tableau system for the linear time mu-calculus. In ICALP'96, Lecture Notes in Computer Science 1099, pages 98-109. Springer-Verlag, July 1996.
@inproceedings{icalp1996-BEM,
  author =              {Bradfield, Julian C. and Esparza, Javier and Mader,
                         Angelika},
  title =               {An effective tableau system for the linear time
                         mu-calculus},
  editor =              {Meyer auf der Heide, Friedhelm and Monien, Burkhard},
  booktitle =           {{P}roceedings of the 23rd {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'96)},
  acronym =             {{ICALP}'96},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1099},
  pages =               {98-109},
  year =                {1996},
  month =               jul,
}
[Ber00] Gérard Berry. The foundations of Esterel. In Gordon D. Plotkin, Colin Stirling, and Mads Tofte (eds.), Proof, Language, and Interaction – Essays in Honour of Robin Milner. MIT Press, 2000.
@incollection{PLI2000-Ber,
  author =              {Berry, G{\'e}rard},
  title =               {The foundations of {E}sterel},
  editor =              {Plotkin, Gordon D. and Stirling, Colin and Tofte,
                         Mads},
  booktitle =           {Proof, Language, and Interaction~-- Essays in Honour
                         of {R}obin {M}ilner},
  publisher =           {MIT Press},
  pages =               {425-454},
  year =                {2000},
}
[Bey01] Dirk Beyer. Rabbit: Verification of Real-Time Systems. In RT-TOOLS'01, pages 13-21. August 2001.
@inproceedings{rttools2001-Bey,
  author =              {Beyer, Dirk},
  title =               {Rabbit: Verification of Real-Time Systems},
  editor =              {Pettersson, Paul and Yovine, Sergio},
  booktitle =           {{P}roceedings of the 1st {W}orkshop on {R}eal-Time
                         {T}ools ({RT-TOOLS}'01)},
  acronym =             {{RT-TOOLS}'01},
  pages =               {13-21},
  year =                {2001},
  month =               aug,
}
[BF09] Patricia Bouyer and Vojtěch Forejt. Reachability in Stochastic Timed Games. In ICALP'09, Lecture Notes in Computer Science 5556, pages 103-114. Springer-Verlag, July 2009.
@inproceedings{icalp2009-BF,
  author =              {Bouyer, Patricia and Forejt, Vojt{{e}}ch},
  title =               {Reachability in Stochastic Timed Games},
  editor =              {Albers, Susanne and Marchetti{-}Spaccamela Alberto
                         and Matias, Yossi and Nikoletseas, Sotiris and
                         Thomas, Wolfgang},
  booktitle =           {{P}roceedings of the 36th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'09)~-- Part~{II}},
  acronym =             {{ICALP}'09},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {5556},
  pages =               {103-114},
  year =                {2009},
  month =               jul,
}
[BFD02] Alexander Bolotov, Michael J. Fischer, and Clare Dixon. On the Relationship between ω-Automata and Temporal Logic Normal Forms. Journal of Logic and Computation 12(4):561-581. Oxford University Press, August 2002.
@article{jlc12(4)-BFD,
  author =              {Bolotov, Alexander and Fischer, Michael J. and
                         Dixon, Clare},
  title =               {On the Relationship between {\(\omega\)}-Automata
                         and Temporal Logic Normal Forms},
  publisher =           {Oxford University Press},
  journal =             {Journal of Logic and Computation},
  volume =              {12},
  number =              {4},
  pages =               {561-581},
  year =                {2002},
  month =               aug,
}
[BFH09] Felix Brandt, Felix Fischer, and Markus Holzer. Symmetries and the complexity of pure Nash equilibrium. Journal of Computer and System Sciences 75(3):163-177. Elsevier, May 2009.
@article{jcss75(3)-BFH,
  author =              {Brandt, Felix and Fischer, Felix and Holzer, Markus},
  title =               {Symmetries and the complexity of pure {N}ash
                         equilibrium},
  publisher =           {Elsevier},
  journal =             {Journal of Computer and System Sciences},
  volume =              {75},
  number =              {3},
  pages =               {163-177},
  year =                {2009},
  month =               may,
}
[BFH+03] Albert Benveniste, Éric Fabre, Stefan Haar, and Claude Jard. Diagnosis of asynchronous discrete event systems: A net-unfolding approach. IEEE Transactions on Automatic Control 48(5):714–727. IEEE Comp. Soc. Press, 2003.
@article{tac48(5)-BFHJ,
  author =              {Benveniste, Albert and Fabre, {\'E}ric and Haar,
                         Stefan and Jard, Claude},
  title =               {Diagnosis of asynchronous discrete event systems:
                         A~net-unfolding approach},
  publisher =           {IEEE Comp. Soc. Press},
  journal =             {IEEE Transactions on Automatic Control},
  volume =              {48},
  number =              {5},
  pages =               {714–727},
  year =                {2003},
  doi =                 {10.1109/TAC.2003.811249},
}
[BFH+01] Gerd Behrmann, Ansgar Fehnker, Thomas Hune, Kim Guldstrand Larsen, Paul Pettersson, and Judi Romijn. Efficient Guiding Towards Cost-Optimality in UPPAAL. In TACAS'01, Lecture Notes in Computer Science 2031, pages 174-188. Springer-Verlag, April 2001.
@inproceedings{tacas2001-BFHLPR,
  author =              {Behrmann, Gerd and Fehnker, Ansgar and Hune, Thomas
                         and Larsen, Kim Guldstrand and Pettersson, Paul and
                         Romijn, Judi},
  title =               {Efficient Guiding Towards Cost-Optimality in UPPAAL},
  editor =              {Margaria, Tiziana and Yi, Wang},
  booktitle =           {{P}roceedings of the 7th {I}nternational
                         {C}onference on {T}ools and {A}lgorithms for
                         {C}onstruction and {A}nalysis of {S}ystems
                         ({TACAS}'01)},
  acronym =             {{TACAS}'01},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2031},
  pages =               {174-188},
  year =                {2001},
  month =               apr,
}
[BFH+01] Gerd Behrmann, Ansgar Fehnker, Thomas Hune, Kim Guldstrand Larsen, Paul Pettersson, Judi Romijn, and Frits Vaandrager. Minimum-Cost Reachability for Priced Timed Automata. In HSCC'01, Lecture Notes in Computer Science 2034, pages 147-161. Springer-Verlag, March 2001.
@inproceedings{hscc2001-BFHLPRV,
  author =              {Behrmann, Gerd and Fehnker, Ansgar and Hune, Thomas
                         and Larsen, Kim Guldstrand and Pettersson, Paul and
                         Romijn, Judi and Vaandrager, Frits},
  title =               {Minimum-Cost Reachability for Priced Timed Automata},
  editor =              {Di{~}Benedetto, Maria Domenica and
                         Sangiovani{-}Vincentelli, Alberto L.},
  booktitle =           {{P}roceedings of the 4th {I}nternational {W}orkshop
                         on {H}ybrid {S}ystems: {C}omputation and {C}ontrol
                         ({HSCC}'01)},
  acronym =             {{HSCC}'01},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2034},
  pages =               {147-161},
  year =                {2001},
  month =               mar,
  doi =                 {10.1007/3-540-45351-2_15},
}
[BFL+10] Patricia Bouyer, Uli Fahrenberg, Kim Guldstrand Larsen, and Nicolas Markey. Timed Automata with Observers under Energy Constraints. In HSCC'10, pages 61-70. ACM Press, April 2010.
Abstract

In this paper, we study one-clock priced timed automata in which prices can grow linearly (dp/dt=k) or exponentially (dp/dt=kp), with discontinuous updates on edges. We propose EXPTIME algorithms to decide the existence of controllers that ensure existence of infinite runs or reachability of some goal location with non-negative observer value all along the run. These algorithms consist in computing the optimal delays that should be elapsed in each location along a run, so that the final observer value is maximized (and never goes below zero).

@inproceedings{hscc2010-BFLM,
  author =              {Bouyer, Patricia and Fahrenberg, Uli and Larsen, Kim
                         Guldstrand and Markey, Nicolas},
  title =               {Timed Automata with Observers under Energy
                         Constraints},
  editor =              {Johansson, Karl Henrik and Yi, Wang},
  booktitle =           {{P}roceedings of the 13th {I}nternational {W}orkshop
                         on {H}ybrid {S}ystems: {C}omputation and {C}ontrol
                         ({HSCC}'10)},
  acronym =             {{HSCC}'10},
  publisher =           {ACM Press},
  pages =               {61-70},
  year =                {2010},
  month =               apr,
  doi =                 {10.1145/1755952.1755963},
  abstract =            {In this paper, we study one-clock priced timed
                         automata in which prices can grow linearly
                         (\(\frac{dp}{dt}=k\)) or exponentially
                         (\(\frac{dp}{dt}=kp\)), with discontinuous updates
                         on edges. We propose EXPTIME algorithms to decide
                         the existence of controllers that ensure existence
                         of infinite runs or reachability of some goal
                         location with non-negative observer value all along
                         the run. These algorithms consist in computing the
                         optimal delays that should be elapsed in each
                         location along a run, so that the final observer
                         value is maximized (and never goes below zero).},
}
[BFL+11] Patricia Bouyer, Uli Fahrenberg, Kim Guldstrand Larsen, and Nicolas Markey. Quantitative analysis of real-time systems using priced timed automata. Communications of the ACM 54(9):78-87. ACM Press, September 2011.
Abstract

The problems of time-dependent behavior in general, and dynamic resource allocation in particular, pervade many aspects of modern life. Prominent examples range from reliability and efficient use of communication resources in a telecommunication network to the allocation of tracks in a continental railway network, from scheduling the usage of computational resources on a chip for durations of nano-seconds to the weekly, monthly, or longer-range reactive planning in a factory or a supply chain.

@article{cacm54(9)-BFLM,
  author =              {Bouyer, Patricia and Fahrenberg, Uli and Larsen, Kim
                         Guldstrand and Markey, Nicolas},
  title =               {Quantitative analysis of real-time systems using
                         priced timed automata},
  publisher =           {ACM Press},
  journal =             {Communications of the ACM},
  volume =              {54},
  number =              {9},
  pages =               {78-87},
  year =                {2011},
  month =               sep,
  doi =                 {10.1145/1995376.1995396},
  abstract =            {The problems of time-dependent behavior in general,
                         and dynamic resource allocation in particular,
                         pervade many aspects of modern life. Prominent
                         examples range from reliability and efficient use of
                         communication resources in a telecommunication
                         network to the allocation of tracks in a continental
                         railway network, from scheduling the usage of
                         computational resources on a chip for durations of
                         nano-seconds to the weekly, monthly, or longer-range
                         reactive planning in a factory or a supply chain.},
}
[BFL+18] Patricia Bouyer, Uli Fahrenberg, Kim Guldstrand Larsen, Nicolas Markey, Joël Ouaknine, and James Worrell. Model Checking Real-Time Systems. In Edmund M. Clarke, Thomas A. Henzinger, Helmut Veith, and Roderick Bloem (eds.), Handbook of Model Checking. Springer-Verlag, May 2018.
Abstract

This chapter surveys timed automata as a formalism for model checking real-time systems. We begin with introducing the model, as an extension of finite-state automata with real-valued variables for measuring time. We then present the main model-checking results in this framework, and give a hint about some recent extensions (namely weighted timed automata and timed games).

@incollection{hmc2018-BFLMOW,
  author =              {Bouyer, Patricia and Fahrenberg, Uli and Larsen, Kim
                         Guldstrand and Markey, Nicolas and Ouaknine,
                         Jo{\"e}l and Worrell, James},
  title =               {Model Checking Real-Time Systems},
  editor =              {Clarke, Edmund M. and Henzinger, Thomas A. and
                         Veith, Helmut and Bloem, Roderick},
  booktitle =           {Handbook of Model Checking},
  publisher =           {Springer-Verlag},
  pages =               {1001-1046},
  chapter =             {29},
  year =                {2018},
  month =               may,
  doi =                 {10.1007/978-3-319-10575-8_29},
  abstract =            {This chapter surveys timed automata as a formalism
                         for model checking real-time systems. We begin with
                         introducing the model, as an extension of
                         finite-state automata with real-valued variables for
                         measuring time. We then present the main
                         model-checking results in this framework, and give a
                         hint about some recent extensions (namely weighted
                         timed automata and timed games).},
}
[BFL+08] Patricia Bouyer, Uli Fahrenberg, Kim Guldstrand Larsen, Nicolas Markey, and Jiří Srba. Infinite Runs in Weighted Timed Automata with Energy Constraints. In FORMATS'08, Lecture Notes in Computer Science 5215, pages 33-47. Springer-Verlag, September 2008.
Abstract

We study the problems of existence and construction of infinite schedules for finite weighted automata and one-clock weighted timed automata, subject to boundary constraints on the accumulated weight. More specifically, we consider automata equipped with positive and negative weights on transitions and locations, corresponding to the production and consumption of some resource (e.g. energy). We ask the question whether there exists an infinite path for which the accumulated weight for any finite prefix satisfies certain constraints (e.g. remains between 0 and some given upper-bound). We also consider a game version of the above, where certain transitions may be uncontrollable.

@inproceedings{formats2008-BFLMS,
  author =              {Bouyer, Patricia and Fahrenberg, Uli and Larsen, Kim
                         Guldstrand and Markey, Nicolas and Srba, Ji{
                         r}{\'\i}},
  title =               {Infinite Runs in Weighted Timed Automata with Energy
                         Constraints},
  editor =              {Cassez, Franck and Jard, Claude},
  booktitle =           {{P}roceedings of the 6th {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'08)},
  acronym =             {{FORMATS}'08},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {5215},
  pages =               {33-47},
  year =                {2008},
  month =               sep,
  doi =                 {10.1007/978-3-540-85778-5_4},
  abstract =            {We~study the problems of existence and construction
                         of infinite schedules for finite weighted automata
                         and one-clock weighted timed automata, subject to
                         boundary constraints on the accumulated weight. More
                         specifically, we~consider automata equipped with
                         positive and negative weights on transitions and
                         locations, corresponding to the production and
                         consumption of some resource (\emph{e.g.}~energy).
                         We~ask the question whether there exists an infinite
                         path for which the accumulated weight for any finite
                         prefix satisfies certain constraints
                         (\emph{e.g.}~remains between~\(0\) and some given
                         upper-bound). We~also consider a game version of the
                         above, where certain transitions may be
                         uncontrollable.},
}
[BFM15] Patricia Bouyer, Erwin Fang, and Nicolas Markey. Permissive strategies in timed automata and games. In AVOCS'15, Electronic Communications of the EASST 72. European Association of Software Science and Technology, September 2015.
Abstract

Timed automata are a convenient framework for modelling and reasoning about real-time systems. While these models are now very well-understood, they do not offer a convenient way of taking timing imprecisions into account. Several solutions (e.g. parametric guard enlargement) have recently been proposed over the last ten years to take such imprecisions into account. In this paper, we propose a new approach for handling robust reachability, based on permissive strategies. While classical strategies propose to play an action at an exact point in time, permissive strategies return an interval of possible dates when to play the selected action. With such a permissive strategy, we associate a penalty, which is the inverse of the length of the proposed interval, and accumulates along the run. We show that in that setting, optimal strategies can be computed in polynomial time for one-clock timed automata.

@inproceedings{avocs2015-BFM,
  author =              {Bouyer, Patricia and Fang, Erwin and Markey,
                         Nicolas},
  title =               {Permissive strategies in timed automata and games},
  editor =              {Grov, Gudmund and Ireland, Andrew},
  booktitle =           {{P}roceedings of the 15th {I}nternational {W}orkshop
                         on {A}utomated {V}erification of {C}ritical
                         {S}ystems ({AVOCS}'15)},
  acronym =             {{AVOCS}'15},
  publisher =           {European Association of Software Science and
                         Technology},
  series =              {Electronic Communications of the EASST},
  volume =              {72},
  year =                {2015},
  month =               sep,
  doi =                 {10.14279/tuj.eceasst.72.1015},
  abstract =            {Timed automata are a convenient framework for
                         modelling and reasoning about real-time systems.
                         While these models are now very well-understood,
                         they do not offer a convenient way of taking timing
                         imprecisions into account. Several solutions (e.g.
                         parametric guard enlargement) have recently been
                         proposed over the last ten years to take such
                         imprecisions into account. In this paper, we propose
                         a new approach for handling robust reachability,
                         based on permissive strategies. While classical
                         strategies propose to play an action at an exact
                         point in time, permissive strategies return an
                         interval of possible dates when to play the selected
                         action. With such a permissive strategy, we
                         associate a penalty, which is the inverse of the
                         length of the proposed interval, and accumulates
                         along the run. We show that in that setting, optimal
                         strategies can be computed in polynomial time for
                         one-clock timed automata.},
}
[BFP+73] Manuel Blum, Robert W. Floyd, Vaughan R. Pratt, Ronald L. Rivest, and Robert Endre Tarjan. Time Bounds for Selection. Journal of Computer and System Sciences 7(4):448-461. Academic Press, August 1973.
@article{jcss7(4)-BFPRT,
  author =              {Blum, Manuel and Floyd, Robert W. and Pratt, Vaughan
                         R. and Rivest, Ronald L. and Tarjan, Robert Endre},
  title =               {Time Bounds for Selection},
  publisher =           {Academic Press},
  journal =             {Journal of Computer and System Sciences},
  volume =              {7},
  number =              {4},
  pages =               {448-461},
  year =                {1973},
  month =               aug,
}
[BFS14] Nathalie Bertrand, Paulin Fournier, and Arnaud Sangnier. Playing with Probabilities in Reconfigurable Broadcast Networks. In FoSSaCS'14, Lecture Notes in Computer Science 8412, pages 134-148. Springer-Verlag, April 2014.
@inproceedings{fossacs2014-BFS,
  author =              {Bertrand, Nathalie and Fournier, Paulin and
                         Sangnier, Arnaud},
  title =               {Playing with Probabilities in Reconfigurable
                         Broadcast Networks},
  editor =              {Muscholl, Anca},
  booktitle =           {{P}roceedings of the 17th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'14)},
  acronym =             {{FoSSaCS}'14},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8412},
  pages =               {134-148},
  year =                {2014},
  month =               apr,
  doi =                 {10.1007/978-3-642-54830-7_9},
}
[BFS15] Nathalie Bertrand, Paulin Fournier, and Arnaud Sangnier. Distributed Local Strategies in Broadcast Networks. In CONCUR'15, Leibniz International Proceedings in Informatics 42, pages 44-57. Leibniz-Zentrum für Informatik, September 2015.
@inproceedings{concur2015-BFS,
  author =              {Bertrand, Nathalie and Fournier, Paulin and
                         Sangnier, Arnaud},
  title =               {Distributed Local Strategies in Broadcast Networks},
  editor =              {Aceto, Luca and de Frutos{-}Escrig, David},
  booktitle =           {{P}roceedings of the 26th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'15)},
  acronym =             {{CONCUR}'15},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {42},
  pages =               {44-57},
  year =                {2015},
  month =               sep,
  doi =                 {10.4230/LIPIcs.CONCUR.2015.44},
}
[BG85] John P. Burgess and Yuri Gurevich. The Decision Problem for Linear Temporal Logic. Notre Dame Journal of Formal Logic 26(2):115-128. April 1985.
@article{ndjfl26(2)-BG,
  author =              {Burgess,John P. and Gurevich, Yuri},
  title =               {The Decision Problem for Linear Temporal Logic},
  journal =             {Notre Dame Journal of Formal Logic},
  volume =              {26},
  number =              {2},
  pages =               {115-128},
  year =                {1985},
  month =               apr,
}
[BG93] Orna Bernholtz and Orna Grumberg. Branching Time Temporal Logic and A m o r p H O u s Tree Automata. In CONCUR'93, Lecture Notes in Computer Science 715, pages 262-277. Springer-Verlag, August 1993.
@inproceedings{concur1993-BG,
  author =              {Bernholtz, Orna and Grumberg, Orna},
  title =               {Branching Time Temporal Logic and {{\(\mathcal{A}
                         \textup{m} \textbf{o} r \textsc{p} \mathcal{H}
                         \texttt{O} u \textbf{s}\)}} Tree Automata},
  editor =              {Best, Eike},
  booktitle =           {{P}roceedings of the 4th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'93)},
  acronym =             {{CONCUR}'93},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {715},
  pages =               {262-277},
  year =                {1993},
  month =               aug,
}
[BG99] Glenn Bruns and Patrice Godefroid. Model Checking Partial State Spaces with 3-Valued Temporal Logics. In CAV'99, Lecture Notes in Computer Science 1633, pages 274-287. Springer-Verlag, July 1999.
@inproceedings{cav1999-BG,
  author =              {Bruns, Glenn and Godefroid, Patrice},
  title =               {Model Checking Partial State Spaces with
                         {\(3\)}-Valued Temporal Logics},
  editor =              {Halbwachs, Nicolas and Peled, Doron A.},
  booktitle =           {{P}roceedings of the 11th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'99)},
  acronym =             {{CAV}'99},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1633},
  pages =               {274-287},
  year =                {1999},
  month =               jul,
}
[BG01] Glenn Bruns and Patrice Godefroid. Temporal Logic Query Checking (Extended Abstract). In LICS'01, pages 409-417. IEEE Comp. Soc. Press, June 2001.
@inproceedings{lics2001-BG,
  author =              {Bruns, Glenn and Godefroid, Patrice},
  title =               {Temporal Logic Query Checking (Extended Abstract)},
  booktitle =           {{P}roceedings of the 16th {A}nnual {S}ymposium on
                         {L}ogic in {C}omputer {S}cience ({LICS}'01)},
  acronym =             {{LICS}'01},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {409-417},
  year =                {2001},
  month =               jun,
}
[BG02] Robert Baumgartner and Georg Gottlob. Propositional Default Logics made Easier: Computational Complexity of Model Checking. Theoretical Computer Science 289(1):591-627. Elsevier, October 2002.
@article{tcs289(1)-BG,
  author =              {Baumgartner, Robert and Gottlob, Georg},
  title =               {Propositional Default Logics made Easier:
                         Computational Complexity of Model Checking},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {289},
  number =              {1},
  pages =               {591-627},
  year =                {2002},
  month =               oct,
}
[BG03] Régis Barbanchon and Étienne Grandjean. The Minimal Logically-Defined NP-Complete Problem. In STACS'04, Lecture Notes in Computer Science 2996, pages 338-349. Springer-Verlag, March 2003.
@inproceedings{stacs2004-BG,
  author =              {Barbanchon, R{\'e}gis and Grandjean, {\'E}tienne},
  title =               {The Minimal Logically-Defined {NP}-Complete Problem},
  editor =              {Diekert, Volker and Habib, Michel},
  booktitle =           {{P}roceedings of the 21st {S}ymposium on
                         {T}heoretical {A}spects of {C}omputer {S}cience
                         ({STACS}'04)},
  acronym =             {{STACS}'04},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2996},
  pages =               {338-349},
  year =                {2003},
  month =               mar,
}
[BG04] Dietmar Berwanger and Erich Grädel. Fixed-Point Logics and Solitaire Games. Theory of Computing Systems 37(6):675-694. Springer-Verlag, December 2004.
@article{TCSyst37(6)-BG,
  author =              {Berwanger, Dietmar and Gr{\"a}del, Erich},
  title =               {Fixed-Point Logics and Solitaire Games},
  publisher =           {Springer-Verlag},
  journal =             {Theory of Computing Systems},
  volume =              {37},
  number =              {6},
  pages =               {675-694},
  year =                {2004},
  month =               dec,
  doi =                 {10.1007/s00224-004-1147-5},
}
[BG05] Dietmar Berwanger and Erich Grädel. Entanglement –-mdash; A Measure for the Complexity of Directed Graphs with Applications to Logic and Games. In LPAR'04, Lecture Notes in Computer Science 3452, pages 209-223. Springer-Verlag, March 2005.
@inproceedings{lpar2004-BG,
  author =              {Berwanger, Dietmar and Gr{\"a}del, Erich},
  title =               {Entanglement~--- {A} Measure for the Complexity of
                         Directed Graphs with Applications to Logic and
                         Games},
  editor =              {Baader, Franz and Voronkov, Andrei},
  booktitle =           {{P}roceedings of the 11th {I}nternational
                         {C}onference {L}ogic {P}rogramming and {A}utomated
                         {R}easoning ({LPAR}'04)},
  acronym =             {{LPAR}'04},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3452},
  pages =               {209-223},
  year =                {2005},
  month =               mar,
}
[BGH13] Olivier Bournez, Daniel S. Graça, and Emmanuel Hainry. Computation with perturbed dynamical systems. Journal of Computer and System Sciences 79(5):714-724. Elsevier, August 2013.
@article{jcss79(5)-BGH,
  author =              {Bournez, Olivier and Gra{\c c}a, Daniel S. and
                         Hainry, Emmanuel},
  title =               {Computation with perturbed dynamical systems},
  publisher =           {Elsevier},
  journal =             {Journal of Computer and System Sciences},
  volume =              {79},
  number =              {5},
  pages =               {714-724},
  year =                {2013},
  month =               aug,
}
[BGH+09] Roderick Bloem, Karin Greimel, Thomas A. Henzinger, and Barbara Jobstmann. Synthesizing robust systems. In FMCAD'09, pages 85-92. IEEE Comp. Soc. Press, November 2009.
@inproceedings{fmcad2009-BGHJ,
  author =              {Bloem, Roderick and Greimel, Karin and Henzinger,
                         Thomas A. and Jobstmann, Barbara},
  title =               {Synthesizing robust systems},
  booktitle =           {{P}roceedings of the 9th {I}nternational
                         {C}onference on {F}ormal {M}ethods in
                         {C}omputer-{A}ided {D}esign ({FMCAD}'09)},
  acronym =             {{FMCAD}'09},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {85-92},
  year =                {2009},
  month =               nov,
}
[BGH+15] Thomas Brihaye, Gilles Geeraerts, Axel Haddad, Engel Lefaucheux, and Benjamin Monmege. Simple priced timed games are not that simple. In FSTTCS'15, Leibniz International Proceedings in Informatics 45, pages 278-292. Leibniz-Zentrum für Informatik, December 2015.
@inproceedings{fsttcs2005-BGHLM,
  author =              {Brihaye, {\relax Th}omas and Geeraerts, Gilles and
                         Haddad, Axel and Lefaucheux, Engel and Monmege,
                         Benjamin},
  title =               {Simple priced timed games are not that simple},
  editor =              {Harsha, Prahladh and Ramalingam, G.},
  booktitle =           {{P}roceedings of the 35th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'15)},
  acronym =             {{FSTTCS}'15},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {45},
  pages =               {278-292},
  year =                {2015},
  month =               dec,
  doi =                 {10.4230/LIPIcs.FSTTCS.2015.278},
}
[BGH+17] Thomas Brihaye, Gilles Geeraerts, Axel Haddad, and Benjamin Monmege. Pseudopolynomial iterative algorithm to solve total-payoff games and min-cost reachability game. Acta Informatica 54(1):85-125. Springer-Verlag, February 2017.
@article{acta54(1)-BGHM,
  author =              {Brihaye, {\relax Th}omas and Geeraerts, Gilles and
                         Haddad, Axel and Monmege, Benjamin},
  title =               {Pseudopolynomial iterative algorithm to solve
                         total-payoff games and min-cost reachability game},
  publisher =           {Springer-Verlag},
  journal =             {Acta Informatica},
  volume =              {54},
  number =              {1},
  pages =               {85-125},
  year =                {2017},
  month =               feb,
  doi =                 {10.1007/s00236-016-0276-z},
}
[BGI+01] Boaz Barak, Oded Goldreich, Rusell Impagliazzo, Steven Rudich, Amit Sahai, Salil Vadhan, and Ke Yang. On the (Im)possibility of Obfuscating Programs. In CRYPTO'01, Lecture Notes in Computer Science 2139, pages 1-18. Springer-Verlag, August 2001.
@inproceedings{crypto2001-BGIRSVY,
  author =              {Barak, Boaz and Goldreich, Oded and Impagliazzo,
                         Rusell and Rudich, Steven and Sahai, Amit and
                         Vadhan, Salil and Yang, Ke},
  title =               {On the (Im)possibility of Obfuscating Programs},
  editor =              {Kilian, Joe},
  booktitle =           {{P}roceedings of the 21st {A}nnual {I}nternational
                         {C}ryptology {C}onference ({CRYPTO}'01)},
  acronym =             {{CRYPTO}'01},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2139},
  pages =               {1-18},
  year =                {2001},
  month =               aug,
}
[BGI+01] Boaz Barak, Oded Goldreich, Rusell Impagliazzo, Steven Rudich, Amit Sahai, Salil Vadhan, and Ke Yang. On the (Im)possibility of Obfuscating Programs. Research Report 01-057, Electronic Colloquium on Computational Complexity, August 2001.
@techreport{eccc2001-BGIRSVY,
  author =              {Barak, Boaz and Goldreich, Oded and Impagliazzo,
                         Rusell and Rudich, Steven and Sahai, Amit and
                         Vadhan, Salil and Yang, Ke},
  title =               {On the (Im)possibility of Obfuscating Programs},
  number =              {01-057},
  year =                {2001},
  month =               aug,
  institution =         {Electronic Colloquium on Computational Complexity},
  type =                {Research Report},
}
[BGK+96] Johan Bengtsson, W. O. David Griffioen, Kåre J. Kristoffersen, Kim Guldstrand Larsen, Fredrik Larsson, Paul Pettersson, and Wang Yi. Verification of an Audio Protocol with Bus Collision Using Uppaal. In CAV'96, Lecture Notes in Computer Science 1102, pages 244-256. Springer-Verlag, July 1996.
@inproceedings{cav1996-BGKLLPY,
  author =              {Bengtsson, Johan and Griffioen, W. O. David and
                         Kristoffersen, K{\aa}re J. and Larsen, Kim
                         Guldstrand and Larsson, Fredrik and Pettersson, Paul
                         and Yi, Wang},
  title =               {Verification of an Audio Protocol with Bus Collision
                         Using {\scshape {U}ppaal}},
  editor =              {Alur, Rajeev and Henzinger, Thomas A.},
  booktitle =           {{P}roceedings of the 8th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'96)},
  acronym =             {{CAV}'96},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1102},
  pages =               {244-256},
  year =                {1996},
  month =               jul,
}
[BGM14] Patricia Bouyer, Patrick Gardy, and Nicolas Markey. Quantitative verification of weighted Kripke structures. In ATVA'14, Lecture Notes in Computer Science 8837, pages 64-80. Springer-Verlag, November 2014.
Abstract

Extending formal verification techniques to handle quantitative aspects, both for the models and for the properties to be checked, has become a central research topic over the last twenty years. Following several recent works, we study model checking for (one-dimensional) weighted Kripke structures with positive and negative weights, and temporal logics constraining the total and/or average weight. We prove decidability when only accumulated weight is constrained, while allowing average-weight constraints alone already is undecidable.

@inproceedings{atva2014-BGM,
  author =              {Bouyer, Patricia and Gardy, Patrick and Markey,
                         Nicolas},
  title =               {Quantitative verification of weighted {K}ripke
                         structures},
  editor =              {Cassez, Franck and Raskin, Jean-Fran{\c c}ois},
  booktitle =           {{P}roceedings of the 12th {I}nternational
                         {S}ymposium on {A}utomated {T}echnology for
                         {V}erification and {A}nalysis ({ATVA}'14)},
  acronym =             {{ATVA}'14},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8837},
  pages =               {64-80},
  year =                {2014},
  month =               nov,
  doi =                 {10.1007/978-3-319-11936-6_6},
  abstract =            {Extending formal verification techniques to handle
                         quantitative aspects, both for the models and for
                         the properties to be checked, has become a central
                         research topic over the last twenty years. Following
                         several recent works, we study model checking for
                         (one-dimensional) weighted Kripke structures with
                         positive and negative weights, and temporal logics
                         constraining the total and/or average weight. We
                         prove decidability when only accumulated weight is
                         constrained, while allowing average-weight
                         constraints alone already is undecidable.},
}
[BGM15] Patricia Bouyer, Patrick Gardy, and Nicolas Markey. Weighted strategy logic with boolean goals over one-counter games. In FSTTCS'15, Leibniz International Proceedings in Informatics 45, pages 69-83. Leibniz-Zentrum für Informatik, December 2015.
Abstract

Strategy Logic is a powerful specification language for expressing non-zero-sum properties of multi-player games. SL conveniently extends the logic ATL with explicit quantification and assignment of strategies. In this paper, we consider games over one-counter automata, and a quantitative extension 1cSL of SL with assertions over the value of the counter. We prove two results: we first show that, if decidable, model checking the so-called Boolean-goal fragment of 1cSL has non-elementary complexity; we actually prove the result for the Boolean-goal fragment of SL over finite-state games, which was an open question in (Mogavero et al. Reasoning about strategies: On the model-checking problem. 2014). As a first step towards proving decidability, we then show that the Boolean-goal fragment of 1cSL over one-counter games enjoys a nice periodicity property.

@inproceedings{fsttcs2015-BGM,
  author =              {Bouyer, Patricia and Gardy, Patrick and Markey,
                         Nicolas},
  title =               {Weighted strategy logic with boolean goals over
                         one-counter games},
  editor =              {Harsha, Prahladh and Ramalingam, G.},
  booktitle =           {{P}roceedings of the 35th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'15)},
  acronym =             {{FSTTCS}'15},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {45},
  pages =               {69-83},
  year =                {2015},
  month =               dec,
  doi =                 {10.4230/LIPIcs.FSTTCS.2015.69},
  abstract =            {Strategy Logic is a powerful specification language
                         for expressing non-zero-sum properties of
                         multi-player games. SL conveniently extends the
                         logic ATL with explicit quantification and
                         assignment of strategies. In this paper, we consider
                         games over one-counter automata, and a quantitative
                         extension 1cSL of SL with assertions over the value
                         of the counter. We prove two results: we first show
                         that, if decidable, model checking the so-called
                         Boolean-goal fragment of 1cSL has non-elementary
                         complexity; we actually prove the result for the
                         Boolean-goal fragment of SL over finite-state games,
                         which was an open question in (Mogavero
                         \emph{et~al.} Reasoning about strategies: On the
                         model-checking problem. 2014). As a first step
                         towards proving decidability, we then show that the
                         Boolean-goal fragment of 1cSL over one-counter games
                         enjoys a nice periodicity property.},
}
[BGM16] Patricia Bouyer, Patrick Gardy, and Nicolas Markey. On the semantics of Strategy Logic. Information Processing Letters 116(2):75-79. Elsevier, February 2016.
Abstract

We define and study a slight variation on the semantics of Strategy Logic: while in the classical semantics, all strategies are shifted during the evaluation of temporal modalities, we propose to only shift the strategies that have been assigned to a player, thus matching the intuition that we can assign the very same strategy to the players at different points in time. We prove that surprisingly, this renders the model-checking problem undecidable.

@article{ipl116(2)-BGM,
  author =              {Bouyer, Patricia and Gardy, Patrick and Markey,
                         Nicolas},
  title =               {On the semantics of Strategy Logic},
  publisher =           {Elsevier},
  journal =             {Information Processing Letters},
  volume =              {116},
  number =              {2},
  pages =               {75-79},
  year =                {2016},
  month =               feb,
  doi =                 {10.1016/j.ipl.2015.10.004},
  abstract =            {We define and study a slight variation on the
                         semantics of Strategy Logic: while in the classical
                         semantics, all strategies are shifted during the
                         evaluation of temporal modalities, we propose to
                         only shift the strategies that have been assigned to
                         a player, thus matching the intuition that we can
                         assign the very same strategy to the players at
                         different points in time. We prove that
                         surprisingly, this renders the model-checking
                         problem undecidable.},
}
[BGM+08] Thomas Brihaye, Mohamed Ghannem, Nicolas Markey, and Lionel Rieg. Good friends are hard to find!. In TIME'08, pages 32-40. IEEE Comp. Soc. Press, June 2008.
Abstract

We focus on the problem of finding (the size of) a minimal winning coalition in a multi-player game. More precisely, we prove that deciding whether there is a winning coalition of size at most k is NP-complete, while deciding whether k is the optimal size is DP-complete. We also study different variants of our original problem: the function problem, where the aim is to effectively compute the coalition; more succinct encoding of the game; and richer families of winning objectives.

@inproceedings{time2008-BGMR,
  author =              {Brihaye, {\relax Th}omas and Ghannem, Mohamed and
                         Markey, Nicolas and Rieg, Lionel},
  title =               {Good friends are hard to find!},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {S}ymposium on {T}emporal {R}epresentation and
                         {R}easoning ({TIME}'08)},
  acronym =             {{TIME}'08},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {32-40},
  year =                {2008},
  month =               jun,
  doi =                 {10.1109/TIME.2008.10},
  abstract =            {We focus on the problem of finding (the~size of)
                         a~minimal winning coalition in a multi-player game.
                         More precisely, we~prove that deciding whether there
                         is a winning coalition of size at most~\(k\) is
                         NP-complete, while deciding whether \(k\) is the
                         optimal size is DP-complete. We~also study different
                         variants of our original problem: the function
                         problem, where the aim is to effectively compute the
                         coalition; more succinct encoding of the game; and
                         richer families of winning objectives.},
}
[BGM+18] Patricia Bouyer, Mauricio González, Nicolas Markey, and Mickael Randour. Multi-weighted Markov Decision Processes with Reachability Objectives. In GandALF'18, Electronic Proceedings in Theoretical Computer Science. September 2018. To appear.
Abstract

In this paper, we are interested in the synthesis of schedulers in double-weighted Markov decision processes, which satisfy both a percentile constraint over a weighted reachability condition, and a quantitative constraint on the expected value of a random variable defined using a weighted reachability condition. This problem is inspired by the modelization of an electric-vehicle charging problem. We study the cartography of the problem, when one parameter varies, and show how a partial cartography can be obtained via two sequences of opimization problems. We discuss completeness and feasability of the method.

@inproceedings{gandalf2018-BGMR,
  author =              {Bouyer, Patricia and Gonz{\'a}lez, Mauricio and
                         Markey, Nicolas and Randour, Mickael},
  title =               {Multi-weighted Markov Decision Processes with
                         Reachability Objectives},
  editor =              {Orlandini, Andrea and Zimmermann, Martin},
  booktitle =           {{P}roceedings of the 9th {I}nternational {S}ymposium
                         on {G}ames, {A}utomata, {L}ogics and {F}ormal
                         {V}erification ({GandALF}'18)},
  acronym =             {{GandALF}'18},
  series =              {Electronic Proceedings in Theoretical Computer
                         Science},
  year =                {2018},
  month =               sep,
  note =                {To~appear},
  abstract =            {In~this paper, we~are interested in the synthesis of
                         schedulers in double-weighted Markov decision
                         processes, which satisfy both a percentile
                         constraint over a weighted reachability condition,
                         and a quantitative constraint on the expected value
                         of a random variable defined using a weighted
                         reachability condition. This~problem is inspired by
                         the modelization of an electric-vehicle charging
                         problem. We study the cartography of the problem,
                         when one parameter varies, and show how a partial
                         cartography can be obtained via two sequences of
                         opimization problems. We~discuss completeness and
                         feasability of the method.},
}
[BGN+14] Thomas Brihaye, Gilles Geeraerts, Shankara Narayanan Krishna, Lakshmi Manasa, Benjamin Monmege, and Ashutosh Trivedi. Adding negative prices to priced timed games. In CONCUR'14, Lecture Notes in Computer Science 8704, pages 560-575. Springer-Verlag, September 2014.
@inproceedings{concur2014-BGKMMT,
  author =              {Brihaye, {\relax Th}omas and Geeraerts, Gilles and
                         Narayanan Krishna, Shankara and Manasa, Lakshmi and
                         Monmege, Benjamin and Trivedi, Ashutosh},
  title =               {Adding negative prices to priced timed games},
  editor =              {Baldan, Paolo and Gorla, Daniele},
  booktitle =           {{P}roceedings of the 25th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'14)},
  acronym =             {{CONCUR}'14},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8704},
  pages =               {560-575},
  year =                {2014},
  month =               sep,
  doi =                 {10.1007/978-3-662-44584-6_38},
}
[BGP96] Béatrice Bérard, Paul Gastin, and Antoine Petit. Timed Automata with non Observable Actions: Expressive power and refinement. In STACS'96, Lecture Notes in Computer Science 1046, pages 257-268. Springer-Verlag, February 1996.
@inproceedings{stacs1996-BGP,
  author =              {B{\'e}rard, B{\'e}atrice and Gastin, Paul and Petit,
                         Antoine},
  title =               {Timed Automata with non Observable Actions:
                         Expressive power and refinement},
  editor =              {Puech, Claude and Reischuk, R{\"u}diger},
  booktitle =           {{P}roceedings of the 13th {S}ymposium on
                         {T}heoretical {A}spects of {C}omputer {S}cience
                         ({STACS}'96)},
  acronym =             {{STACS}'96},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1046},
  pages =               {257-268},
  year =                {1996},
  month =               feb,
}
[BGS92] José Balcázar, Joaquim Gabarró, and Miklós Sántha. Deciding bisimilarity is P-complete. Formal Aspects of Computing 4(1 supplement):638-648. Springer-Verlag, November 1992.
@article{fac4(1sup)-BGSC,
  author =              {Balc{\'a}zar, Jos{\'e} and Gabarr{\'o}, Joaquim and
                         S{\'a}ntha, Mikl{\'o}s},
  title =               {Deciding bisimilarity is {P}-complete},
  publisher =           {Springer-Verlag},
  journal =             {Formal Aspects of Computing},
  volume =              {4},
  number =              {1 supplement},
  pages =               {638-648},
  year =                {1992},
  month =               nov,
}
[BGS14] Benedikt Bollig, Paul Gastin, and Jana Schubert. Parameterized verification of communicating automata under context bounds. In RP'14, Lecture Notes in Computer Science 8762, pages 45-57. Springer-Verlag, September 2014.
@inproceedings{rp2014-BGS,
  author =              {Bollig, Benedikt and Gastin, Paul and Schubert,
                         Jana},
  title =               {Parameterized verification of communicating automata
                         under context bounds},
  editor =              {Ouaknine, Jo{\"e}l and Potapov, Igor and Worrell,
                         James},
  booktitle =           {{P}roceedings of the 8th {W}orkshop on
                         {R}eachability {P}roblems in {C}omputational
                         {M}odels ({RP}'14)},
  acronym =             {{RP}'14},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8762},
  pages =               {45-57},
  year =                {2014},
  month =               sep,
  doi =                 {10.1007/978-3-319-11439-2_4},
}
[BGW01] Nikita Borisov, Ian Goldberg, and Klaus W. Wagner. Intercepting Mobile Communications: The Insecurity of 802.11. In MOBICOM'01, pages 180-189. ACM Press, July 2001.
@inproceedings{mobicom2001-BGW,
  author =              {Borisov, Nikita and Goldberg, Ian and Wagner, Klaus
                         W.},
  title =               {Intercepting Mobile Communications: The Insecurity
                         of~{\(802.11\)}},
  booktitle =           {{P}roceedings of the 7th {A}nnual {I}nternational
                         {C}onference on {M}obile {C}omputing and
                         {N}etworking ({MOBICOM}'01)},
  acronym =             {{MOBICOM}'01},
  publisher =           {ACM Press},
  pages =               {180-189},
  year =                {2001},
  month =               jul,
}
[BH67] Manuel Blum and Carl Hewitt. Automata on a 2-Dimensional Tape. In FOCS'67, pages 155-160. IEEE Comp. Soc. Press, October 1967.
@inproceedings{focs1967-BH,
  author =              {Blum, Manuel and Hewitt, Carl},
  title =               {Automata on a 2-Dimensional Tape},
  booktitle =           {{P}roceedings of the 8th {A}nnual {S}ymposium on
                         {F}oundations of {C}omputer {S}cience ({FOCS}'67)},
  acronym =             {{FOCS}'67},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {155-160},
  year =                {1967},
  month =               oct,
}
[BH81] Arthur J. Bernstein and Jr. Harter. Proving Real-Time Properties of Programs with Temporal Logic. In SOSP'81, Operating System Review 15(5), pages 1-11. ACM Press, December 1981.
@inproceedings{sosp1981-BH,
  author =              {Bernstein, Arthur J. and Harter, Paul K., Jr.},
  title =               {Proving Real-Time Properties of Programs with
                         Temporal Logic},
  booktitle =           {{P}roceedings of the 8th {ACM} {S}ymposium on
                         {O}perating {S}ystems {P}rinciples ({SOSP}'81)},
  acronym =             {{SOSP}'81},
  publisher =           {ACM Press},
  series =              {Operating System Review},
  volume =              {15},
  number =              {5},
  pages =               {1-11},
  year =                {1981},
  month =               dec,
}
[BH04] Anne Bergeron and Sylvie Hamel. From Cascade Decompositions to Bit-Vector Algotithms. Theoretical Computer Science 313(1):3-16. Elsevier, February 2004.
@article{tcs313(1)-BH,
  author =              {Bergeron, Anne and Hamel, Sylvie},
  title =               {From Cascade Decompositions to Bit-Vector
                         Algotithms},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {313},
  number =              {1},
  pages =               {3-16},
  year =                {2004},
  month =               feb,
}
[BH05] Dragan Bosnacki and Gerard J. Holzmann. Improving Spin's Partial-Order Reduction for Breadth-First Search. In SPIN'05, Lecture Notes in Computer Science 3639, pages 91-105. Springer-Verlag, April 2005.
@inproceedings{spin2005-BH,
  author =              {Bosnacki, Dragan and Holzmann, Gerard J.},
  title =               {Improving {S}pin's Partial-Order Reduction for
                         Breadth-First Search},
  editor =              {Godefroid, Patrice},
  booktitle =           {{P}roceedings of the 12th {I}nternational {SPIN}
                         {W}orkshop ({SPIN}'05)},
  acronym =             {{SPIN}'05},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3639},
  pages =               {91-105},
  year =                {2005},
  month =               apr,
  doi =                 {10.1007/11537328_10},
}
[BH06] Bernard Boigelot and Frédéric Herbreteau. The Power of Hybrid Acceleration. In CAV'06, Lecture Notes in Computer Science 4144, pages 438-451. Springer-Verlag, July 2006.
@inproceedings{cav2006-BH,
  author =              {Boigelot, Bernard and Herbreteau, Fr{\'e}d{\'e}ric},
  title =               {The Power of Hybrid Acceleration},
  editor =              {Ball, Thomas and Jones, Robert},
  booktitle =           {{P}roceedings of the 18th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'06)},
  acronym =             {{CAV}'06},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4144},
  pages =               {438-451},
  year =                {2006},
  month =               jul,
  doi =                 {10.1007/11817963_40},
}
[Bhe09] Hans Bherer. Controller Synthesis for Parameterized Discrete Event Systems. PhD thesis, Université Laval, Québec, Canada, 2009.
@phdthesis{bherer09,
  author =              {Bherer, Hans},
  title =               {Controller Synthesis for Parameterized Discrete
                         Event Systems},
  year =                {2009},
  school =              {Universit{\'e} Laval, Qu\'ebec, Canada},
}
[BHJ03] Bernard Boigelot, Frédéric Herbreteau, and Sébastien Jodogne. Hybrid Acceleration using Real Vector Automata (extended abstract). In CAV'03, Lecture Notes in Computer Science 2725, pages 193-205. Springer-Verlag, July 2003.
@inproceedings{cav2003-BHJ,
  author =              {Boigelot, Bernard and Herbreteau, Fr{\'e}d{\'e}ric
                         and Jodogne, S{\'e}bastien},
  title =               {Hybrid Acceleration using Real Vector Automata
                         (extended abstract)},
  editor =              {Hunt, Jr, Warren A. and Somenzi, Fabio},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'03)},
  acronym =             {{CAV}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2725},
  pages =               {193-205},
  year =                {2003},
  month =               jul,
  doi =                 {10.1007/978-3-540-45069-6_19},
}
[BHJ03] Bernard Boigelot, Frédéric Herbreteau, and Sébastien Jodogne. Hybrid Acceleration using Real Vector Automata. Technical Report 2003.18, Centre Fédéré en Vérification, Bruxelles, Belgium, 2003.
@techreport{TR-cfv0318,
  author =              {Boigelot, Bernard and Herbreteau, Fr{\'e}d{\'e}ric
                         and Jodogne, S{\'e}bastien},
  title =               {Hybrid Acceleration using Real Vector Automata},
  number =              {2003.18},
  year =                {2003},
  institution =         {Centre F\'ed\'er\'e en V\'erification, Bruxelles,
                         Belgium},
  type =                {Technical Report},
}
[BHK00] Steven Bradley, William Henderson, and David Kendall. Using Timed Automata for Response Time Analysis and Distributer Real-Time Systems. In WRTP'99, pages 143-148. Pergamon Press, January 2000.
@inproceedings{wrtp1999-BHK,
  author =              {Bradley, Steven and Henderson, William and Kendall,
                         David},
  title =               {Using Timed Automata for Response Time Analysis and
                         Distributer Real-Time Systems},
  editor =              {Frigeri, Alceu Heinke and Halang, Wolfgang A. and
                         Son, Sang H.},
  booktitle =           {{P}roceedings of the 24th {IFAC}/{IFIP} {W}orkshop
                         on {R}eal-{T}ime {P}rogramming ({WRTP}'99)},
  acronym =             {{WRTP}'99},
  publisher =           {Pergamon Press},
  pages =               {143-148},
  year =                {2000},
  month =               jan,
  confyear =            {1999},
  confmonth =           {5},
}
[BHL14] Nathalie Bertrand, Serge Haddad, and Engel Lefaucheux. Foundation of Diagnosis and Predictability in Probabilistic Systems. In FSTTCS'14, Leibniz International Proceedings in Informatics 29, pages 417-429. Leibniz-Zentrum für Informatik, December 2014.
@inproceedings{fsttcs2014-BHL,
  author =              {Bertrand, Nathalie and Haddad, Serge and Lefaucheux,
                         Engel},
  title =               {Foundation of Diagnosis and Predictability in
                         Probabilistic Systems},
  editor =              {Raman, Venkatesh and Suresh, S. P.},
  booktitle =           {{P}roceedings of the 34th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'14)},
  acronym =             {{FSTTCS}'14},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {29},
  pages =               {417-429},
  year =                {2014},
  month =               dec,
  doi =                 {10.4230/LIPIcs.FSTTCS.2014.417},
}
[BHM08] Aske Wiid Brekling, Michael R. Hansen, and Jan Madsen. Models and formal verification of multiprocessor system-on-chips. Journal of Logic and Algebraic Programming 77(1-2):1-19. Elsevier, October 2008.
@article{jlap77(1-2)-BHM,
  author =              {Brekling, Aske Wiid and Hansen, Michael R. and
                         Madsen, Jan},
  title =               {Models and formal verification of multiprocessor
                         system-on-chips},
  publisher =           {Elsevier},
  journal =             {Journal of Logic and Algebraic Programming},
  volume =              {77},
  number =              {1-2},
  pages =               {1-19},
  year =                {2008},
  month =               oct,
}
[BHM14] Bernard Boigelot, Frédéric Herbreteau, and Isabelle Mainz. Acceleration of Affine Hybrid Transformations. In ATVA'14, Lecture Notes in Computer Science 8837, pages 31-46. Springer-Verlag, November 2014.
@inproceedings{atva2014-BHM,
  author =              {Boigelot, Bernard and Herbreteau, Fr{\'e}d{\'e}ric
                         and Mainz, Isabelle},
  title =               {Acceleration of Affine Hybrid Transformations},
  editor =              {Cassez, Franck and Raskin, Jean-Fran{\c c}ois},
  booktitle =           {{P}roceedings of the 12th {I}nternational
                         {S}ymposium on {A}utomated {T}echnology for
                         {V}erification and {A}nalysis ({ATVA}'14)},
  acronym =             {{ATVA}'14},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8837},
  pages =               {31-46},
  year =                {2014},
  month =               nov,
  doi =                 {10.1007/978-3-319-11936-6_4},
}
[BHM17] Béatrice Bérard, Loïc Hélouët, and John Mullins. Non-interference in partial order models. ACM Transactions on Embedded Computing Systems. ACM Press, 2017. To appear.
@article{tecs-BHM,
  author =              {B{\'e}rard, B{\'e}atrice and H{\'e}lou{\"e}t,
                         Lo{\"\i}c and Mullins, John},
  title =               {Non-interference in partial order models},
  publisher =           {ACM Press},
  journal =             {ACM Transactions on Embedded Computing Systems},
  year =                {2017},
  note =                {To~appear},
}
[BHM+17] Patricia Bouyer, Piotr Hofman, Nicolas Markey, Mickael Randour, and Martin Zimmermann. Bounding Average-energy Games. In FoSSaCS'17, Lecture Notes in Computer Science 10203, pages 179-195. Springer-Verlag, April 2017.
Abstract

We consider average-energy games, where the goal is to minimize the long-run average of the accumulated energy. Decidability of average-energy games with a lower-bound constraint on the energy level (but no upper bound) is an open problem; in particular, there is no known upper bound on the memory that is required for winning strategies.

By reducing average-energy games with lower-bounded energy to infinite-state mean-payoff games and analyzing the frequency of low-energy configurations, we show an almost tight doubly-exponential upper bound on the necessary memory, and that the winner of average-energy games with lower-bounded energy can be determined in doubly-exponential time. We also prove EXPSPACE-hardness of this problem.

Finally, we consider multi-dimensional extensions of all types of average-energy games: without bounds, with only a lower bound, and with both a lower and an upper bound on the energy. We show that the fully-bounded version is the only case to remain decidable in multiple dimensions.

@inproceedings{fossacs2017-BHMRZ,
  author =              {Bouyer, Patricia and Hofman, Piotr and Markey,
                         Nicolas and Randour, Mickael and Zimmermann, Martin},
  title =               {Bounding Average-energy Games},
  editor =              {Esparza, Javier and Murawski, Andrzej},
  booktitle =           {{P}roceedings of the 20th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'17)},
  acronym =             {{FoSSaCS}'17},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {10203},
  pages =               {179-195},
  year =                {2017},
  month =               apr,
  doi =                 {10.1007/978-3-662-54458-7_11},
  abstract =            {We~consider average-energy games, where the goal is
                         to minimize the long-run average of the accumulated
                         energy. Decidability of average-energy games with a
                         lower-bound constraint on the energy level (but~no
                         upper bound) is an open problem; in~particular,
                         there is no known upper bound on the memory that is
                         required for winning strategies.\par By~reducing
                         average-energy games with lower-bounded energy to
                         infinite-state mean-payoff games and analyzing the
                         frequency of low-energy configurations, we~show an
                         almost tight doubly-exponential upper bound on the
                         necessary memory, and that the winner of
                         average-energy games with lower-bounded energy can
                         be determined in doubly-exponential time. We~also
                         prove EXPSPACE-hardness of this problem.\par
                         Finally, we~consider multi-dimensional extensions of
                         all types of average-energy games: without bounds,
                         with only a lower bound, and with both a lower and
                         an upper bound on the energy. We show that the
                         fully-bounded version is the only case to remain
                         decidable in multiple dimensions.},
}
[BHP+07] Thomas Brihaye, Thomas A. Henzinger, Vinayak S. Prabhu, and Jean-François Raskin. Minimum-Time Reachability in Timed Games. In ICALP'07, Lecture Notes in Computer Science 4596, pages 825-837. Springer-Verlag, July 2007.
@inproceedings{icalp2007-BHPR,
  author =              {Brihaye, {\relax Th}omas and Henzinger, Thomas A.
                         and Prabhu, Vinayak S. and Raskin, Jean-Fran{\c
                         c}ois},
  title =               {Minimum-Time Reachability in Timed Games},
  editor =              {Arge, Lars and Cachin, Christian and Jurdzi{\'n}ski,
                         Tomasz and Tarlecki, Andrzej},
  booktitle =           {{P}roceedings of the 34th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'07)},
  acronym =             {{ICALP}'07},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4596},
  pages =               {825-837},
  year =                {2007},
  month =               jul,
}
[BHR06] Patricia Bouyer, Serge Haddad, and Pierre-Alain Reynier. Timed Unfoldings for Networks of Timed Automata. In ATVA'06, Lecture Notes in Computer Science 4218, pages 292-306. Springer-Verlag, October 2006.
@inproceedings{atva2006-BHR,
  author =              {Bouyer, Patricia and Haddad, Serge and Reynier,
                         Pierre-Alain},
  title =               {Timed Unfoldings for Networks of Timed Automata},
  editor =              {Graf, Susanne and Zhang, Wenhui},
  booktitle =           {{P}roceedings of the 4th {I}nternational {S}ymposium
                         on {A}utomated {T}echnology for {V}erification and
                         {A}nalysis ({ATVA}'06)},
  acronym =             {{ATVA}'06},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4218},
  pages =               {292-306},
  year =                {2006},
  month =               oct,
  doi =                 {10.1007/11901914_23},
}
[BHR09] Patricia Bouyer, Serge Haddad, and Pierre-Alain Reynier. Undecidability Results for Timed Automata with Silent Transitions. Fundamenta Informaticae 92(1-2):1-25. IOS Press, 2009.
@article{fundi92(1-2)-BHR,
  author =              {Bouyer, Patricia and Haddad, Serge and Reynier,
                         Pierre-Alain},
  title =               {Undecidability Results for Timed Automata with
                         Silent Transitions},
  publisher =           {IOS Press},
  journal =             {Fundamenta Informaticae},
  volume =              {92},
  number =              {1-2},
  pages =               {1-25},
  year =                {2009},
}
[BHR+02] Danièle Beauquier, Yoram Hirshfeld, Alexander Rabinovich, and Anatol Slissenko. The Probablity Nesting Game. In EXPRESS'02, Electronic Notes in Theoretical Computer Science 68(2), pages 3-13. Elsevier, August 2002.
@inproceedings{express2002-BHRS,
  author =              {Beauquier, Dani{\`e}le and Hirshfeld, Yoram and
                         Rabinovich, Alexander and Slissenko, Anatol},
  title =               {The Probablity Nesting Game},
  editor =              {Nestmann, Uwe and Panangaden, Prakash},
  booktitle =           {{P}roceedings of the 9th {I}nternational {W}orkshop
                         on {E}xpressiveness in {C}oncurrency ({EXPRESS}'02)},
  acronym =             {{EXPRESS}'02},
  publisher =           {Elsevier},
  series =              {Electronic Notes in Theoretical Computer Science},
  volume =              {68},
  number =              {2},
  pages =               {3-13},
  year =                {2002},
  month =               aug,
}
[BHT07] Dirk Beyer, Thomas A. Henzinger, and Grégory Théoduloz. Configurable Software Verification: Concretizing the Convergence of Model Checking and Program Analysis. In CAV'07, Lecture Notes in Computer Science 4590. Springer-Verlag, July 2007.
@inproceedings{cav2007-BHT,
  author =              {Beyer, Dirk and Henzinger, Thomas A. and
                         Th{\'e}oduloz, Gr{\'e}gory},
  title =               {Configurable Software Verification: Concretizing the
                         Convergence of Model Checking and Program Analysis},
  editor =              {Damm, Werner and Hermanns, Holger},
  booktitle =           {{P}roceedings of the 19th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'07)},
  acronym =             {{CAV}'07},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4590},
  year =                {2007},
  month =               jul,
  doi =                 {10.1007/978-3-540-73368-3_51},
}
[BIL06] Marius Bozga, Radu Iosif, and Yassine Lakhnech. Flat Parametric Counter Automata. Research Report TR-2005-15, Lab. VERIMAG, Grenoble, France, March 2006.
@techreport{TR-verimag2005-15,
  author =              {Bozga, Marius and Iosif, Radu and Lakhnech, Yassine},
  title =               {Flat Parametric Counter Automata},
  number =              {TR-2005-15},
  year =                {2006},
  month =               mar,
  institution =         {Lab. VERIMAG, Grenoble, France},
  type =                {Research Report},
}
[BIL06] Marius Bozga, Radu Iosif, and Yassine Lakhnech. Flat Parametric Counter Automata. In ICALP'06, Lecture Notes in Computer Science 4052, pages 577-588. Springer-Verlag, July 2006.
@inproceedings{icalp2006-BIL,
  author =              {Bozga, Marius and Iosif, Radu and Lakhnech, Yassine},
  title =               {Flat Parametric Counter Automata},
  editor =              {Bugliesi, Michele and Preneel, Bart and Sassone,
                         Vladimiro and Wegener, Ingo},
  booktitle =           {{P}roceedings of the 33rd {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'06))~-- Part~{II}},
  acronym =             {{ICALP}'06},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4052},
  pages =               {577-588},
  year =                {2006},
  month =               jul,
  doi =                 {10.1007/11787006_49},
}
[BIS90] David A. Mix Barrington, Neil Immerman, and Howard Straubing. On Uniformity Within NC1. Journal of Computer and System Sciences 41(3):274-306. Academic Press, December 1990.
@article{jcss41(3)-BIS,
  author =              {Barrington, David A. Mix and Immerman, Neil and
                         Straubing, Howard},
  title =               {On Uniformity Within {{\(\mathit{NC}^1\)}}},
  publisher =           {Academic Press},
  journal =             {Journal of Computer and System Sciences},
  volume =              {41},
  number =              {3},
  pages =               {274-306},
  year =                {1990},
  month =               dec,
}
[BJ10] Nils Bulling and Wojciech Jamroga. Model Checking Agents with Memory Is Harder than It Seemed. In AAMAS'10, pages 633-640. International Foundation for Autonomous Agents and Multiagent Systems, May 2010.
@inproceedings{aamas2010-BJ,
  author =              {Bulling, Nils and Jamroga, Wojciech},
  title =               {Model Checking Agents with Memory Is Harder than It
                         Seemed},
  editor =              {van der Hoek, Wiebe and Kaminka, Gal A. and
                         Lesp{\'e}rance, Yves and Luck, Michael and Sen,
                         Sandeep},
  booktitle =           {{P}roceedings of the 9th {I}nternational
                         {C}onference on {A}utonomous {A}gents and
                         {M}ultiagent {S}ystems ({AAMAS}'10)},
  acronym =             {{AAMAS}'10},
  publisher =           {International Foundation for Autonomous Agents and
                         Multiagent Systems},
  pages =               {633-640},
  year =                {2010},
  month =               may,
}
[BJK10] Tomáš Brázdil, Petr Jančar, and Antonín Kučera. Reachability Games on Extended Vector Addition Systems with States. In ICALP'10, Lecture Notes in Computer Science 6199, pages 478-489. Springer-Verlag, July 2010.
@inproceedings{icalp2010-BJK,
  author =              {Br{\'a}zdil, Tom{\'a}{ s} and Jan{ c}ar, Petr
                         and Ku{ c}era, Anton{\'\i}n},
  title =               {Reachability Games on Extended Vector Addition
                         Systems with States},
  editor =              {Abramsky, Samson and Gavoille, Cyril and Kirchner,
                         Claude and Meyer auf der Heide, Friedhelm and
                         Spirakis, Paul G.},
  booktitle =           {{P}roceedings of the 37th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'10)~-- Part~{II}},
  acronym =             {{ICALP}'10},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6199},
  pages =               {478-489},
  year =                {2010},
  month =               jul,
  doi =                 {10.1007/978-3-642-14162-1_40},
}
[BJK+05] Manfred Broy, Bengt Jonsson, Joost-Pieter Katoen, Martin Leucker, and Alexander Pretschner. Model-Based Testing of Reactive Systems – Advanced lectures. Lecture Notes in Computer Science 3472. Springer-Verlag, 2005.
@book{mbt-bjklp,
  author =              {Broy, Manfred and Jonsson, Bengt and Katoen,
                         Joost-Pieter and Leucker, Martin and Pretschner,
                         Alexander},
  title =               {Model-Based Testing of Reactive Systems~-- Advanced
                         lectures},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3472},
  year =                {2005},
}
[BJL+10] Thomas Brihaye, Marc Jungers, Samson Lasaulce, Nicolas Markey, and Ghassan Oreiby. Using Model Checking for Analyzing Distributed Power Control Problems. EURASIP Journal on Wireless Communications and Networking 2010(861472). Hindawi Publishing Corp., June 2010.
Abstract

Model checking (MC) is a formal verification technique which has been known and still knows a resounding success in the computer science community. Realizing that the distributed power control (PC) problem can be modeled by a timed game between a given transmitter and its environment, the authors wanted to know whether this approach can be applied to distributed PC. It turns out that it can be applied successfully and allows one to analyze realistic scenarios including the case of discrete transmit powers and games with incomplete information. The proposed methodology is as follows. We state some objectives a transmitter-receiver pair would like to reach. The network is modeled by a game where transmitters are considered as timed automata interacting with each other. The objectives are then translated into timed alternating-time temporal logic formulae and MC is exploited to know whether the desired properties are verified and determine a winning strategy.

@article{jwcn2010(861472)-BJLMO,
  author =              {Brihaye, {\relax Th}omas and Jungers, Marc and
                         Lasaulce, Samson and Markey, Nicolas and Oreiby,
                         Ghassan},
  title =               {Using Model Checking for Analyzing Distributed Power
                         Control Problems},
  publisher =           {Hindawi Publishing Corp.},
  journal =             {EURASIP Journal on Wireless Communications and
                         Networking},
  volume =              {2010},
  number =              {861472},
  year =                {2010},
  month =               jun,
  doi =                 {10.1155/2010/861472},
  abstract =            {Model checking~(MC) is a formal verification
                         technique which has been known and still knows a
                         resounding success in the computer science
                         community. Realizing that the distributed power
                         control~(PC) problem can be modeled by a timed game
                         between a given transmitter and its environment, the
                         authors wanted to know whether this approach can be
                         applied to distributed~PC. It~turns out that it can
                         be applied successfully and allows one to analyze
                         realistic scenarios including the case of discrete
                         transmit powers and games with incomplete
                         information. The proposed methodology is as follows.
                         We state some objectives a transmitter-receiver pair
                         would like to reach. The network is modeled by a
                         game where transmitters are considered as timed
                         automata interacting with each other. The objectives
                         are then translated into timed alternating-time
                         temporal logic formulae and MC is exploited to know
                         whether the desired properties are verified and
                         determine a winning strategy.},
}
[BJL+98] Johan Bengtsson, Bengt Jonsson, Johan Lilius, and Wang Yi. Partial-Order Reductions for Timed Systems. In CONCUR'98, Lecture Notes in Computer Science 1466, pages 485-500. Springer-Verlag, September 1998.
@inproceedings{concur1998-BJLY,
  author =              {Bengtsson, Johan and Jonsson, Bengt and Lilius,
                         Johan and Yi, Wang},
  title =               {Partial-Order Reductions for Timed Systems},
  editor =              {Sangiorgi, Davide and de Simone, Robert},
  booktitle =           {{P}roceedings of the 9th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'98)},
  acronym =             {{CONCUR}'98},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1466},
  pages =               {485-500},
  year =                {1998},
  month =               sep,
  doi =                 {10.1007/BFb0055643},
}
[BJM15] Patricia Bouyer, Samy Jaziri, and Nicolas Markey. On the Value Problem in Weighted Timed Games. In CONCUR'15, Leibniz International Proceedings in Informatics 42, pages 311-324. Leibniz-Zentrum für Informatik, September 2015.
Abstract

A weighted timed game is a timed game with extra quantitative information representing e.g. energy consumption. Optimizing the cost for reaching a target is a natural question, which has been investigated for ten years. Existence of optimal strategies is known to be undecidable in general, and only very restricted classes of games have been described for which optimal cost and almost-optimal strategies can be computed.

In this paper, we show that the value problem is undecidable in general weighted timed games. The undecidability proof relies on that for the existence of optimal strategies and on a diagonalization construction recently designed in the context of quantitative temporal logics. We then provide an algorithm to compute arbitrary approximations of the value in a game, and almost-optimal strategies. The algorithm applies in a large subclass of weighted timed games, and is the first approximation scheme which is designed in the current context.

@inproceedings{concur2015-BJM,
  author =              {Bouyer, Patricia and Jaziri, Samy and Markey,
                         Nicolas},
  title =               {On~the Value Problem in Weighted Timed Games},
  editor =              {Aceto, Luca and de Frutos{-}Escrig, David},
  booktitle =           {{P}roceedings of the 26th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'15)},
  acronym =             {{CONCUR}'15},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {42},
  pages =               {311-324},
  year =                {2015},
  month =               sep,
  doi =                 {10.4230/LIPIcs.CONCUR.2015.311},
  abstract =            {A~weighted timed game is a timed game with extra
                         quantitative information representing e.g. energy
                         consumption. Optimizing the cost for reaching a
                         target is a natural question, which has been
                         investigated for ten years. Existence of optimal
                         strategies is known to be undecidable in general,
                         and only very restricted classes of games have been
                         described for which optimal cost and almost-optimal
                         strategies can be computed.\par In this paper, we
                         show that the value problem is undecidable in
                         general weighted timed games. The undecidability
                         proof relies on that for the existence of optimal
                         strategies and on a diagonalization construction
                         recently designed in the context of quantitative
                         temporal logics. We then provide an algorithm to
                         compute arbitrary approximations of the value in a
                         game, and almost-optimal strategies. The algorithm
                         applies in a large subclass of weighted timed games,
                         and is the first approximation scheme which is
                         designed in the current context.},
}
[BJM17] Patricia Bouyer, Samy Jaziri, and Nicolas Markey. On the determinization of timed systems. In FORMATS'17, Lecture Notes in Computer Science 10419, pages 25-41. Springer-Verlag, September 2017.
Abstract

We introduce a new formalism called automata over a timed domain which provides an adequate framework for the determinization of timed systems. In this formalism, determinization w.r.t. timed language is always possible at the cost of changing the timed domain. We give a condition for determinizability of automata over a timed domain without changing the timed domain, which allows us to recover several known determinizable classes of timed systems, such as strongly-non-zeno timed automata, integer-reset timed automata, perturbed timed automata, etc. Moreover in the case of timed automata this condition encompasses most determinizability conditions from the literature.

@inproceedings{formats2017-BJM,
  author =              {Bouyer, Patricia and Jaziri, Samy and Markey,
                         Nicolas},
  title =               {On the determinization of timed systems},
  editor =              {Abate, Alessandro and Geeraerts, Gilles},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'17)},
  acronym =             {{FORMATS}'17},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {10419},
  pages =               {25-41},
  year =                {2017},
  month =               sep,
  doi =                 {10.1007/978-3-319-65765-3 2},
  abstract =            {We introduce a new formalism called \emph{automata
                         over a timed domain} which provides an adequate
                         framework for the determinization of timed systems.
                         In~this formalism, determinization w.r.t. timed
                         language is always possible at the cost of changing
                         the timed domain. We~give a condition for
                         determinizability of automata over a timed domain
                         without changing the timed domain, which allows us
                         to recover several known determinizable classes of
                         timed systems, such as strongly-non-zeno timed
                         automata, integer-reset timed automata, perturbed
                         timed automata, etc. Moreover in the case of timed
                         automata this condition encompasses most
                         determinizability conditions from the literature.},
}
[BJM17] Patricia Bouyer, Vincent Jugé, and Nicolas Markey. Courcelle's Theorem Made Dynamic. Technical Report 1702.05183, arXiv, February 2017.
Abstract

Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of model checking a fixed monadic second-order formula over evolving subgraphs of a fixed maximal graph having bounded tree-width; here the subgraph evolves by losing or gaining edges (from the maximal graph). We show that this problem is in DynFO (with LOGSPACE precomputation), via a reduction to a Dyck reachability problem on an acyclic automaton.

@techreport{arxiv1702.05183-BJM,
  author =              {Bouyer, Patricia and Jug{\'e}, Vincent and Markey,
                         Nicolas},
  title =               {{C}ourcelle's Theorem Made Dynamic},
  number =              {1702.05183},
  year =                {2017},
  month =               feb,
  institution =         {arXiv},
  abstract =            {Dynamic complexity is concerned with updating the
                         output of a problem when the input is slightly
                         changed. We study the dynamic complexity of model
                         checking a fixed monadic second-order formula over
                         evolving subgraphs of a fixed maximal graph having
                         bounded tree-width; here the subgraph evolves by
                         losing or gaining edges (from the maximal graph).
                         We~show that this problem is in DynFO (with LOGSPACE
                         precomputation), via a reduction to a Dyck
                         reachability problem on an acyclic automaton.},
}
[BJM+02] Marius Bozga, Hou Jianmin, Oded Maler, and Sergio Yovine. Verification of Asynchronous Circuits using Timed Automata. In TPTS'02, Electronic Notes in Theoretical Computer Science 65(6), pages 47-59. Elsevier, April 2002.
@inproceedings{tpts2002-BJMY,
  author =              {Bozga, Marius and Jianmin, Hou and Maler, Oded and
                         Yovine, Sergio},
  title =               {Verification of Asynchronous Circuits using Timed
                         Automata},
  editor =              {Asarin, Eugene and Maler, Oded and Yovine, Sergio},
  booktitle =           {{P}roceedings of the 1st {W}orkshop on {T}heory and
                         {P}ractice of {T}imed {S}ystems ({TPTS}'02)},
  acronym =             {{TPTS}'02},
  publisher =           {Elsevier},
  series =              {Electronic Notes in Theoretical Computer Science},
  volume =              {65},
  number =              {6},
  pages =               {47-59},
  year =                {2002},
  month =               apr,
}
[BJS09] Joakim Byg, Kenneth Yrke Jørgensen, and Jiří Srba. TAPAAL: Editor, Simulator and Verifier of Timed-Arc Petri Nets. In ATVA'09, Lecture Notes in Computer Science 5799, pages 84-89. Springer-Verlag, October 2009.
@inproceedings{atva2009-BJS,
  author =              {Byg, Joakim and J{\o}rgensen, Kenneth Yrke and Srba,
                         Ji{ r}{\'\i}},
  title =               {{TAPAAL}: Editor, Simulator and Verifier of
                         Timed-Arc {P}etri Nets},
  editor =              {Liu, Zhiming and Ravn, Anders P.},
  booktitle =           {{P}roceedings of the 7th {I}nternational {S}ymposium
                         on {A}utomated {T}echnology for {V}erification and
                         {A}nalysis ({ATVA}'09)},
  acronym =             {{ATVA}'09},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {5799},
  pages =               {84-89},
  year =                {2009},
  month =               oct,
  doi =                 {10.1007/978-3-642-04761-9_7},
}
[BJS+12] Nathalie Bertrand, Thierry Jéron, Amélie Stainer, and Moez Krichen. Off-line test selection with test purposes for non-deterministic timed automata. Logical Methods in Computer Science 8(4). 2012.
@article{lmcs8(4)-BJSK,
  author =              {Bertrand, Nathalie and J{\'e}ron, Thierry and
                         Stainer, Am{\'e}lie and Krichen, Moez},
  title =               {Off-line test selection with test purposes for
                         non-deterministic timed automata},
  journal =             {Logical Methods in Computer Science},
  volume =              {8},
  number =              {4},
  year =                {2012},
  doi =                 {10.2168/LMCS-8(4:8)2012},
}
[BJV10] Jasper Berendsen, David N. Jansen, and Frits Vaandrager. Fortuna: model checking priced probabilistic timed automata. In QEST'10, pages 273-281. IEEE Comp. Soc. Press, September 2010.
@inproceedings{qest2010-BJV,
  author =              {Berendsen, Jasper and Jansen, David N. and
                         Vaandrager, Frits},
  title =               {Fortuna: model checking priced probabilistic timed
                         automata},
  booktitle =           {{P}roceedings of the 7th {I}nternational
                         {C}onference on {Q}uantitative {E}valuation of
                         {S}ystems ({QEST}'10)},
  acronym =             {{QEST}'10},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {273-281},
  year =                {2010},
  month =               sep,
}
[BJW02] Julien Bernet, David Janin, and Igor Walukiewicz. Permissive strategies: from parity games to safety games. RAIRO – Theoretical Informatics and Applications 36(3):261-275. EDP Sciences, 2002.
@article{rairo-tia36(3)-BJW,
  author =              {Bernet, Julien and Janin, David and Walukiewicz,
                         Igor},
  title =               {Permissive strategies: from parity games to safety
                         games},
  publisher =           {EDP Sciences},
  journal =             {RAIRO~-- Theoretical Informatics and Applications},
  volume =              {36},
  number =              {3},
  pages =               {261-275},
  year =                {2002},
}
[BK08] Christel Baier and Joost-Pieter Katoen. Principles of Model-Checking. MIT Press, May 2008.
@book{PoMC2008-BK,
  author =              {Baier, {\relax Ch}ristel and Katoen, Joost-Pieter},
  title =               {Principles of Model-Checking},
  publisher =           {MIT Press},
  year =                {2008},
  month =               may,
}
[BKK11] Christel Baier, Joachim Klein, and Sascha Klüppelholz. A Compositional Framework for Controller Synthesis. In CONCUR'11, Lecture Notes in Computer Science 6901, pages 512-527. Springer-Verlag, September 2011.
@inproceedings{concur2011-BKK,
  author =              {Baier, {\relax Ch}ristel and Klein, Joachim and
                         Kl{\"u}ppelholz, Sascha},
  title =               {A Compositional Framework for Controller Synthesis},
  editor =              {Katoen, Joost-Pieter and K{\"o}nig, Barbara},
  booktitle =           {{P}roceedings of the 22nd {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'11)},
  acronym =             {{CONCUR}'11},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6901},
  pages =               {512-527},
  year =                {2011},
  month =               sep,
}
[BKK+14] Tomáš Brázdil, David Klaška, Antonín Kučera, and Petr Novotný. Minimizing Running Costs in Consumption Systems. In CAV'14, Lecture Notes in Computer Science 8559, pages 457-472. Springer-Verlag, July 2014.
@inproceedings{cav2014-BKKN,
  author =              {Br{\'a}zdil, Tom{\'a}{ s} and Kla{{s}}ka, David
                         and Ku{ c}era, Anton{\'\i}n and Novotn{\'y}, Petr},
  title =               {Minimizing Running Costs in Consumption Systems},
  editor =              {Biere, Armin and Bloem, Roderick},
  booktitle =           {{P}roceedings of the 26th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'14)},
  acronym =             {{CAV}'14},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8559},
  pages =               {457-472},
  year =                {2014},
  month =               jul,
}
[BKM+06] Jeremy W. Bryans, Maciej Koutny, Laurent Mazaré, and Peter Y. A. Ryan. Opacity generalised to transition systems. In FAST'05, Lecture Notes in Computer Science 3866, pages 81-95. Springer-Verlag, 2006.
@inproceedings{fast2005-BKMR,
  author =              {Bryans, Jeremy W. and Koutny, Maciej and Mazar{\'e},
                         Laurent and Ryan, Peter Y. A.},
  title =               {Opacity generalised to transition systems},
  editor =              {Dimitrakos, Theo and Martinelli, Fabio and Ryan,
                         Peter Y. A. and Schneider, Steve},
  booktitle =           {{R}evised {S}elected {P}apers of the 3rd
                         {I}nternational {W}orkshop on {F}ormal {A}spects in
                         {S}ecurity and {T}rust ({FAST}'05)},
  acronym =             {{FAST}'05},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3866},
  pages =               {81-95},
  year =                {2006},
  confyear =            {2005},
  confmonth =           {7},
  doi =                 {10.1007/11679219_7},
}
[BKO+08] Thomas Bøgholm, Henrik Kargh-hansen, Petur Olsen, Bent Thomsen, and Kim Guldstrand Larsen. Model-based schedulability analysis of safety critical hard real-time Java programs. In JTRES'08, ACM International Conference Proceeding Series 343, pages 106-114. ACM Press, September 2008.
@inproceedings{jtres2008-BKOTL,
  author =              {B{\o}gholm, Thomas and Kargh{-}hansen, Henrik and
                         Olsen, Petur and Thomsen, Bent and Larsen, Kim
                         Guldstrand},
  title =               {Model-based schedulability analysis of safety
                         critical hard real-time Java programs},
  editor =              {Bollella, Gregory and Locke, C. Douglas},
  booktitle =           {{P}roceedings of the 6th {I}nternational {W}orkshop
                         on {J}ava {T}echnologies for {R}eal-time and
                         {E}mbedded {S}ystems ({JTRES}'08)},
  acronym =             {{JTRES}'08},
  publisher =           {ACM Press},
  series =              {ACM International Conference Proceeding Series},
  volume =              {343},
  pages =               {106-114},
  year =                {2008},
  month =               sep,
  doi =                 {10.1145/1434790.1434807},
}
[BKP86] Howard Barringer, Ruurd Kuiper, and Amir Pnueli. A Really Abstract Concurrent Model and its Temporal Logic. In POPL'86, pages 173-183. ACM Press, January 1986.
@inproceedings{popl1986-BKP,
  author =              {Barringer, Howard and Kuiper, Ruurd and Pnueli,
                         Amir},
  title =               {A Really Abstract Concurrent Model and its Temporal
                         Logic},
  booktitle =           {Conference Record of the 13th {ACM} {S}ymposium on
                         {P}rinciples of {P}rogramming {L}anguages
                         ({POPL}'86)},
  acronym =             {{POPL}'86},
  publisher =           {ACM Press},
  pages =               {173-183},
  year =                {1986},
  month =               jan,
}
[BKP11] Dietmar Berwanger, łukasz Kaiser, and Bernd Puchala. A Perfect-Information Construction for Coordination in Games. In FSTTCS'11, Leibniz International Proceedings in Informatics 13, pages 387-398. Leibniz-Zentrum für Informatik, December 2011.
@inproceedings{fsttcs2011-BKP,
  author =              {Berwanger, Dietmar and Kaiser, {\L}ukasz and
                         Puchala, Bernd},
  title =               {A~Perfect-Information Construction for Coordination
                         in Games},
  editor =              {Chakraborty, Supratik and Kumar, Amit},
  booktitle =           {{P}roceedings of the 31st {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'11)},
  acronym =             {{FSTTCS}'11},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {13},
  pages =               {387-398},
  year =                {2011},
  month =               dec,
}
[BL69] Julius R. Büchi and Lawrence H. Landweber. Solving Sequential Conditions by Finite-State Strategies. Transactions of the American Mathematical Society 138:295-311. American Mathematical Society, April 1969.
@article{TAMS138-BL,
  author =              {B{\"u}chi, Julius R. and Landweber, Lawrence H.},
  title =               {Solving Sequential Conditions by Finite-State
                         Strategies},
  publisher =           {American Mathematical Society},
  journal =             {Transactions of the American Mathematical Society},
  volume =              {138},
  pages =               {295-311},
  year =                {1969},
  month =               apr,
}
[BL95] Ahmed Bouajjani and Yassine Lakhnech. Temporal Logic + Timed Automata: Expressiveness and Decidability. In CONCUR'95, Lecture Notes in Computer Science 962, pages 531-545. Springer-Verlag, August 1995.
@inproceedings{concur1995-BL,
  author =              {Bouajjani, Ahmed and Lakhnech, Yassine},
  title =               {Temporal Logic + Timed Automata: Expressiveness and
                         Decidability},
  editor =              {Lee, Insup and Smolka, Scott A.},
  booktitle =           {{P}roceedings of the 6th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'95)},
  acronym =             {{CONCUR}'95},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {962},
  pages =               {531-545},
  year =                {1995},
  month =               aug,
}
[BL12] Mikołaj Bojańczyk and Sławomir Lasota. A Machine-Independent Characterization of Timed Languages. In ICALP'12, Lecture Notes in Computer Science 7392, pages 92-103. Springer-Verlag, July 2012.
@inproceedings{icalp2012-BL,
  author =              {Boja{\'n}czyk, Miko{\l}aj and Lasota, S{\l}awomir},
  title =               {A~Machine-Independent Characterization of Timed
                         Languages},
  editor =              {Czumaj, Artur and Mehlhorn, Kurt and Pitts, Andrew
                         and Wattenhofer, Roger},
  booktitle =           {{P}roceedings of the 39th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'12)~-- Part~{II}},
  acronym =             {{ICALP}'12},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {7392},
  pages =               {92-103},
  year =                {2012},
  month =               jul,
  doi =                 {10.1007/978-3-642-31585-5_12},
}
[Bla00] Patrick Blackburn. Representations, Reasoning and Relational Structures: a Hybrid Logic Manifesto. Logic Journal of the IGPL 8(3):339-365. Oxford University Press, May 2000.
@article{jigpl8(3)-Bla,
  author =              {Blackburn, Patrick},
  title =               {Representations, Reasoning and Relational
                         Structures: a Hybrid Logic Manifesto},
  publisher =           {Oxford University Press},
  journal =             {Logic Journal of the IGPL},
  volume =              {8},
  number =              {3},
  pages =               {339-365},
  year =                {2000},
  month =               may,
}
[BLM07] Patricia Bouyer, Kim Guldstrand Larsen, and Nicolas Markey. Model Checking One-clock Priced Timed Automata. In FoSSaCS'07, Lecture Notes in Computer Science 4423, pages 108-122. Springer-Verlag, March 2007.
Abstract

We consider the model of priced (a.k.a. weighted) timed automata, an extension of timed automata with cost information on both locations and transitions. We prove that model-checking this class of models against the logic WCTL, CTL with cost-constrained modalities, is PSPACE-complete under the "single-clock" assumption. In contrast, it has been recently proved that the model-checking problem is undecidable for this model as soon as the system has three clocks. We also prove that the model-checking of WCTL becomes undecidable, even under this "single-clock" assumption.

@inproceedings{fossacs2007-BLM,
  author =              {Bouyer, Patricia and Larsen, Kim Guldstrand and
                         Markey, Nicolas},
  title =               {Model Checking One-clock Priced Timed Automata},
  editor =              {Seidl, Helmut},
  booktitle =           {{P}roceedings of the 10th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'07)},
  acronym =             {{FoSSaCS}'07},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4423},
  pages =               {108-122},
  year =                {2007},
  month =               mar,
  doi =                 {10.1007/978-3-540-71389-0_9},
  abstract =            {We consider the model of priced (a.k.a.~weighted)
                         timed automata, an extension of timed automata with
                         cost information on both locations and transitions.
                         We prove that model-checking this class of models
                         against the logic~WCTL, CTL~with cost-constrained
                         modalities, is PSPACE-complete under the
                         {"}single-clock{"} assumption. In~contrast, it~has
                         been recently proved that the model-checking problem
                         is undecidable for this model as soon as the system
                         has three clocks. We also prove that the
                         model-checking of~WCTL becomes undecidable, even
                         under this {"}single-clock{"} assumption.},
}
[BLM08] Patricia Bouyer, Kim Guldstrand Larsen, and Nicolas Markey. Model Checking One-clock Priced Timed Automata. Logical Methods in Computer Science 4(2). May 2008.
Abstract

We consider the model of priced (a.k.a. weighted) timed automata, an extension of timed automata with cost information on both locations and transitions, and we study various model-checking problems for that model based on extensions of classical temporal logics with cost constraints on modalities. We prove that, under the assumption that the model has only one clock, model-checking this class of models against the logic WCTL, CTL with cost-constrained modalities, is PSPACE-complete (while it has been shown undecidable as soon as the model has three clocks). We also prove that model checking WMTL (LTL with cost-constrained modalities) is decidable only if there is a single clock in the model and a single stopwatch cost variable (i.e., whose slopes lie in {0,1}).

@article{lmcs4(2)-BLM,
  author =              {Bouyer, Patricia and Larsen, Kim Guldstrand and
                         Markey, Nicolas},
  title =               {Model Checking One-clock Priced Timed Automata},
  journal =             {Logical Methods in Computer Science},
  volume =              {4},
  number =              {2},
  year =                {2008},
  month =               may,
  doi =                 {10.2168/LMCS-4(2:9)2008},
  abstract =            {We consider the model of priced (a.k.a.~weighted)
                         timed automata, an extension of timed automata with
                         cost information on both locations and transitions,
                         and we study various model-checking problems for
                         that model based on extensions of classical temporal
                         logics with cost constraints on modalities. We prove
                         that, under the assumption that the model has only
                         one clock, model-checking this class of models
                         against the logic~WCTL, CTL with cost-constrained
                         modalities, is PSPACE-complete (while it has been
                         shown undecidable as soon as the model has three
                         clocks). We~also prove that model checking WMTL (LTL
                         with cost-constrained modalities) is decidable only
                         if there is a single clock in the model and a single
                         stopwatch cost variable (\textit{i.e.}, whose slopes
                         lie in~\(\{0,1\}\)).},
}
[BLM12] Patricia Bouyer, Kim Guldstrand Larsen, and Nicolas Markey. Lower-Bound Constrained Runs in Weighted Timed Automata. In QEST'12, pages 128-137. IEEE Comp. Soc. Press, September 2012.
Abstract

We investigate a number of problems related to infinite runs of weighted timed automata, subject to lower-bound constraints on the accumulated weight. Closing an open problem from [Bouyer et al., "Infinite runs in weighted timed automata with energy constraints", FORMATS'08], we show that the existence of an infinite lower-bound-constrained run is–-mdash;for us somewhat unexpectedly–-mdash;undecidable for weighted timed automata with four or more clocks.

This undecidability result assumes a fixed and know initial credit. We show that the related problem of existence of an initial credit for which there ex- ist a feasible run is decidable in PSPACE. We also investigate the variant of these problems where only bounded-duration runs are considered, showing that this restriction makes our original problem decidable in NEXPTIME. Finally, we prove that the universal versions of all those problems (i.e, checking that all the considered runs satisfy the lower-bound constraint) are decidable in PSPACE.

@inproceedings{qest2012-BLM,
  author =              {Bouyer, Patricia and Larsen, Kim Guldstrand and
                         Markey, Nicolas},
  title =               {Lower-Bound Constrained Runs in Weighted Timed
                         Automata},
  booktitle =           {{P}roceedings of the 9th {I}nternational
                         {C}onference on {Q}uantitative {E}valuation of
                         {S}ystems ({QEST}'12)},
  acronym =             {{QEST}'12},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {128-137},
  year =                {2012},
  month =               sep,
  doi =                 {10.1109/QEST.2012.28},
  abstract =            {We investigate a number of problems related to
                         infinite runs of weighted timed automata, subject to
                         lower-bound constraints on the accumulated weight.
                         Closing an open problem from [Bouyer
                         \textit{et~al.}, {"}Infinite runs in weighted timed
                         automata with energy constraints{"}, FORMATS'08], we
                         show that the existence of an infinite
                         lower-bound-constrained run is---for us somewhat
                         unexpectedly---undecidable for weighted timed
                         automata with four or more clocks.\par This
                         undecidability result assumes a fixed and know
                         initial credit. We show that the related problem of
                         existence of an initial credit for which there ex-
                         ist a feasible run is decidable in PSPACE. We also
                         investigate the variant of these problems where only
                         bounded-duration runs are considered, showing that
                         this restriction makes our original problem
                         decidable in NEXPTIME. Finally, we prove that the
                         universal versions of all those problems (i.e,
                         checking that all the considered runs satisfy the
                         lower-bound constraint) are decidable in PSPACE.},
}
[BLM14] Patricia Bouyer, Kim Guldstrand Larsen, and Nicolas Markey. Lower-Bound Constrained Runs in Weighted Timed Automata. Performance Evaluation 73:91-109. Elsevier, March 2014.
Abstract

We investigate a number of problems related to infinite runs of weighted timed automata (with a single weight variable), subject to lower-bound constraints on the accumulated weight. Closing an open problem from an earlier paper, we show that the existence of an infinite lower-bound-constrained run is–for us somewhat unexpectedly–undecidable for weighted timed automata with four or more clocks.

This undecidability result assumes a fixed and known initial credit. We show that the related problem of existence of an initial credit for which there exists a feasible run is decidable in PSPACE. We also investigate the variant of these problems where only bounded-duration runs are considered, showing that this restriction makes our original problem decidable in NEXPTIME. We prove that the universal versions of all those problems (i.e, checking that all the considered runs satisfy the lower-bound constraint) are decidable in PSPACE.

Finally, we extend this study to multi-weighted timed automata: the existence of a feasible run becomes undecidable even for bounded duration, but the existence of initial credits remains decidable (in PSPACE).

@article{peva73()-BLM,
  author =              {Bouyer, Patricia and Larsen, Kim Guldstrand and
                         Markey, Nicolas},
  title =               {Lower-Bound Constrained Runs in Weighted Timed
                         Automata},
  publisher =           {Elsevier},
  journal =             {Performance Evaluation},
  volume =              {73},
  pages =               {91-109},
  year =                {2014},
  month =               mar,
  doi =                 {10.1016/j.peva.2013.11.002},
  abstract =            {We investigate a number of problems related to
                         infinite runs of weighted timed automata (with a
                         single weight variable), subject to lower-bound
                         constraints on the accumulated weight. Closing an
                         open problem from an earlier paper, we show that the
                         existence of an infinite lower-bound-constrained run
                         is--for us somewhat unexpectedly--undecidable for
                         weighted timed automata with four or more
                         clocks.\par This undecidability result assumes a
                         fixed and known initial credit. We show that the
                         related problem of existence of an initial credit
                         for which there exists a feasible run is decidable
                         in PSPACE. We also investigate the variant of these
                         problems where only bounded-duration runs are
                         considered, showing that this restriction makes our
                         original problem decidable in NEXPTIME. We prove
                         that the universal versions of all those problems
                         (i.e, checking that all the considered runs satisfy
                         the lower-bound constraint) are decidable in
                         PSPACE.\par Finally, we extend this study to
                         multi-weighted timed automata: the existence of a
                         feasible run becomes undecidable even for bounded
                         duration, but the existence of initial credits
                         remains decidable (in~PSPACE).},
}
[BLM+07] Thomas Brihaye, François Laroussinie, Nicolas Markey, and Ghassan Oreiby. Timed Concurrent Game Structures. In CONCUR'07, Lecture Notes in Computer Science 4703, pages 445-459. Springer-Verlag, September 2007.
Abstract

We propose a new model for timed games, based on concurrent game structures (CGSs). Compared to the classical timed game automata of Asarin et al., our timed CGSs are "more concurrent", in the sense that they always allow all the agents to act on the system, independently of the delay they want to elapse before their action. Timed CGSs weaken the "element of surprise" of timed game automata reported by de Alfaro et al.

We prove that our model has nice properties, in particular that model-checking timed CGSs against timed ATL is decidable via region abstraction, and in particular that strategies are "region-stable" if winning objectives are. We also propose a new extension of TATL, containing ATL*, which we call TALTL. We prove that model-checking this logic remains decidable on timed CGSs. Last, we explain how our algorithms can be adapted in order to rule out Zeno (co-)strategies, based on the ideas of Henzinger et al.

@inproceedings{concur2007-BLMO,
  author =              {Brihaye, {\relax Th}omas and Laroussinie, Fran{\c
                         c}ois and Markey, Nicolas and Oreiby, Ghassan},
  title =               {Timed Concurrent Game Structures},
  editor =              {Caires, Lu{\'\i}s and Vasconcelos, Vasco T.},
  booktitle =           {{P}roceedings of the 18th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'07)},
  acronym =             {{CONCUR}'07},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4703},
  pages =               {445-459},
  year =                {2007},
  month =               sep,
  doi =                 {10.1007/978-3-540-74407-8_30},
  abstract =            {We propose a new model for timed games, based on
                         concurrent game structures~(CGSs). Compared to the
                         classical \emph{timed game automata} of~Asarin
                         \emph{et~al.}, our timed~CGSs are {"}more
                         concurrent{"}, in the sense that they always allow
                         all the agents to act on the system, independently
                         of the delay they want to elapse before their
                         action. Timed CGSs weaken the {"}element of
                         surprise{"} of timed game automata reported by
                         de~Alfaro \emph{et~al.}\par We prove that our model
                         has nice properties, in particular that
                         model-checking timed CGSs against timed
                         \(\textsf{ATL}\) is decidable \emph{via} region
                         abstraction, and in particular that strategies are
                         {"}region-stable{"} if winning objectives are. We
                         also propose a new extension of \(\textsf{TATL}\),
                         containing~\(\textsf{ATL}^{*}\), which we
                         call~\(\textsf{TALTL}\). We~prove that
                         model-checking this logic remains decidable on timed
                         CGSs. Last, we explain how our algorithms can be
                         adapted in order to rule out Zeno (co-)strategies,
                         based on the ideas of Henzinger \emph{et~al.}},
}
[BLM+17] Patricia Bouyer, François Laroussinie, Nicolas Markey, Joël Ouaknine, and James Worrell. Timed temporal logics. In Luca Aceto, Giorgio Bacci, Giovanni Bacci, Anna Ingólfsdóttir, Axel Legay, and Radu Mardare (eds.), Models, Algorithms, Logics and Tools: Essays Dedicated to Kim Guldstrand Larsen on the Occasion of His 60th Birthday, Lecture Notes in Computer Science 10460, pages 211-230. Springer-Verlag, August 2017.
Abstract

Since the early 1990's, classical temporal logics have been extended with timing constraints. While temporal logics only express contraints on the order of events, their timed extensions can add quantitative constraints on delays between those events. We survey expressiveness and algorithmic results on those logics, and discuss semantic choices that may look unimportant but do have an impact on the questions we consider.

@incollection{kimfest2017-BLMOW,
  author =              {Bouyer, Patricia and Laroussinie, Fran{\c c}ois and
                         Markey, Nicolas and Ouaknine, Jo{\"e}l and Worrell,
                         James},
  title =               {Timed temporal logics},
  editor =              {Aceto, Luca and Bacci, Giorgio and Bacci, Giovanni
                         and Ing{\'o}lfsd{\'o}ttir, Anna and Legay, Axel and
                         Mardare, Radu},
  booktitle =           {Models, Algorithms, Logics and Tools: Essays
                         Dedicated to Kim Guldstrand Larsen on the Occasion
                         of His 60th Birthday},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {10460},
  pages =               {211-230},
  year =                {2017},
  month =               aug,
  doi =                 {10.1007/978-3-319-65764-6_11},
  abstract =            {Since the early 1990's, classical temporal logics
                         have been extended with timing constraints. While
                         temporal logics only express contraints on the order
                         of events, their timed extensions can add
                         quantitative constraints on delays between those
                         events. We survey expressiveness and algorithmic
                         results on those logics, and discuss semantic
                         choices that may look unimportant but do have an
                         impact on the questions we consider.},
}
[BLM+06] Patricia Bouyer, Kim Guldstrand Larsen, Nicolas Markey, and Jacob Illum Rasmussen. Almost Optimal Strategies in One-Clock Priced Timed Automata. In FSTTCS'06, Lecture Notes in Computer Science 4337, pages 345-356. Springer-Verlag, December 2006.
Abstract

We consider timed games extended with cost information, and prove computability of the optimal cost and of ε-optimal memoryless strategies in timed games with one clock. In contrast, this problem has recently been proved undecidable for timed games with three clocks.

@inproceedings{fsttcs2006-BLMR,
  author =              {Bouyer, Patricia and Larsen, Kim Guldstrand and
                         Markey, Nicolas and Rasmussen, Jacob Illum},
  title =               {Almost Optimal Strategies in One-Clock Priced Timed
                         Automata},
  editor =              {Arun-Kumar, S. and Garg, Naveen},
  booktitle =           {{P}roceedings of the 26th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'06)},
  acronym =             {{FSTTCS}'06},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4337},
  pages =               {345-356},
  year =                {2006},
  month =               dec,
  doi =                 {10.1007/11944836_32},
  abstract =            {We consider timed games extended with cost
                         information, and prove computability of the optimal
                         cost and of \(\epsilon\)-optimal memoryless
                         strategies in timed games with one~clock.
                         In~contrast, this problem has recently been proved
                         undecidable for timed games with three clocks.},
}
[BLM+97] David A. Mix Barrington, Chi-Jen Lu, Peter Bro Miltersen, and Sven Skyum. On Uniformity Within NC1. Research Report 97-044, Electronic Colloquium on Computational Complexity, September 1997.
@techreport{eccc1997-BLMS,
  author =              {Barrington, David A. Mix and Lu, Chi-Jen and
                         Miltersen, Peter Bro and Skyum, Sven},
  title =               {On Uniformity Within {{\(\mathit{NC}^1\)}}},
  number =              {97-044},
  year =                {1997},
  month =               sep,
  institution =         {Electronic Colloquium on Computational Complexity},
  type =                {Research Report},
}
[BLM+11] Patricia Bouyer, Kim Guldstrand Larsen, Nicolas Markey, Ocan Sankur, and Claus Thrane. Timed automata can always be made implementable. In CONCUR'11, Lecture Notes in Computer Science 6901, pages 76-91. Springer-Verlag, September 2011.
Abstract

Timed automata follow a mathematical semantics, which assumes perfect precision and synchrony of clocks. Since this hypothesis does not hold in digital systems, properties proven formally on a timed automaton may be lost at implementation. In order to ensure implementability, several approaches have been considered, corresponding to different hypotheses on the implementation platform. We address two of these: a timed automaton is samplable if its semantics is preserved under a discretization of time; it is robust if its semantics is preserved when all timing constraints are relaxed by some small positive parameter. We propose a construction which makes timed automata implementable in the above sense: From any timed automaton A, we build a timed automaton A' that exhibits the same behaviour as A, and moreover is both robust and samplable by construction.

@inproceedings{concur2011-BLMST,
  author =              {Bouyer, Patricia and Larsen, Kim Guldstrand and
                         Markey, Nicolas and Sankur, Ocan and Thrane, Claus},
  title =               {Timed automata can always be made implementable},
  editor =              {Katoen, Joost-Pieter and K{\"o}nig, Barbara},
  booktitle =           {{P}roceedings of the 22nd {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'11)},
  acronym =             {{CONCUR}'11},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6901},
  pages =               {76-91},
  year =                {2011},
  month =               sep,
  doi =                 {10.1007/978-3-642-23217-6_6},
  abstract =            {Timed automata follow a mathematical semantics,
                         which assumes perfect precision and synchrony of
                         clocks. Since this hypothesis does not hold in
                         digital systems, properties proven formally on a
                         timed automaton may be lost at implementation. In
                         order to ensure implementability, several approaches
                         have been considered, corresponding to different
                         hypotheses on the implementation platform. We
                         address two of these: a~timed automaton is samplable
                         if its semantics is preserved under a discretization
                         of time; it is robust if its semantics is preserved
                         when all timing constraints are relaxed by some
                         small positive parameter. We propose a construction
                         which makes timed automata implementable in the
                         above sense: From any timed
                         automaton~\(\mathcal{A}\), we build a timed
                         automaton~\(\mathcal{A}'\) that exhibits the same
                         behaviour as~\(\mathcal{A}\), and moreover is both
                         robust and samplable by construction.},
}
[BLN03] Dirk Beyer, Claus Lewerentz, and Andreas Noack. Rabbit: A Tool for BDD-Based Verification of Real-Time Systems. In CAV'03, Lecture Notes in Computer Science 2725, pages 122-125. Springer-Verlag, July 2003.
@inproceedings{cav2003-BLN,
  author =              {Beyer, Dirk and Lewerentz, Claus and Noack, Andreas},
  title =               {Rabbit: A~Tool for {BDD}-Based Verification of
                         Real-Time Systems},
  editor =              {Hunt, Jr, Warren A. and Somenzi, Fabio},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'03)},
  acronym =             {{CAV}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2725},
  pages =               {122-125},
  year =                {2003},
  month =               jul,
  doi =                 {10.1007/978-3-540-45069-6_13},
}
[Blo93] Johannes Blömer. Computing Sums of Radicals in Polynomial Time. Technical Report B93-13, Department of Computer Science, University of Paderborn, Germany, August 1993.
@techreport{UPad-B93-13-Blo,
  author =              {Bl{\"o}mer, Johannes},
  title =               {Computing Sums of Radicals in Polynomial Time},
  number =              {B93-13},
  year =                {1993},
  month =               aug,
  institution =         {Department of Computer Science, University of
                         Paderborn, Germany},
}
[BLP03] Gerd Behrmann, Kim Guldstrand Larsen, and Radek Pelánek. To Store or not to Store. In CAV'03, Lecture Notes in Computer Science 2725, pages 433-445. Springer-Verlag, July 2003.
@inproceedings{cav2003-BLP,
  author =              {Behrmann, Gerd and Larsen, Kim Guldstrand and
                         Pel{\'a}nek, Radek},
  title =               {To Store or not to Store},
  editor =              {Hunt, Jr, Warren A. and Somenzi, Fabio},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'03)},
  acronym =             {{CAV}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2725},
  pages =               {433-445},
  year =                {2003},
  month =               jul,
}
[BLP+98] Gerd Behrmann, Kim Guldstrand Larsen, Justin Pearson, Carsten Weise, and Wang Yi. Efficient Timed Reachability Analysis using Clock Difference Diagrams. Research Report RS-98-47, Basic Research in Computer Science, Aalborg University, Denmark, December 1998.
@techreport{TR-brics9847,
  author =              {Behrmann, Gerd and Larsen, Kim Guldstrand and
                         Pearson, Justin and Weise, Carsten and Yi, Wang},
  title =               {Efficient Timed Reachability Analysis using Clock
                         Difference Diagrams},
  number =              {RS-98-47},
  year =                {1998},
  month =               dec,
  institution =         {Basic Research in Computer Science, Aalborg
                         University, Denmark},
  type =                {Research Report},
}
[BLP+99] Gerd Behrmann, Kim Guldstrand Larsen, Justin Pearson, Carsten Weise, and Wang Yi. Efficient Timed Reachability Analysis using Clock Difference Diagrams. In CAV'99, Lecture Notes in Computer Science 1633, pages 341-353. Springer-Verlag, July 1999.
@inproceedings{cav1999-BLPWY,
  author =              {Behrmann, Gerd and Larsen, Kim Guldstrand and
                         Pearson, Justin and Weise, Carsten and Yi, Wang},
  title =               {Efficient Timed Reachability Analysis using Clock
                         Difference Diagrams},
  editor =              {Halbwachs, Nicolas and Peled, Doron A.},
  booktitle =           {{P}roceedings of the 11th {I}nternational
                         {C}onference on {C}omputer {A}ided {V}erification
                         ({CAV}'99)},
  acronym =             {{CAV}'99},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1633},
  pages =               {341-353},
  year =                {1999},
  month =               jul,
}
[BLR05] Patricia Bouyer, François Laroussinie, and Pierre-Alain Reynier. Diagonal Constraints in Timed Automata: Forward Analysis of Timed Systems. In FORMATS'05, Lecture Notes in Computer Science 3829, pages 112-126. Springer-Verlag, September 2005.
@inproceedings{formats2005-BLR,
  author =              {Bouyer, Patricia and Laroussinie, Fran{\c c}ois and
                         Reynier, Pierre-Alain},
  title =               {Diagonal Constraints in Timed Automata: Forward
                         Analysis of Timed Systems},
  editor =              {Pettersson, Paul and Yi, Wang},
  booktitle =           {{P}roceedings of the 3rd {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'05)},
  acronym =             {{FORMATS}'05},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3829},
  pages =               {112-126},
  year =                {2005},
  month =               sep,
  doi =                 {10.1007/11603009_10},
}
[BLR05] Gerd Behrmann, Kim Guldstrand Larsen, and Jacob Illum Rasmussen. Optimal scheduling using priced timed automata. SIGMETRICS Performance Evaluation Review 32(4):34-40. March 2005.
@article{SIGper32(4)-BLR,
  author =              {Behrmann, Gerd and Larsen, Kim Guldstrand and
                         Rasmussen, Jacob Illum},
  title =               {Optimal scheduling using priced timed automata},
  journal =             {SIGMETRICS Performance Evaluation Review},
  volume =              {32},
  number =              {4},
  pages =               {34-40},
  year =                {2005},
  month =               mar,
}
[BLW01] Benedikt Bollig, Martin Leucker, and Michael Weber. Local Parallel Model Checking for the Alternation Free μ-Calculus. Technical Report AIB-2001-04, Aachener Informatik Berichte, Aachen, Germany, March 2001.
@techreport{TR-Aachen0104,
  author =              {Bollig, Benedikt and Leucker, Martin and Weber,
                         Michael},
  title =               {Local Parallel Model Checking for the Alternation
                         Free {\(\mu\)}-Calculus},
  number =              {AIB-2001-04},
  year =                {2001},
  month =               mar,
  institution =         {Aachener Informatik Berichte, Aachen, Germany},
  type =                {Technical Report},
}
[BM83] Bernard Berthomieu and Miguel Menasche. An Enumerative Approach for Analyzing Time Petri Nets. In WCC'83, pages 41-46. North-Holland/IFIP, September 1983.
@inproceedings{ifipwcc1983-BM,
  author =              {Berthomieu, Bernard and Menasche, Miguel},
  title =               {An Enumerative Approach for Analyzing Time {P}etri
                         Nets},
  editor =              {Mason, R. E. A.},
  booktitle =           {{I}nformation {P}rocessing~83~-- {P}roceedings of
                         the 9th {IFIP} {W}orld {C}omputer {C}ongress
                         ({WCC}'83)},
  acronym =             {{WCC}'83},
  publisher =           {North-Holland/IFIP},
  pages =               {41-46},
  year =                {1983},
  month =               sep,
}
[BM95] Michael S. Branicky and Sanjoy K. Mitter. Algorithms for Optimal Hybrid Control. In CDC'95, pages 2661-2666. IEEE Comp. Soc. Press, December 1995.
@inproceedings{cdc1995-BM,
  author =              {Branicky, Michael S. and Mitter, Sanjoy K.},
  title =               {Algorithms for Optimal Hybrid Control},
  booktitle =           {{P}roceedings of the 34th {IEEE} {C}onference on
                         {D}ecision and {C}ontrol ({CDC}'95)},
  acronym =             {{CDC}'95},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {2661-2666},
  year =                {1995},
  month =               dec,
}
[BM00] Ed Brinksma and Angelika Mader. Verification and Optimization of a PLC Control Schedule. In SPIN'00, Lecture Notes in Computer Science 1885, pages 73-92. Springer-Verlag, August 2000.
@inproceedings{spin2000-BM,
  author =              {Brinksma, Ed and Mader, Angelika},
  title =               {Verification and Optimization of a {PLC} Control
                         Schedule},
  editor =              {Havelund, Klaus and Penix, John and Viser, Willem},
  booktitle =           {{P}roceedings of the 7th {I}nternational {SPIN}
                         {W}orkshop ({SPIN}'00)},
  acronym =             {{SPIN}'00},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {1885},
  pages =               {73-92},
  year =                {2000},
  month =               aug,
}
[BM07] Patricia Bouyer and Nicolas Markey. Costs are Expensive!. In FORMATS'07, Lecture Notes in Computer Science 4763, pages 53-68. Springer-Verlag, October 2007.
Abstract

We study the model-checking problem for WMTL, a cost-extension of the linear-time timed temporal logic MTL, that is interpreted over weighted timed automata. We draw a complete picture of the decidability for that problem: it is decidable only for the class of one-clock weighted timed automata with a restricted stopwatch cost, and any slight extension of this model leads to undecidability. We finally give some consequences on the undecidability of linear hybrid automata.

@inproceedings{formats2007-BM,
  author =              {Bouyer, Patricia and Markey, Nicolas},
  title =               {Costs are Expensive!},
  editor =              {Raskin, Jean-Fran{\c c}ois and Thiagarajan, P. S.},
  booktitle =           {{P}roceedings of the 5th {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'07)},
  acronym =             {{FORMATS}'07},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4763},
  pages =               {53-68},
  year =                {2007},
  month =               oct,
  doi =                 {10.1007/978-3-540-75454-1_6},
  abstract =            {We study the model-checking problem for WMTL,
                         a~cost-extension of the linear-time timed temporal
                         logic MTL, that is interpreted over weighted timed
                         automata. We~draw a complete picture of the
                         decidability for that problem: it~is decidable only
                         for the class of one-clock weighted timed automata
                         with a restricted stopwatch cost, and any slight
                         extension of this model leads to undecidability.
                         We~finally give some consequences on the
                         undecidability of linear hybrid automata.},
}
[BMF02] Ed Brinksma, Angelika Mader, and Ansgar Fehnker. Verification and Optimization of a PLC Control Schedule. International Journal on Software Tools for Technology Transfer 4(1):21-33. Springer-Verlag, 2002.
@article{sttt4(1)-BMF,
  author =              {Brinksma, Ed and Mader, Angelika and Fehnker,
                         Ansgar},
  title =               {Verification and Optimization of a {PLC} Control
                         Schedule},
  publisher =           {Springer-Verlag},
  journal =             {International Journal on Software Tools for
                         Technology Transfer},
  volume =              {4},
  number =              {1},
  pages =               {21-33},
  year =                {2002},
}
[BMM14] Patricia Bouyer, Nicolas Markey, and Raj Mohan Matteplackel. Averaging in LTL. In CONCUR'14, Lecture Notes in Computer Science 8704, pages 266-280. Springer-Verlag, September 2014.
Abstract

For the accurate analysis of computerized systems, powerful quantitative formalisms have been designed, together with efficient verification algorithms. However, verification has mostly remained boolean–-mdash;either a property is true, or it is false. We believe that this is too crude in a context where quantitative information and constraints are crucial: correctness should be quantified!

In a recent line of works, several authors have proposed quantitative semantics for temporal logics, using e.g. discounting modalities (which give less importance to distant events). In the present paper, we define and study a quantitative semantics of LTL with averaging modalities, either on the long run or within an until modality. This, in a way, relaxes the classical Boolean semantics of LTL, and provides a measure of certain properties of a model. We prove that computing and even approximating the value of a formula in this logic is undecidable.

@inproceedings{concur2014-BMM,
  author =              {Bouyer, Patricia and Markey, Nicolas and
                         Matteplackel, Raj~Mohan},
  title =               {Averaging in~{LTL}},
  editor =              {Baldan, Paolo and Gorla, Daniele},
  booktitle =           {{P}roceedings of the 25th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'14)},
  acronym =             {{CONCUR}'14},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8704},
  pages =               {266-280},
  year =                {2014},
  month =               sep,
  doi =                 {10.1007/978-3-662-44584-6_19},
  abstract =            {For the accurate analysis of computerized systems,
                         powerful quantitative formalisms have been designed,
                         together with efficient verification algorithms.
                         However, verification has mostly remained
                         boolean---either a property is~true, or it~is false.
                         We~believe that this is too crude in a context where
                         quantitative information and constraints are
                         crucial: correctness should be quantified!\par In a
                         recent line of works, several authors have proposed
                         quantitative semantics for temporal logics, using
                         e.g. \emph{discounting} modalities (which give less
                         importance to distant events). In~the present paper,
                         we define and study a quantitative semantics of~LTL
                         with \emph{averaging} modalities, either on the long
                         run or within an until modality. This, in a way,
                         relaxes the classical Boolean semantics of~LTL, and
                         provides a measure of certain properties of a model.
                         We~prove that computing and even approximating the
                         value of a formula in this logic is undecidable.},
}
[BMM+00] Jean-Camille Birget, Stuart W. Margolis, John C. Meakin, and Pascal Weil. PSPACE-Completeness of Certain Algorithmic Problems on the Subgroups of Free Groups. Theoretical Computer Science 242(1-2):247-281. Elsevier, July 2000.
@article{tcs242(1-2)-BMMW,
  author =              {Birget, Jean-Camille and Margolis, Stuart W. and
                         Meakin, John C. and Weil, Pascal},
  title =               {{PSPACE}-Completeness of Certain Algorithmic
                         Problems on the Subgroups of Free Groups},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {242},
  number =              {1-2},
  pages =               {247-281},
  year =                {2000},
  month =               jul,
}
[BMO+08] Patricia Bouyer, Nicolas Markey, Joël Ouaknine, Philippe Schnoebelen, and James Worrell. On Termination for Faulty Channel Machines. In STACS'08, Leibniz International Proceedings in Informatics 1, pages 121-132. Leibniz-Zentrum für Informatik, February 2008.
Abstract

A channel machine consists of a finite controller together with several fifo channels; the controller can read messages from the head of a channel and write messages to the tail of a channel. In this paper, we focus on channel machines with insertion errors, i.e., machines in whose channels messages can spontaneously appear. Such devices have been previously introduced in the study of Metric Temporal Logic. We consider the termination problem: are all the computations of a given insertion channel machine finite? We show that this problem has non-elementary, yet primitive recursive complexity.

@inproceedings{stacs2008-BMOSW,
  author =              {Bouyer, Patricia and Markey, Nicolas and Ouaknine,
                         Jo{\"e}l and Schnoebelen, {\relax Ph}ilippe and
                         Worrell, James},
  title =               {On Termination for Faulty Channel Machines},
  editor =              {Albers, Susanne and Weil, Pascal},
  booktitle =           {{P}roceedings of the 25th {S}ymposium on
                         {T}heoretical {A}spects of {C}omputer {S}cience
                         ({STACS}'08)},
  acronym =             {{STACS}'08},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {1},
  pages =               {121-132},
  year =                {2008},
  month =               feb,
  doi =                 {10.4230/LIPIcs.STACS.2008.1339},
  abstract =            {A channel machine consists of a finite controller
                         together with several fifo channels; the controller
                         can read messages from the head of a channel and
                         write messages to the tail of a channel. In this
                         paper, we focus on channel machines with
                         \emph{insertion errors}, \textit{i.e.}, machines in
                         whose channels messages can spontaneously appear.
                         Such devices have been previously introduced in the
                         study of Metric Temporal Logic. We~consider the
                         termination problem: are all the computations of a
                         given insertion channel machine finite? We~show that
                         this problem has non-elementary, yet primitive
                         recursive complexity.},
}
[BMO+12] Patricia Bouyer, Nicolas Markey, Joël Ouaknine, Philippe Schnoebelen, and James Worrell. On Termination and Invariance for Faulty Channel Systems. Formal Aspects of Computing 24(4-6):595-607. Springer-Verlag, July 2012.
Abstract

channel machine consists of a finite controller together with several fifo channels; the controller can read messages from the head of a channel and write messages to the tail of a channel. In this paper we focus on channel machines with insertion errors, i.e., machines in whose channels messages can spontaneously appear. We consider the invariance problem: does a given insertion channel machine have an infinite computation all of whose configurations satisfy a given predicate? We show that this problem is primitive-recursive if the predicate is closed under message losses. We also give a non-elementary lower bound for the invariance problem under this restriction. Finally, using the previous result, we show that the satisfiability problem for the safety fragment of Metric Temporal Logic is non-elementary.

@article{fac24(4-6)-BMOSW,
  author =              {Bouyer, Patricia and Markey, Nicolas and Ouaknine,
                         Jo{\"e}l and Schnoebelen, {\relax Ph}ilippe and
                         Worrell, James},
  title =               {On Termination and Invariance for Faulty Channel
                         Systems},
  publisher =           {Springer-Verlag},
  journal =             {Formal Aspects of Computing},
  volume =              {24},
  number =              {4-6},
  pages =               {595-607},
  year =                {2012},
  month =               jul,
  doi =                 {10.1007/s00165-012-0234-7},
  abstract =            {A~\emph{channel machine} consists of a finite
                         controller together with several fifo channels; the
                         controller can read messages from the head of a
                         channel and write messages to the tail of a channel.
                         In this paper we focus on channel machines with
                         \emph{insertion errors}, i.e., machines in whose
                         channels messages can spontaneously appear. We
                         consider the invariance problem: does a given
                         insertion channel machine have an infinite
                         computation all of whose configurations satisfy a
                         given predicate? We show that this problem is
                         primitive-recursive if the predicate is closed under
                         message losses. We also give a non-elementary lower
                         bound for the invariance problem under this
                         restriction. Finally, using the previous result, we
                         show that the satisfiability problem for the safety
                         fragment of Metric Temporal Logic is
                         non-elementary.},
}
[BMO+11] Patricia Bouyer, Nicolas Markey, Jörg Olschewski, and Michael Ummels. Measuring Permissiveness in Parity Games: Mean-Payoff Parity Games Revisited. In ATVA'11, Lecture Notes in Computer Science 6996, pages 135-149. Springer-Verlag, October 2011.
Abstract

We study nondeterministic strategies in parity games with the aim of computing a most permissive winning strategy. Following earlier work, we measure permissiveness in terms of the average number/weight of transitions blocked by a strategy. Using a translation into mean-payoff parity games, we prove that deciding (the permissiveness of) a most permissive winning strategy is in NP∩coNP. Along the way, we provide a new study of mean-payoff parity games. In particular, we give a new algorithm for solving these games, which beats all previously known algorithms for this problem.

@inproceedings{atva2011-BMOU,
  author =              {Bouyer, Patricia and Markey, Nicolas and Olschewski,
                         J{\"o}rg and Ummels, Michael},
  title =               {Measuring Permissiveness in Parity Games:
                         Mean-Payoff Parity Games Revisited},
  editor =              {Bultan, Tevfik and Hsiung, Pao-Ann},
  booktitle =           {{P}roceedings of the 9th {I}nternational {S}ymposium
                         on {A}utomated {T}echnology for {V}erification and
                         {A}nalysis ({ATVA}'11)},
  acronym =             {{ATVA}'11},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6996},
  pages =               {135-149},
  year =                {2011},
  month =               oct,
  doi =                 {10.1007/978-3-642-24372-1_11},
  abstract =            {We study nondeterministic strategies in parity games
                         with the aim of computing a most permissive winning
                         strategy. Following earlier work, we measure
                         permissiveness in terms of the average
                         number{\slash}weight of transitions blocked by a
                         strategy. Using a translation into mean-payoff
                         parity games, we prove that deciding (the
                         permissiveness~of) a~most permissive winning
                         strategy is in \(\textsf{NP}\cap\textsf{coNP}\).
                         Along the way, we~provide a new study of mean-payoff
                         parity games. In particular, we give a new algorithm
                         for solving these games, which beats all previously
                         known algorithms for this problem.},
}
[BMO+07] Patricia Bouyer, Nicolas Markey, Joël Ouaknine, and James Worrell. The Cost of Punctuality. In LICS'07, pages 109-118. IEEE Comp. Soc. Press, July 2007.
Abstract

In an influential paper titled "The Benefits of Relaxing Punctuality", Alur, Feder, and Henzinger introduced Metric Interval Temporal Logic (MITL) as a fragment of the real-time logic Metric Temporal Logic (MTL) in which exact or punctual timing constraints are banned. Their main result showed that model checking and satisfiability for MITL are both EXPSPACE-Complete.

Until recently, it was widely believed that admitting even the simplest punctual specifications in any linear-time temporal logic would automatically lead to undecidability. Although this was recently disproved, until now no punctual fragment of MTL was known to have even primitive recursive complexity (with certain decidable fragments having provably non-primitive recursive complexity).

In this paper we identify a `co-flat' subset of MTL that is capable of expressing a large class of punctual specifications and for which model checking (although not satisfiability) has no complexity cost over MITL. Our logic is moreover qualitatively different from MITL in that it can express properties that are not timed-regular. Correspondingly, our decision procedures do not involve translating formulas into finite-state automata, but rather into certain kinds of reversal-bounded Turing machines. Using this translation we show that the model checking problem for our logic is EXPSPACE-Complete, and is even PSPACE-Complete if timing constraints are encoded in unary.

@inproceedings{lics2007-BMOW,
  author =              {Bouyer, Patricia and Markey, Nicolas and Ouaknine,
                         Jo{\"e}l and Worrell, James},
  title =               {The Cost of Punctuality},
  booktitle =           {{P}roceedings of the 22nd {A}nnual {S}ymposium on
                         {L}ogic in {C}omputer {S}cience ({LICS}'07)},
  acronym =             {{LICS}'07},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {109-118},
  year =                {2007},
  month =               jul,
  doi =                 {10.1109/LICS.2007.49},
  abstract =            {In an influential paper titled {"}The Benefits of
                         Relaxing Punctuality{"}, Alur, Feder, and~Henzinger
                         introduced Metric Interval Temporal Logic~(MITL) as
                         a fragment of the real-time logic Metric Temporal
                         Logic~(MTL) in which exact or punctual timing
                         constraints are banned. Their main result showed
                         that model checking and satisfiability for~MITL are
                         both EXPSPACE-Complete.\par Until recently, it was
                         widely believed that admitting even the simplest
                         punctual specifications in any linear-time temporal
                         logic would automatically lead to undecidability.
                         Although this was recently disproved, until now no
                         punctual fragment of~MTL was known to have even
                         primitive recursive complexity (with certain
                         decidable fragments having provably non-primitive
                         recursive complexity).\par In this paper we identify
                         a `co-flat' subset of~MTL that is capable of
                         expressing a large class of punctual specifications
                         and for which model checking (although not
                         satisfiability) has no complexity cost over~MITL.
                         Our logic is moreover qualitatively different
                         from~MITL in that it can express properties that are
                         not timed-regular. Correspondingly, our decision
                         procedures do not involve translating formulas into
                         finite-state automata, but rather into certain kinds
                         of reversal-bounded Turing machines. Using this
                         translation we show that the model checking problem
                         for our logic is EXPSPACE-Complete, and is even
                         PSPACE-Complete if timing constraints are encoded in
                         unary.},
}
[BMO+08] Patricia Bouyer, Nicolas Markey, Joël Ouaknine, and James Worrell. On Expressiveness and Complexity in Real-time Model Checking. In ICALP'08, Lecture Notes in Computer Science 5126, pages 124-135. Springer-Verlag, July 2008.
Abstract

Metric Interval Temporal Logic (MITL) is a popular formalism for expressing real-time specifications. This logic achieves decidability by restricting the precision of timing constraints, in particular, by banning so-called punctual specifications. In this paper we introduce a significantly more expressive logic that can express a wide variety of punctual specifications, but whose model-checking problem has the same complexity as that of MITL. We conclude that for model checking the most commonly occurring specifications, such as invariance and bounded response, punctuality can be accommodated at no cost.

@inproceedings{icalp2008-BMOW,
  author =              {Bouyer, Patricia and Markey, Nicolas and Ouaknine,
                         Jo{\"e}l and Worrell, James},
  title =               {On Expressiveness and Complexity in Real-time Model
                         Checking},
  editor =              {Aceto, Luca and Damg{\aa}rd, Ivan and Goldberg,
                         Leslie Ann and Halld{\'o}rsson, Magn{\'u}s M. and
                         Ing{\'o}lfsd{\'o}ttir, Anna and Walukiewicz, Igor},
  booktitle =           {{P}roceedings of the 35th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'08)~-- Part~{II}},
  acronym =             {{ICALP}'08},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {5126},
  pages =               {124-135},
  year =                {2008},
  month =               jul,
  doi =                 {10.1007/978-3-540-70583-3_11},
  abstract =            {Metric Interval Temporal Logic (MITL) is a popular
                         formalism for expressing real-time specifications.
                         This logic achieves decidability by restricting the
                         precision of timing constraints, in particular, by
                         banning so-called \emph{punctual} specifications.
                         In~this paper we~introduce a significantly more
                         expressive logic that can express a wide variety of
                         punctual specifications, but whose model-checking
                         problem has the same complexity as that of~MITL.
                         We~conclude that for model checking the most
                         commonly occurring specifications, such as
                         invariance and bounded response, punctuality can be
                         accommodated at no cost.},
}
[BMP10] Laura Bozzelli, Aniello Murano, and Adriano Peron. Pushdown module checking. Formal Methods in System Design 36(1):65-95. Springer-Verlag, February 2010.
@article{fmsd36(1)-BMP,
  author =              {Bozzelli, Laura and Murano, Aniello and Peron,
                         Adriano},
  title =               {Pushdown module checking},
  publisher =           {Springer-Verlag},
  journal =             {Formal Methods in System Design},
  volume =              {36},
  number =              {1},
  pages =               {65-95},
  year =                {2010},
  month =               feb,
}
[BMP+15] Patricia Bouyer, Nicolas Markey, Nicolas Perrin, and Philipp Schlehuber-Caissier. Timed automata abstraction of switched dynamical systems using control funnels. In FORMATS'15, Lecture Notes in Computer Science 9268, pages 60-75. Springer-Verlag, September 2015.
Abstract

The development of formal methods for control design is an important challenge with potential applications in a wide range of safety-critical cyber-physical systems. Focusing on switched dynamical systems, we propose a new abstraction, based on time-varying regions of invariance (the control funnels), that models behaviors of systems as timed automata. The main advantage of this method is that it allows automated verification of formal specifications and reactive controller synthesis without discretizing the evolution of the state of the system. Efficient constructions are possible in the case of linear dynamics. We demonstrate the potential of our approach with two examples.

@inproceedings{formats2015-BMPS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Perrin,
                         Nicolas and Schlehuber{-}Caissier, Philipp},
  title =               {Timed automata abstraction of switched dynamical
                         systems using control funnels},
  editor =              {Sankaranarayanan, Sriram and Vicario, Enrico},
  booktitle =           {{P}roceedings of the 13th {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'15)},
  acronym =             {{FORMATS}'15},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {9268},
  pages =               {60-75},
  year =                {2015},
  month =               sep,
  doi =                 {10.1007/978-3-319-22975-1_5},
  abstract =            {The~development of formal methods for control design
                         is an important challenge with potential
                         applications in a wide range of safety-critical
                         cyber-physical systems. Focusing on switched
                         dynamical systems, we~propose a new abstraction,
                         based on time-varying regions of invariance
                         (the~\emph{control funnels}), that models behaviors
                         of systems as timed automata. The main advantage of
                         this method is that it allows automated verification
                         of formal specifications and reactive controller
                         synthesis without discretizing the evolution of the
                         state of the system. Efficient constructions are
                         possible in the case of linear dynamics.
                         We~demonstrate the potential of our approach with
                         two examples.},
}
[BMP+17] Patricia Bouyer, Nicolas Markey, Nicolas Perrin, and Philipp Schlehuber-Caissier. Timed automata abstraction of switched dynamical systems using control funnels. Real-Time Systems 53(3):327-353. Kluwer Academic, May 2017.
Abstract

The development of formal methods for control design is an important challenge with potential applications in a wide range of safety-critical cyber-physical systems. Focusing on switched dynamical systems, we propose a new abstraction, based on time-varying regions of invariance (control funnels), that models behaviors of systems as timed automata. The main advantage of this method is that it allows for the automated verification and reactive controller synthesis without discretizing the evolution of the state of the system. Efficient and analytic constructions are possible in the case of linear dynamics whereas bounding funnels with conjectured properties based on numerical simulations can be used for general nonlinear dynamics. We demonstrate the potential of our approach with three examples.

@article{rts53(3)-BMPS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Perrin,
                         Nicolas and Schlehuber{-}Caissier, Philipp},
  title =               {Timed automata abstraction of switched dynamical
                         systems using control funnels},
  publisher =           {Kluwer Academic},
  journal =             {Real-Time Systems},
  volume =              {53},
  number =              {3},
  pages =               {327-353},
  year =                {2017},
  month =               may,
  doi =                 {10.1007/s11241-016-9262-3},
  abstract =            {The development of formal methods for control design
                         is an important challenge with potential
                         applications in a wide range of safety-critical
                         cyber-physical systems. Focusing on switched
                         dynamical systems, we propose a new abstraction,
                         based on time-varying regions of invariance (control
                         funnels), that models behaviors of systems as timed
                         automata. The~main advantage of this method is that
                         it allows for the automated verification and
                         reactive controller synthesis without discretizing
                         the evolution of the state of the system. Efficient
                         and analytic constructions are possible in the case
                         of linear dynamics whereas bounding funnels with
                         conjectured properties based on numerical
                         simulations can be used for general nonlinear
                         dynamics. We~demonstrate the potential of our
                         approach with three examples.},
}
[BMP+97] Martin Beaudry, Pierre McKenzie, Pierre Péladeau, and Denis Thérien. Finite Monoids: From Word to Circuit Evaluation. SIAM Journal on Computing 26(1):138-152. Society for Industrial and Applied Math., February 1997.
@article{siamcomp26(1)-BMPT,
  author =              {Beaudry, Martin and McKenzie, Pierre and
                         P{\'e}ladeau, Pierre and Th{\'e}rien, Denis},
  title =               {Finite Monoids: From Word to Circuit Evaluation},
  publisher =           {Society for Industrial and Applied Math.},
  journal =             {SIAM Journal on Computing},
  volume =              {26},
  number =              {1},
  pages =               {138-152},
  year =                {1997},
  month =               feb,
}
[BMR06] Patricia Bouyer, Nicolas Markey, and Pierre-Alain Reynier. Robust Model-Checking of Linear-Time Properties in Timed Automata. In LATIN'06, Lecture Notes in Computer Science 3887, pages 238-249. Springer-Verlag, March 2006.
Abstract

Formal verification of timed systems is well understood, but their implementation is still challenging. Recent works by Raskin et al. have brought out a model of parameterized timed automata that can be used to prove implementability of timed systems for safety properties. We define here a more general notion of robust model-checking for linear-time properties, which consists in verifying whether a property still holds even if the transitions are slightly delayed or expedited. We provide PSPACE algorithms for the robust model-checking of Büchi-like and LTL properties. We also verify bounded-response-time properties.

@inproceedings{latin2006-BMR,
  author =              {Bouyer, Patricia and Markey, Nicolas and Reynier,
                         Pierre-Alain},
  title =               {Robust Model-Checking of Linear-Time Properties in
                         Timed Automata},
  editor =              {Correa, Jos{\'e} R. and Hevia, Alejandro and Kiwi,
                         Marcos},
  booktitle =           {{P}roceedings of the 7th {L}atin {A}merican
                         {S}ymposium on {T}heoretical {IN}formatics
                         ({LATIN}'06)},
  acronym =             {{LATIN}'06},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {3887},
  pages =               {238-249},
  year =                {2006},
  month =               mar,
  doi =                 {10.1007/11682462_25},
  abstract =            {Formal verification of timed systems is well
                         understood, but their \emph{implementation} is still
                         challenging. Recent works by Raskin \emph{et al.}
                         have brought out a model of parameterized timed
                         automata that can be used to prove
                         \emph{implementability} of timed systems for safety
                         properties. We define here a more general notion of
                         robust model-checking for linear-time properties,
                         which consists in verifying whether a property still
                         holds even if the transitions are slightly delayed
                         or expedited. We provide PSPACE algorithms for the
                         robust model-checking of B{\"u}chi-like and LTL
                         properties. We also verify bounded-response-time
                         properties.},
}
[BMR08] Patricia Bouyer, Nicolas Markey, and Pierre-Alain Reynier. Robust Analysis of Timed Automata via Channel Machines. In FoSSaCS'08, Lecture Notes in Computer Science 4962, pages 157-171. Springer-Verlag, March 2008.
Abstract

Whereas formal verification of timed systems has become a very active field of research, the idealised mathematical semantics of timed automata cannot be faithfully implemented. Several works have thus focused on a modified semantics of timed automata which ensures implementability, and robust model-checking algorithms for safety, and later LTL properties have been designed. Recently, a new approach has been proposed, which reduces (standard) model-checking of timed automata to other verification problems on channel machines. Thanks to a new encoding of the modified semantics as a network of timed systems, we propose an original combination of both approaches, and prove that robust model-checking for coFlat-MTL, a large fragment of MTL, is EXPSPACE-Complete.

@inproceedings{fossacs2008-BMR,
  author =              {Bouyer, Patricia and Markey, Nicolas and Reynier,
                         Pierre-Alain},
  title =               {Robust Analysis of Timed Automata via Channel
                         Machines},
  editor =              {Amadio, Roberto},
  booktitle =           {{P}roceedings of the 11th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'08)},
  acronym =             {{FoSSaCS}'08},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4962},
  pages =               {157-171},
  year =                {2008},
  month =               mar,
  doi =                 {10.1007/978-3-540-78499-9_12},
  abstract =            {Whereas formal verification of timed systems has
                         become a very active field of research, the
                         idealised mathematical semantics of timed automata
                         cannot be faithfully implemented. Several works have
                         thus focused on a modified semantics of timed
                         automata which ensures implementability, and robust
                         model-checking algorithms for safety, and later LTL
                         properties have been designed. Recently, a~new
                         approach has been proposed, which reduces (standard)
                         model-checking of timed automata to other
                         verification problems on channel machines. Thanks to
                         a new encoding of the modified semantics as a
                         network of timed systems, we propose an original
                         combination of both approaches, and prove that
                         robust model-checking for coFlat-MTL, a large
                         fragment of~MTL, is EXPSPACE-Complete.},
}
[BMR14] Véronique Bruyère, Noëmie Meunier, and Jean-François Raskin. Secure Equilibria in Weighted Games. In CSL/ LICS'14. ACM Press, July 2014.
@inproceedings{csllics2014-BMR,
  author =              {Bruy{\`e}re, V{\'e}ronique and Meunier, No{\"e}mie
                         and Raskin, Jean-Fran{\c c}ois},
  title =               {Secure Equilibria in Weighted Games},
  booktitle =           {{P}roceedings of the Joint Meeting of the 23rd
                         {EACSL} {A}nnual {C}onference on {C}omputer
                         {S}cience {L}ogic and the 29th {A}nnual {ACM\slash
                         IEEE} {S}ymposium on {L}ogic {I}n {C}omputer
                         {S}cience ({CSL\slash LICS}'14)},
  acronym =             {{CSL\slash LICS}'14},
  publisher =           {ACM Press},
  chapter =             {26},
  year =                {2014},
  month =               jul,
  doi =                 {10.1145/2603088.2603109},
}
[BMR+15] Patricia Bouyer, Nicolas Markey, Mickael Randour, Kim Guldstrand Larsen, and Simon Laursen. Average-energy games. In GandALF'15, Electronic Proceedings in Theoretical Computer Science 193, pages 1-15. September 2015.
Abstract

Two-player quantitative zero-sum games provide a natural framework to synthesize controllers with performance guarantees for reactive systems within an uncontrollable environment. Classical settings include mean-payoff games, where the objective is to optimize the long-run average gain per action, and energy games, where the system has to avoid running out of energy.

We study average-energy games, where the goal is to optimize the long-run average of the accumulated energy. We show that this objective arises naturally in several applications, and that it yields interesting connections with previous concepts in the literature. We prove that deciding the winner in such games is in NPcoNP and at least as hard as solving mean-payoff games, and we establish that memoryless strategies suffice to win. We also consider the case where the system has to minimize the average-energy while maintaining the accumulated energy within predefined bounds at all times: this corresponds to operating with a finite-capacity storage for energy. We give results for one-player and two-player games, and establish complexity bounds and memory requirements.

@inproceedings{gandalf2015-BMRLL,
  author =              {Bouyer, Patricia and Markey, Nicolas and Randour,
                         Mickael and Larsen, Kim Guldstrand and Laursen,
                         Simon},
  title =               {Average-energy games},
  editor =              {Esparza, Javier and Tronci, Enrico},
  booktitle =           {{P}roceedings of the 6th {I}nternational {S}ymposium
                         on {G}ames, {A}utomata, {L}ogics and {F}ormal
                         {V}erification ({GandALF}'15)},
  acronym =             {{GandALF}'15},
  series =              {Electronic Proceedings in Theoretical Computer
                         Science},
  volume =              {193},
  pages =               {1-15},
  year =                {2015},
  month =               sep,
  doi =                 {10.4204/EPTCS.193.1},
  abstract =            {Two-player quantitative zero-sum games provide a
                         natural framework to synthesize controllers with
                         performance guarantees for reactive systems within
                         an uncontrollable environment. Classical settings
                         include mean-payoff games, where the objective is to
                         optimize the long-run average gain per action, and
                         energy games, where the system has to avoid running
                         out of energy.\par We study \emph{average-energy}
                         games, where the goal is to optimize the long-run
                         average of the accumulated energy. We show that this
                         objective arises naturally in several applications,
                         and that it yields interesting connections with
                         previous concepts in the literature. We prove that
                         deciding the winner in such games is in
                         \textsf{NP}{{\(\cap\)}}\textsf{coNP} and at least as
                         hard as solving mean-payoff games, and we establish
                         that memoryless strategies suffice to win. We also
                         consider the case where the system has to minimize
                         the average-energy while maintaining the accumulated
                         energy within predefined bounds at all times: this
                         corresponds to operating with a finite-capacity
                         storage for energy. We give results for one-player
                         and two-player games, and establish complexity
                         bounds and memory requirements.},
}
[BMR+18] Patricia Bouyer, Nicolas Markey, Mickael Randour, Kim Guldstrand Larsen, and Simon Laursen. Average-energy games. Acta Informatica 55(2). Springer-Verlag, March 2018.
Abstract

Two-player quantitative zero-sum games provide a natural framework to synthesize controllers with performance guarantees for reactive systems within an uncontrollable environment. Classical settings include mean-payoff games, where the objective is to optimize the long-run average gain per action, and energy games, where the system has to avoid running out of energy. We study average-energy games, where the goal is to optimize the long-run average of the accumulated energy. We show that this objective arises naturally in several applications, and that it yields interesting connections with previous concepts in the literature. We prove that deciding the winner in such games is in NP∩coNP and at least as hard as solving mean-payoff games, and we establish that memoryless strategies suffice to win. We also consider the case where the system has to minimize the average-energy while maintaining the accumulated energy within predefined bounds at all times: this corresponds to operating with a finite-capacity storage for energy. We give results for one-player and two-player games, and establish complexity bounds and memory requirements.

@article{acta55(2)-BMRLL,
  author =              {Bouyer, Patricia and Markey, Nicolas and Randour,
                         Mickael and Larsen, Kim Guldstrand and Laursen,
                         Simon},
  title =               {Average-energy games},
  publisher =           {Springer-Verlag},
  journal =             {Acta Informatica},
  volume =              {55},
  number =              {2},
  year =                {2018},
  month =               mar,
  doi =                 {10.1007/s00236-016-0274-1},
  abstract =            {Two-player quantitative zero-sum games provide a
                         natural framework to synthesize controllers with
                         performance guarantees for reactive systems within
                         an uncontrollable environment. Classical settings
                         include mean-payoff games, where the objective is to
                         optimize the long-run average gain per action, and
                         energy games, where the system has to avoid running
                         out of energy. We study average-energy games, where
                         the goal is to optimize the long-run average of the
                         accumulated energy. We show that this objective
                         arises naturally in several applications, and that
                         it yields interesting connections with previous
                         concepts in the literature. We prove that deciding
                         the winner in such games is in
                         \(\textsf{NP}\cap\textsf{coNP}\) and at least as
                         hard as solving mean-payoff games, and we establish
                         that memoryless strategies suffice to win. We also
                         consider the case where the system has to minimize
                         the average-energy while maintaining the accumulated
                         energy within predefined bounds at all times: this
                         corresponds to operating with a finite-capacity
                         storage for energy. We give results for one-player
                         and two-player games, and establish complexity
                         bounds and memory requirements.},
}
[BMR+16] Patricia Bouyer, Nicolas Markey, Mickael Randour, Arnaud Sangnier, and Daniel Stan. Reachability in Networks of Register Protocols under Stochastic Schedulers. In ICALP'16, Leibniz International Proceedings in Informatics 55, pages 106:1-106:14. Leibniz-Zentrum für Informatik, July 2016.
Abstract

We study the almost-sure reachability problem in a distributed system obtained as the asynchronous composition of N copies (called processes) of the same automaton (called protocol), that can communicate via a shared register with finite domain. The automaton has two types of transitions: write-transitions update the value of the register, while read-transitions move to a new state depending on the content of the register. Non-determinism is resolved by a stochastic scheduler. Given a protocol, we focus on almost-sure reachability of a target state by one of the processes. The answer to this problem naturally depends on the number N of processes. However, we prove that our setting has a cut-off property : the answer to the almost-sure reachability problem is constant when N is large enough; we then develop an EXPSPACE algorithm deciding whether this constant answer is positive or negative.

@inproceedings{icalp2016-BMRSS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Randour,
                         Mickael and Sangnier, Arnaud and Stan, Daniel},
  title =               {Reachability in Networks of Register Protocols under
                         Stochastic Schedulers},
  editor =              {Chatzigiannakis, Ioannis and Mitzenmacher, Michael
                         and Rabani, Yuval and Sangiorgi, Davide},
  booktitle =           {{P}roceedings of the 43rd {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'16)~-- Part~{II}},
  acronym =             {{ICALP}'16},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {55},
  pages =               {106:1-106:14},
  year =                {2016},
  month =               jul,
  doi =                 {10.4230/LIPIcs.ICALP.2016.106},
  abstract =            {We study the almost-sure reachability problem in a
                         distributed system obtained as the asynchronous
                         composition of~\(N\) copies (called
                         \emph{processes}) of the same automaton (called
                         \emph{protocol}), that can communicate via a shared
                         register with finite domain. The automaton has two
                         types of transitions: write-transitions update the
                         value of the register, while read-transitions move
                         to a new state depending on the content of the
                         register. Non-determinism is resolved by a
                         stochastic scheduler. Given a protocol, we focus on
                         almost-sure reachability of a target state by one of
                         the processes. The answer to this problem naturally
                         depends on the number~\(N\) of processes. However,
                         we prove that our setting has a cut-off property :
                         the answer to the almost-sure reachability problem
                         is constant when \(N\) is large enough; we~then
                         develop an EXPSPACE algorithm deciding whether this
                         constant answer is positive or negative.},
}
[BMS11] Patricia Bouyer, Nicolas Markey, and Ocan Sankur. Robust Model-Checking of Timed Automata via Pumping in Channel Machines. In FORMATS'11, Lecture Notes in Computer Science 6919, pages 97-112. Springer-Verlag, September 2011.
Abstract

Timed automata are governed by a mathematical semantics which assumes perfectly continuous and precise clocks. This requirement is not satised by digital hardware on which the models are implemented. In fact, it was shown that the presence of imprecisions, however small they may be, may yield extra behaviours. Therefore correctness proven on the formal model does not imply correctness of the real system.

The problem of robust model-checking was then dened to circumvent this inconsistency. It consists in computing a bound on the imprecision under which the system will be correct.

In this work, we show that robust model-checking against ω-regular properties for timed automata can be reduced to standard model-checking of timed automata, by computing an adequate bound on the imprecision. This yields a new algorithm for robust model-checking of ω-regular properties, which is both optimal and valid for general timed automata.

@inproceedings{formats2011-BMS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Sankur,
                         Ocan},
  title =               {Robust Model-Checking of Timed Automata via Pumping
                         in Channel Machines},
  editor =              {Fahrenberg, Uli and Tripakis, Stavros},
  booktitle =           {{P}roceedings of the 9th {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'11)},
  acronym =             {{FORMATS}'11},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {6919},
  pages =               {97-112},
  year =                {2011},
  month =               sep,
  doi =                 {10.1007/978-3-642-24310-3_8},
  abstract =            {Timed automata are governed by a mathematical
                         semantics which assumes perfectly continuous and
                         precise clocks. This requirement is not satised by
                         digital hardware on which the models are
                         implemented. In~fact, it~was shown that the presence
                         of imprecisions, however small they may be, may
                         yield extra behaviours. Therefore correctness proven
                         on the formal model does not imply correctness of
                         the real system.\par The problem of robust
                         model-checking was then dened to circumvent this
                         inconsistency. It consists in computing a bound on
                         the imprecision under which the system will be
                         correct.\par In this work, we show that robust
                         model-checking against \(\omega\)-regular properties
                         for timed automata can be reduced to standard
                         model-checking of timed automata, by computing an
                         adequate bound on the imprecision. This yields a new
                         algorithm for robust model-checking of
                         \(\omega\)-regular properties, which is both optimal
                         and valid for general timed automata.},
}
[BMS12] Patricia Bouyer, Nicolas Markey, and Ocan Sankur. Robust reachability in timed automata: a game-based approach. In ICALP'12, Lecture Notes in Computer Science 7392, pages 128-140. Springer-Verlag, July 2012.
Abstract

Reachability checking is one of the most basic problems in verification. By solving this problem, one synthesizes a strategy that dictates the actions to be performed for ensuring that the target location is reached. In this work, we are interested in synthesizing "robust" strategies for ensuring reachability of a location in a timed automaton; with "robust", we mean that it must still ensure reachability even when the delays are perturbed by the environment. We model this perturbed semantics as a game between the controller and its environment, and solve the parameterized robust reachability problem: we show that the existence of an upper bound on the perturbations under which there is a strategy reaching a target location is EXPTIME-complete.

@inproceedings{icalp2012-BMS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Sankur,
                         Ocan},
  title =               {Robust reachability in timed automata: a~game-based
                         approach},
  editor =              {Czumaj, Artur and Mehlhorn, Kurt and Pitts, Andrew
                         and Wattenhofer, Roger},
  booktitle =           {{P}roceedings of the 39th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'12)~-- Part~{II}},
  acronym =             {{ICALP}'12},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {7392},
  pages =               {128-140},
  year =                {2012},
  month =               jul,
  doi =                 {10.1007/978-3-642-31585-5_15},
  abstract =            {Reachability checking is one of the most basic
                         problems in verification. By solving this problem,
                         one synthesizes a strategy that dictates the actions
                         to be performed for ensuring that the target
                         location is reached. In this work, we are interested
                         in synthesizing {"}robust{"} strategies for ensuring
                         reachability of a location in a timed automaton;
                         with {"}robust{"}, we mean that it must still ensure
                         reachability even when the delays are perturbed by
                         the environment. We model this perturbed semantics
                         as a game between the controller and its
                         environment, and solve the parameterized robust
                         reachability problem: we show that the existence of
                         an upper bound on the perturbations under which
                         there is a strategy reaching a target location is
                         EXPTIME-complete.},
}
[BMS13] Patricia Bouyer, Nicolas Markey, and Ocan Sankur. Robustness in timed automata. In RP'13, Lecture Notes in Computer Science 8169, pages 1-18. Springer-Verlag, September 2013.
Abstract

In this paper we survey several approaches to the robustness of timed automata, that is, the ability of a system to resist to slight perturbations or errors. We will concentrate on robustness against timing errors which can be due to measuring errors, imprecise clocks, and unexpected runtime behaviors such as execution times that are longer or shorter than expected.

We consider the perturbation model of guard enlargement and formulate several robust verification problems that have been studied recently, including robustness analysis, robust implementation, and robust control.

@inproceedings{rp2013-BMS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Sankur,
                         Ocan},
  title =               {Robustness in timed automata},
  editor =              {Abdulla, Parosh Aziz and Potapov, Igor},
  booktitle =           {{P}roceedings of the 7th {W}orkshop on
                         {R}eachability {P}roblems in {C}omputational
                         {M}odels ({RP}'13)},
  acronym =             {{RP}'13},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8169},
  pages =               {1-18},
  year =                {2013},
  month =               sep,
  doi =                 {10.1007/978-3-642-41036-9_1},
  abstract =            {In this paper we survey several approaches to the
                         robustness of timed automata, that~is, the ability
                         of a system to resist to slight perturbations or
                         errors. We will concentrate on robustness against
                         timing errors which can be due to measuring errors,
                         imprecise clocks, and unexpected runtime behaviors
                         such as execution times that are longer or shorter
                         than expected.\par We consider the perturbation
                         model of guard enlargement and formulate several
                         robust verification problems that have been studied
                         recently, including robustness analysis, robust
                         implementation, and robust control.},
}
[BMS13] Patricia Bouyer, Nicolas Markey, and Ocan Sankur. Robust Weighted Timed Automata and Games. In FORMATS'13, Lecture Notes in Computer Science 8053, pages 31-46. Springer-Verlag, August 2013.
Abstract

Weighted timed automata extend timed automata with cost variables that can be used to model the evolution of various quantities. Although cost-optimal reachability is decidable (in polynomial space) on this model, it becomes undecidable on weighted timed games. This paper studies cost-optimal reachability problems on weighted timed automata and games under robust semantics. More precisely, we consider two perturbation game semantics that introduce imprecisions in the standard semantics, and bring robustness properties w.r.t. timing imprecisions to controllers. We give a polynomial-space algorithm for weighted timed automata, and prove the undecidability of cost-optimal reachability on weighted timed games, showing that the problem is robustly undecidable.

@inproceedings{formats2013-BMS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Sankur,
                         Ocan},
  title =               {Robust Weighted Timed Automata and Games},
  editor =              {Braberman, V{\'\i}ctor and Fribourg, Laurent},
  booktitle =           {{P}roceedings of the 11th {I}nternational
                         {C}onferences on {F}ormal {M}odelling and {A}nalysis
                         of {T}imed {S}ystems ({FORMATS}'13)},
  acronym =             {{FORMATS}'13},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {8053},
  pages =               {31-46},
  year =                {2013},
  month =               aug,
  doi =                 {10.1007/978-3-642-40229-6_3},
  abstract =            {Weighted timed automata extend timed automata with
                         cost variables that can be used to model the
                         evolution of various quantities. Although
                         cost-optimal reachability is decidable (in
                         polynomial space) on this model, it becomes
                         undecidable on weighted timed games. This paper
                         studies cost-optimal reachability problems on
                         weighted timed automata and games under robust
                         semantics. More precisely, we consider two
                         perturbation game semantics that introduce
                         imprecisions in the standard semantics, and bring
                         robustness properties w.r.t. timing imprecisions to
                         controllers. We give a polynomial-space algorithm
                         for weighted timed automata, and prove the
                         undecidability of cost-optimal reachability on
                         weighted timed games, showing that the problem is
                         robustly undecidable.},
}
[BMS14] Patricia Bouyer, Nicolas Markey, and Daniel Stan. Mixed Nash Equilibria in Concurrent Games. In FSTTCS'14, Leibniz International Proceedings in Informatics 29, pages 351-363. Leibniz-Zentrum für Informatik, December 2014.
Abstract

We study mixed-strategy Nash equilibria in multiplayer deterministic concurrent games played on graphs, with terminal-reward payoffs (that is, absorbing states with a value for each player). We show undecidability of the existence of a constrained Nash equilibrium (the constraint requiring that one player should have maximal payoff), with only three players and 0/1-rewards (i.e., reachability objectives). This has to be compared with the undecidability result by Ummels and Wojtczak for turn-based games which requires 14 players and general rewards. Our proof has various interesting consequences: (i) the undecidability of the existence of a Nash equilibrium with a constraint on the social welfare; (ii) the undecidability of the existence of an (unconstrained) Nash equilibrium in concurrent games with terminal-reward payoffs.

@inproceedings{fsttcs2014-BMS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Stan,
                         Daniel},
  title =               {Mixed {N}ash Equilibria in Concurrent Games},
  editor =              {Raman, Venkatesh and Suresh, S. P.},
  booktitle =           {{P}roceedings of the 34th {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'14)},
  acronym =             {{FSTTCS}'14},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
  volume =              {29},
  pages =               {351-363},
  year =                {2014},
  month =               dec,
  doi =                 {10.4230/LIPIcs.FSTTCS.2014.351},
  abstract =            {We study mixed-strategy Nash equilibria in
                         multiplayer deterministic concurrent games played on
                         graphs, with terminal-reward payoffs (that is,
                         absorbing states with a value for each player). We
                         show undecidability of the existence of a
                         constrained Nash equilibrium (the constraint
                         requiring that one player should have maximal
                         payoff), with only three players and 0/1-rewards
                         (i.e., reachability objectives). This has to be
                         compared with the undecidability result by Ummels
                         and Wojtczak for turn-based games which requires 14
                         players and general rewards. Our proof has various
                         interesting consequences: (i)~the~undecidability of
                         the existence of a Nash equilibrium with a
                         constraint on the social welfare;
                         (ii)~the~undecidability of the existence of an
                         (unconstrained) Nash equilibrium in concurrent games
                         with terminal-reward payoffs.},
}
[BMS15] Patricia Bouyer, Nicolas Markey, and Ocan Sankur. Robust Reachability in Timed Automata and Games: A Game-based Approach. Theoretical Computer Science 563:43-74. Elsevier, January 2015.
Abstract

Reachability checking is one of the most basic problems in verification. By solving this problem in a game, one can synthesize a strategy that dictates the actions to be performed for ensuring that the target location is reached. In this work, we are interested in synthesizing "robust" strategies for ensuring reachability of a location in timed automata. By robust, we mean that it must still ensure reachability even when the delays are perturbed by the environment. We model this perturbed semantics as a game between the controller and its environment, and solve the parameterized robust reachability problem: we show that the existence of an upper bound on the perturbations under which there is a strategy reaching a target location is EXPTIME-complete. We also extend our algorithm, with the same complexity, to turn-based timed games, where the successor state is entirely determined by the environment in some locations.

@article{tcs563()-BMS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Sankur,
                         Ocan},
  title =               {Robust Reachability in Timed Automata and Games:
                         A~Game-based Approach},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {563},
  pages =               {43-74},
  year =                {2015},
  month =               jan,
  doi =                 {10.1016/j.tcs.2014.08.014},
  abstract =            {Reachability checking is one of the most basic
                         problems in verification. By solving this problem in
                         a game, one can synthesize a strategy that dictates
                         the actions to be performed for ensuring that the
                         target location is reached. In this work, we are
                         interested in synthesizing {"}robust{"} strategies
                         for ensuring reachability of a location in timed
                         automata. By robust, we mean that it must still
                         ensure reachability even when the delays are
                         perturbed by the environment. We model this
                         perturbed semantics as a game between the controller
                         and its environment, and solve the parameterized
                         robust reachability problem: we show that the
                         existence of an upper bound on the perturbations
                         under which there is a strategy reaching a target
                         location is EXPTIME-complete. We also extend our
                         algorithm, with the same complexity, to turn-based
                         timed games, where the successor state is entirely
                         determined by the environment in some locations.},
}
[BMS16] Patricia Bouyer, Nicolas Markey, and Daniel Stan. Stochastic Equilibria under Imprecise Deviations in Terminal-Reward Concurrent Games. In GandALF'16, Electronic Proceedings in Theoretical Computer Science 226, pages 61-75. September 2016.
Abstract

We study the existence of mixed-strategy equilibria in concurrent games played on graphs. While existence is guaranteed with safety objectives for each player, Nash equilibria need not exist when players are given arbitrary terminal-reward objectives, and their existence is undecidable with qualitative reachability objectives (and only three players). However, these results rely on the fact that the players can enforce infinite plays while trying to improve their payoffs. In this paper, we introduce a relaxed notion of equilibria, where deviations are imprecise. We prove that contrary to Nash equilibria, such (stationary) equilibria always exist, and we develop a PSPACE algorithm to compute one.

@inproceedings{gandalf2016-BMS,
  author =              {Bouyer, Patricia and Markey, Nicolas and Stan,
                         Daniel},
  title =               {Stochastic Equilibria under Imprecise Deviations in
                         Terminal-Reward Concurrent Games},
  editor =              {Cantone, Domenico and Delzanno, Giorgio},
  booktitle =           {{P}roceedings of the 7th {I}nternational {S}ymposium
                         on {G}ames, {A}utomata, {L}ogics and {F}ormal
                         {V}erification ({GandALF}'16)},
  acronym =             {{GandALF}'16},
  series =              {Electronic Proceedings in Theoretical Computer
                         Science},
  volume =              {226},
  pages =               {61-75},
  year =                {2016},
  month =               sep,
  doi =                 {10.4204/EPTCS.226.5},
  abstract =            {We study the existence of mixed-strategy equilibria
                         in concurrent games played on graphs. While
                         existence is guaranteed with safety objectives for
                         each player, Nash equilibria need not exist when
                         players are given arbitrary terminal-reward
                         objectives, and their existence is undecidable with
                         qualitative reachability objectives (and~only three
                         players). However, these results rely on the fact
                         that the players can enforce infinite plays while
                         trying to improve their payoffs. In this paper, we
                         introduce a relaxed notion of equilibria, where
                         deviations are imprecise. We prove that contrary to
                         Nash equilibria, such (stationary) equilibria always
                         exist, and we develop a PSPACE algorithm to compute
                         one.},
}
[BMT99] Augustin Baziramwabo, Pierre McKenzie, and Denis Thérien. Modular Temporal Logic. In LICS'99, pages 344-351. IEEE Comp. Soc. Press, July 1999.
@inproceedings{lics1999-BMT,
  author =              {Baziramwabo, Augustin and McKenzie, Pierre and
                         Th{\'e}rien, Denis},
  title =               {Modular Temporal Logic},
  booktitle =           {{P}roceedings of the 14th {A}nnual {S}ymposium on
                         {L}ogic in {C}omputer {S}cience ({LICS}'99)},
  acronym =             {{LICS}'99},
  publisher =           {IEEE Comp. Soc. Press},
  pages =               {344-351},
  year =                {1999},
  month =               jul,
}
[BMV14] Patricia Bouyer, Nicolas Markey, and Steen Vester. Nash Equilibria in Symmetric Games with Partial Observation. In SR'14, Electronic Proceedings in Theoretical Computer Science 146, pages 49-55. March 2014.
Abstract

We investigate a model for representing large multiplayer games, which satisfy strong symmetry properties. This model is made of multiple copies of an arena; each player plays in his own arena, and can partially observe what the other players do. Therefore, this game has partial information and symmetry constraints, which make the computation of Nash equilibria difficult. We show several undecidability results, and for bounded-memory strategies, we precisely characterize the complexity of computing pure Nash equilibria (for qualitative objectives) in this game model.

@inproceedings{sr2014-BMV,
  author =              {Bouyer, Patricia and Markey, Nicolas and Vester,
                         Steen},
  title =               {Nash Equilibria in Symmetric Games with Partial
                         Observation},
  booktitle =           {{P}roceedings of the 2nd {I}nternational {W}orkshop
                         on {S}trategic {R}easoning ({SR}'14)},
  acronym =             {{SR}'14},
  series =              {Electronic Proceedings in Theoretical Computer
                         Science},
  volume =              {146},
  pages =               {49-55},
  year =                {2014},
  month =               mar,
  doi =                 {10.4204/EPTCS.146.7},
  abstract =            {We investigate a model for representing large
                         multiplayer games, which satisfy strong symmetry
                         properties. This model is made of multiple copies of
                         an arena; each player plays in his own arena, and
                         can partially observe what the other players do.
                         Therefore, this game has partial information and
                         symmetry constraints, which make the computation of
                         Nash equilibria difficult. We show several
                         undecidability results, and for bounded-memory
                         strategies, we precisely characterize the complexity
                         of computing pure Nash equilibria (for qualitative
                         objectives) in this game model.},
}
[BMV15] Dietmar Berwanger, Anup Basil Mathew, and Marie Van den Bogaard. Hierarchical Information Patterns and Distributed Strategy Synthesis. In ATVA'15, Lecture Notes in Computer Science 9364, pages 378-393. Springer-Verlag, October 2015.
@inproceedings{atva2015-BMV,
  author =              {Berwanger, Dietmar and Mathew, Anup Basil and
                         Van{~}den{ }Bogaard, Marie},
  title =               {Hierarchical Information Patterns and Distributed
                         Strategy Synthesis},
  editor =              {Finkbeiner, Bernd and Pu, Geguang and Zhang, Lijun},
  booktitle =           {{P}roceedings of the 13th {I}nternational
                         {S}ymposium on {A}utomated {T}echnology for
                         {V}erification and {A}nalysis ({ATVA}'15)},
  acronym =             {{ATVA}'15},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {9364},
  pages =               {378-393},
  year =                {2015},
  month =               oct,
}
[BMV17] Patricia Bouyer, Nicolas Markey, and Steen Vester. Nash Equilibria in Symmetric Graph Games with Partial Observation. Information and Computation 254(2):238-258. Elsevier, June 2017.
Abstract

We investigate a model for representing large multiplayer games, which satisfy strong symmetry properties. This model is made of multiple copies of an arena; each player plays in his own arena, and can partially observe what the other players do. Therefore, this game has partial information and symmetry constraints, which make the computation of Nash equilibria difficult. We show several undecidability results, and for bounded-memory strategies, we precisely characterize the complexity of computing pure Nash equilibria (for qualitative objectives) in this game model.

@article{icomp254()-BMV,
  author =              {Bouyer, Patricia and Markey, Nicolas and Vester,
                         Steen},
  title =               {{N}ash Equilibria in Symmetric Graph Games with
                         Partial Observation},
  publisher =           {Elsevier},
  journal =             {Information and Computation},
  volume =              {254},
  number =              {2},
  pages =               {238-258},
  year =                {2017},
  month =               jun,
  doi =                 {10.1016/j.ic.2016.10.010},
  abstract =            {We investigate a model for representing large
                         multiplayer games, which satisfy strong symmetry
                         properties. This model is made of multiple copies of
                         an arena; each player plays in his own arena, and
                         can partially observe what the other players do.
                         Therefore, this game has partial information and
                         symmetry constraints, which make the computation of
                         Nash equilibria difficult. We show several
                         undecidability results, and for bounded-memory
                         strategies, we precisely characterize the complexity
                         of computing pure Nash equilibria (for qualitative
                         objectives) in this game model.},
}
[BO04] Stephen J. Bellantoni and Isabel Oitavem. Separating NC along the δ axis. Theoretical Computer Science 318(1-2):57-78. Elsevier, June 2004.
@article{tcs318(1-2)-BO,
  author =              {Bellantoni, Stephen J. and Oitavem, Isabel},
  title =               {Separating {{\(\mathrm{NC}\)}} along the
                         {{\(\delta\)}} axis},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {318},
  number =              {1-2},
  pages =               {57-78},
  year =                {2004},
  month =               jun,
  doi =                 {10.1016/j.tcs.2003.10.021},
}
[Bod88] Hans L. Bodlaender. Dynamic Programming on Graphs with Bounded Treewidth. In ICALP'88, Lecture Notes in Computer Science 317, pages 105-118. Springer-Verlag, July 1988.
@inproceedings{icalp1988-Bod,
  author =              {Bodlaender, Hans L.},
  title =               {Dynamic Programming on Graphs with Bounded
                         Treewidth},
  editor =              {Lepist{\"o}, Timo and Salomaa, Arto},
  booktitle =           {{P}roceedings of the 15th {I}nternational
                         {C}olloquium on {A}utomata, {L}anguages and
                         {P}rogramming ({ICALP}'88)},
  acronym =             {{ICALP}'88},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {317},
  pages =               {105-118},
  year =                {1988},
  month =               jul,
  doi =                 {10.1007/3-540-19488-6_110},
}
[Bod93] Hans L. Bodlaender. A Tourist Guide through Treewidth. Acta Cybernetica 11(1-2):1-21. Institute of Informatics, University of Szeged, 1993.
@article{actacyb11(1-2)-Bod,
  author =              {Bodlaender, Hans L.},
  title =               {A~Tourist Guide through Treewidth},
  publisher =           {Institute of Informatics, University of Szeged},
  journal =             {Acta Cybernetica},
  volume =              {11},
  number =              {1-2},
  pages =               {1-21},
  year =                {1993},
}
[Bod96] Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25(6):1305-1317. Society for Industrial and Applied Math., December 1996.
@article{siamcomp25(6)-Bod,
  author =              {Bodlaender, Hans L.},
  title =               {A~linear-time algorithm for finding
                         tree-decompositions of small treewidth},
  publisher =           {Society for Industrial and Applied Math.},
  journal =             {SIAM Journal on Computing},
  volume =              {25},
  number =              {6},
  pages =               {1305-1317},
  year =                {1996},
  month =               dec,
  doi =                 {10.1137/S0097539793251219},
}
[Boj08] Mikołaj Bojańczyk. The Common Fragment of ACTL and LTL. In FoSSaCS'08, Lecture Notes in Computer Science 4962, pages 172-185. Springer-Verlag, March 2008.
@inproceedings{fossacs2008-Boj,
  author =              {Boja{\'n}czyk, Miko{\l}aj},
  title =               {The Common Fragment of {ACTL} and~{LTL}},
  editor =              {Amadio, Roberto},
  booktitle =           {{P}roceedings of the 11th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'08)},
  acronym =             {{FoSSaCS}'08},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4962},
  pages =               {172-185},
  year =                {2008},
  month =               mar,
}
[Bou02] Patricia Bouyer. Timed Automata May Cause some Troubles. Research Report LSV-02-09, Lab. Spécification & Vérification, ENS Cachan, France, July 2002.
@techreport{LSV0209-Bou,
  author =              {Bouyer, Patricia},
  title =               {Timed Automata May Cause some Troubles},
  number =              {LSV-02-09},
  year =                {2002},
  month =               jul,
  institution =         {Lab.~Sp\'ecification \& V\'erification, ENS Cachan,
                         France},
  type =                {Research Report},
}
[Bou03] Patricia Bouyer. Untameable Timed Automata!. In STACS'03, Lecture Notes in Computer Science 2607, pages 620-631. Springer-Verlag, February 2003.
@inproceedings{stacs2003-Bou,
  author =              {Bouyer, Patricia},
  title =               {Untameable Timed Automata!},
  editor =              {Alt, Helmut and Habib, Michel},
  booktitle =           {{P}roceedings of the 20th {S}ymposium on
                         {T}heoretical {A}spects of {C}omputer {S}cience
                         ({STACS}'03)},
  acronym =             {{STACS}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2607},
  pages =               {620-631},
  year =                {2003},
  month =               feb,
  doi =                 {10.1007/3-540-36494-3_54},
}
[Bou04] Patricia Bouyer. Forward Analysis of Updatable Timed Automata. Formal Methods in System Design 24(3):281-320. Kluwer Academic, May 2004.
@article{fmsd24(3)-Bou,
  author =              {Bouyer, Patricia},
  title =               {Forward Analysis of Updatable Timed Automata},
  publisher =           {Kluwer Academic},
  journal =             {Formal Methods in System Design},
  volume =              {24},
  number =              {3},
  pages =               {281-320},
  year =                {2004},
  month =               may,
  doi =                 {10.1023/B:FORM.0000026093.21513.31},
}
[Boz08] Laura Bozzelli. The complexity of CTL* + Linear Past. In FoSSaCS'08, Lecture Notes in Computer Science 4962, pages 186-200. Springer-Verlag, March 2008.
@inproceedings{fossacs2008-Boz,
  author =              {Bozzelli, Laura},
  title =               {The complexity of CTL\textsuperscript{*} + Linear
                         Past},
  editor =              {Amadio, Roberto},
  booktitle =           {{P}roceedings of the 11th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'08)},
  acronym =             {{FoSSaCS}'08},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {4962},
  pages =               {186-200},
  year =                {2008},
  month =               mar,
}
[BP03] Bruno Blanchet and Andreas Podelski. Verification of Cryptographic Protocols: Tagging Enforces Temrination. In FoSSaCS'03, Lecture Notes in Computer Science 2620, pages 136-152. Springer-Verlag, April 2003.
@inproceedings{fossacs2003-BP,
  author =              {Blanchet, Bruno and Podelski, Andreas},
  title =               {Verification of Cryptographic Protocols: Tagging
                         Enforces Temrination},
  editor =              {Gordon, Andrew D.},
  booktitle =           {{P}roceedings of the 6th {I}nternational
                         {C}onference on {F}oundations of {S}oftware
                         {S}cience and {C}omputation {S}tructure
                         ({FoSSaCS}'03)},
  acronym =             {{FoSSaCS}'03},
  publisher =           {Springer-Verlag},
  series =              {Lecture Notes in Computer Science},
  volume =              {2620},
  pages =               {136-152},
  year =                {2003},
  month =               apr,
}
[BPM81] Mordechai Ben-Ari, Amir Pnueli, and Zohar Manna. The Temporal Logic of Branching Time. In POPL'81, pages 164-176. ACM Press, January 1981.
@inproceedings{popl1981-BPM,
  author =              {Ben{-}Ari, Mordechai and Pnueli, Amir and Manna,
                         Zohar},
  title =               {The Temporal Logic of Branching Time},
  booktitle =           {Conference Record of the 8th {ACM} {S}ymposium on
                         {P}rinciples of {P}rogramming {L}anguages
                         ({POPL}'81)},
  acronym =             {{POPL}'81},
  publisher =           {ACM Press},
  pages =               {164-176},
  year =                {1981},
  month =               jan,
}
[BPM83] Mordechai Ben-Ari, Amir Pnueli, and Zohar Manna. The Temporal Logic of Branching Time. Acta Informatica 20:207-226. Springer-Verlag, 1983.
@article{acta20()-BPM,
  author =              {Ben{-}Ari, Mordechai and Pnueli, Amir and Manna,
                         Zohar},
  title =               {The Temporal Logic of Branching Time},
  publisher =           {Springer-Verlag},
  journal =             {Acta Informatica},
  volume =              {20},
  pages =               {207-226},
  year =                {1983},
}
[BR02] Danièle Beauquier and Alexander Rabinovich. Monadic Logic of Order over Naturals has no Finite Base. Journal of Logic and Computation 12(2):243-253. Oxford University Press, April 2002.
@article{jlc12(2)-BR,
  author =              {Beauquier, Dani{\`e}le and Rabinovich, Alexander},
  title =               {Monadic Logic of Order over Naturals has no Finite
                         Base},
  publisher =           {Oxford University Press},
  journal =             {Journal of Logic and Computation},
  volume =              {12},
  number =              {2},
  pages =               {243-253},
  year =                {2002},
  month =               apr,
}
[BR03] Véronique Bruyère and Jean-François Raskin. Real-Time Model-Checking: Parameters Everywhere. In FSTTCS'03, Lecture Notes in Computer Science 2914, pages 100-111. Springer-Verlag, December 2003.
@inproceedings{fsttcs2003-BR,
  author =              {Bruy{\`e}re, V{\'e}ronique and Raskin, Jean-Fran{\c
                         c}ois},
  title =               {Real-Time Model-Checking: {P}arameters Everywhere},
  editor =              {Pandya, Paritosh K. and Radhakrishnan, Jaikumar},
  booktitle =           {{P}roceedings of the 23rd {C}onference on
                         {F}oundations of {S}oftware {T}echnology and
                         {T}heoretical {C}omputer {S}cience ({FSTTCS}'03)},
  acronym =             {{FSTTCS}'03},
  publisher =           {Springer-Verlag},
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  author =              {Bradfield, Julian C.},
  title =               {The Modal Mu-Calculus Alternation Hierarchy is
                         Strict},
  publisher =           {Elsevier},
  journal =             {Theoretical Computer Science},
  volume =              {195},
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                         Randal E.},
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                         {C}onference ({DAC}'90)},
  acronym =             {{DAC}'90},
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@phdthesis{phd-brenguier,
  author =              {Brenguier, Romain},
  title =               {Nash Equilibria in Concurrent Games~--Application to
                         Timed Games},
  year =                {2012},
  month =               nov,
  school =              {Lab.~Sp\'ecification \& V\'erification, ENS Cachan,
                         France},
  type =                {Th\`ese de doctorat},
}
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  booktitle =           {{P}roceedings of the 27th {I}nternational
                         {C}onference on {C}oncurrency {T}heory
                         ({CONCUR}'16)},
  acronym =             {{CONCUR}'16},
  publisher =           {Leibniz-Zentrum f{\"u}r Informatik},
  series =              {Leibniz International Proceedings in Informatics},
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  year =                {2016},
  month =               aug,
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}
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                         {EACSL} {A}nnual {C}onference on {C}omputer
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                         IEEE} {S}ymposium on {L}ogic {I}n {C}omputer
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}
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                         Sankur, Ocan},
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}
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                         A.},
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                         {S}ignificant {D}ifference ({COMPOS}'97)},
  acronym =             {{COMPOS}'97},
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                         Daniele},
  booktitle =           {{P}roceedings of the 5th {K}urt {G}{\"o}del
                         {C}olloquium ({KGC}'97)},
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  year =                {2004},
  doi =                 {10.1007/b98282},
}