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A synthesis method for zero-sum mean-payoff asynchronous probabilistic games.

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Titel: A synthesis method for zero-sum mean-payoff asynchronous probabilistic games.
Autoren: Zhao W; The Department of Computer Engineering, Jiangsu University of Technology, Changzhou, 213001, China. zhaowei618@jsut.edu.cn., Liu W; The National University of Defense Technology, Changsha, 410073, China., Liu Z; Southwest University, Chongqing, 400715, China. zhimingliu88@swu.edu.cn., Wang T; The College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, China.
Quelle: Scientific reports [Sci Rep] 2025 Jan 17; Vol. 15 (1), pp. 2291. Date of Electronic Publication: 2025 Jan 17.
Publikationsart: Journal Article
Sprache: English
Info zur Zeitschrift: Publisher: Nature Publishing Group Country of Publication: England NLM ID: 101563288 Publication Model: Electronic Cited Medium: Internet ISSN: 2045-2322 (Electronic) Linking ISSN: 20452322 NLM ISO Abbreviation: Sci Rep Subsets: PubMed not MEDLINE; MEDLINE
Imprint Name(s): Original Publication: London : Nature Publishing Group, copyright 2011-
Abstract: The traditional synthesis problem aims to automatically construct a reactive system (if it exists) satisfying a given Linear Temporal Logic (LTL) specifications, and is often referred to as a qualitative problem. There is also a class of synthesis problems aiming at quantitative properties, such as mean-payoff values, and this type of problem is called a quantitative problem. For the two types of synthesis problems, the research on the former has been relatively mature, and the latter also has received huge amounts of attention. System designers prefer to synthesize systems that satisfy resource constraints. To this end, this paper focuses on the reactive synthesis problem of combining quantitative and qualitative objectives. First, zero-sum mean-payoff asynchronous probabilistic games are proposed, where the system aims at the expected mean payoff in a probabilistic environment while satisfying an LTL winning condition against an adversarial environment. Then, the case of taking the wider class of Generalized Reactivity(1) (GR(1)) formula as an LTL winning condition is studied, that is, the synthesis problem of the expected mean payoffs is studied for the system with the probability of winning. Next, two symbolic algorithms running in polynomial time are proposed to calculate the expected mean payoffs, and both algorithms adopt uniform random strategies. Combining the probability of system winning, the expected mean payoffs of the system when it has the probability of winning is calculated. Finally, our two algorithms are implemented, and their convergence and volatility are demonstrated through experiments.
(© 2025. The Author(s).)
Competing Interests: Declarations. Competing interests: The authors declare no competing interests.
References: Henzinger, T. A. Quantitative reactive models. In Model Driven Engineering Languages and Systems—15th International Conference, MODELS 2012, Innsbruck, Austria, September 30–October 5, 2012. Proceedings. Lecture Notes in Computer Science Vol. 7590 (eds France, R. B. et al.) 1–2 (Springer, 2012). https://doi.org/10.1007/978-3-642-33666-9_1 . (PMID: 10.1007/978-3-642-33666-9_1)
Hunter, P., Pauly, A., Pérez, G. A. & Raskin, J. Mean-payoff games with partial observation. Theor. Comput. Sci. 735, 82–110. https://doi.org/10.1016/j.tcs.2017.03.038 (2018). (PMID: 10.1016/j.tcs.2017.03.038)
Chatterjee, K., Doyen, L., Gimbert, H. & Oualhadj, Y. Perfect-information stochastic mean-payoff parity games. In Foundations of Software Science and Computation Structures—17th International Conference, FOSSACS 2014, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2014, Grenoble, France, April 5–13, 2014, Proceedings. Lecture Notes in Computer Science Vol. 8412 (ed. Muscholl, A.) 210–225 (Springer, 2014). https://doi.org/10.1007/978-3-642-54830-7_14 .
Velner, Y. et al. The complexity of multi-mean-payoff and multi-energy games. Inf. Comput. 241, 177–196. https://doi.org/10.1016/j.ic.2015.03.001 (2015). (PMID: 10.1016/j.ic.2015.03.001)
Brim, L., Chaloupka, J., Doyen, L., Gentilini, R. & Raskin, J. Faster algorithms for mean-payoff games. Formal Methods Syst. Des. 38, 97–118. https://doi.org/10.1007/s10703-010-0105-x (2011). (PMID: 10.1007/s10703-010-0105-x)
Björklund, H., Sandberg, S. & Vorobyov, S. G. Memoryless determinacy of parity and mean payoff games: a simple proof. Theor. Comput. Sci. 310, 365–378. https://doi.org/10.1016/S0304-3975(03)00427-4 (2004). (PMID: 10.1016/S0304-3975(03)00427-4)
Zwick, U. & Paterson, M. The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158, 343–359. https://doi.org/10.1016/0304-3975(95)00188-3 (1996). (PMID: 10.1016/0304-3975(95)00188-3)
Bruyère, V., Filiot, E., Randour, M. & Raskin, J. Meet your expectations with guarantees: Beyond worst-case synthesis in quantitative games. Inf. Comput. 254, 259–295. https://doi.org/10.1016/j.ic.2016.10.011 (2017). (PMID: 10.1016/j.ic.2016.10.011)
Church, A. Logic, arithmetic, and automata. J. Symb. Logic 29, 210 (1964).
Pnueli, A. & Rosner, R. On the synthesis of an asynchronous reactive module. In Automata, Languages and Programming, 16th International Colloquium, ICALP89, Stresa, Italy, July 11–15, 1989, Proceedings. Lecture Notes in Computer Science Vol. 372 (eds Ausiello, G. et al.) 652–671 (Springer, 1989). https://doi.org/10.1007/BFb0035790 . (PMID: 10.1007/BFb0035790)
Wallmeier, N., Hütten, P. & Thomas, W. Symbolic synthesis of finite-state controllers for request-response specifications. In Implementation and Application of Automata, 8th International Conference, CIAA 2003, Santa Barbara, California, USA, July 16–18, 2003, Proceedings. Lecture Notes in Computer Science Vol. 2759 (eds Ibarra, O. H. & Dang, Z.) 11–22 (Springer, 2003). https://doi.org/10.1007/3-540-45089-0_3 . (PMID: 10.1007/3-540-45089-0_3)
Alur, R. & Torre, S. L. Deterministic generators and games for ltl fragments. ACM Trans. Comput. Log. 5, 1–25. https://doi.org/10.1145/963927.963928 (2004). (PMID: 10.1145/963927.963928)
Harding, A., Ryan, M. & Schobbens, P. A new algorithm for strategy synthesis in LTL games. In Tools and Algorithms for the Construction and Analysis of Systems, 11th International Conference, TACAS 2005, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2005, Edinburgh, UK, April 4–8, 2005, Proceedings. Lecture Notes in Computer Science Vol. 3440 (eds Halbwachs, N. & Zuck, L. D.) 477–492 (Springer, 2005). https://doi.org/10.1007/978-3-540-31980-1_31 .
Jobstmann, B., Griesmayer, A. & Bloem, R. Program repair as a game. In Computer Aided Verification, 17th International Conference, CAV 2005, Edinburgh, Scotland, UK, July 6–10, 2005, Proceedings. Lecture Notes in Computer Science Vol. 3576 (eds Etessami, K. & Rajamani, S. K.) 226–238 (Springer, 2005). https://doi.org/10.1007/11513988_23 . (PMID: 10.1007/11513988_23)
Bloem, R., Jobstmann, B., Piterman, N., Pnueli, A. & Sa’ar, Y. Synthesis of reactive(1) designs. J. Comput. Syst. Sci. 78, 911–938. https://doi.org/10.1016/j.jcss.2011.08.007 (2012). (PMID: 10.1016/j.jcss.2011.08.007)
Ehrenfeucht, A. & Mycielski, J. Positional strategies for mean payoff games. Int. J. Game Theory 8, 109–113 (1979). (PMID: 10.1007/BF01768705)
Ummels, M. & Wojtczak, D. The complexity of Nash equilibria in limit-average games. In CONCUR 2011—Concurrency Theory—22nd International Conference, CONCUR 2011, Aachen, Germany, September 6–9, 2011. Proceedings. Lecture Notes in Computer Science Vol. 6901 (eds Katoen, J. & König, B.) 482–496 (Springer, 2011). https://doi.org/10.1007/978-3-642-23217-6_32 . (PMID: 10.1007/978-3-642-23217-6_32)
Bouyer, P., Brenguier, R. & Markey, N. Nash equilibria for reachability objectives in multi-player timed games. In CONCUR 2010—Concurrency Theory, 21st International Conference, CONCUR 2010, Paris, France, August 31–September 3, 2010. Proceedings. Lecture Notes in Computer Science Vol. 6269 (eds Gastin, P. & Laroussinie, F.) 192–206 (Springer, 2010). https://doi.org/10.1007/978-3-642-15375-4_14 . (PMID: 10.1007/978-3-642-15375-4_14)
Chatterjee, K., Henzinger, T. A. & Jurdzinski, M. Mean-payoff parity games. In 20th IEEE Symposium on Logic in Computer Science (LICS 2005), 26-29 June 2005, Chicago, IL, USA, Proceedings 178–187 (IEEE Computer Society, 2005). https://doi.org/10.1109/LICS.2005.26.
Chatterjee, K., Komárková, Z. & Kretínský, J. Unifying two views on multiple mean-payoff objectives in Markov decision processes. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6–10, 2015 244–256 (IEEE Computer Society, 2015). https://doi.org/10.1109/LICS.2015.32.
Almagor, S., Kupferman, O. & Velner, Y. Minimizing expected cost under hard Boolean constraints, with applications to quantitative synthesis. In 27th International Conference on Concurrency Theory, CONCUR 2016, August 23–26, 2016, Québec City, Canada. LIPIcs Vol. 59 (eds Desharnais, J. & Jagadeesan, R.) 9:1-9:15 (Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2016). https://doi.org/10.4230/LIPIcs.CONCUR.2016.9 . (PMID: 10.4230/LIPIcs.CONCUR.2016.9)
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Clemente, L. & Raskin, J. Multidimensional beyond worst-case and almost-sure problems for mean-payoff objectives. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6–10, 2015 257–268 (IEEE Computer Society, 2015). https://doi.org/10.1109/LICS.2015.33.
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Chatterjee, K. & Doyen, L. Energy and mean-payoff parity Markov decision processes. In Mathematical Foundations of Computer Science 2011—36th International Symposium, MFCS 2011, Warsaw, Poland, August 22–26, 2011. Proceedings. Lecture Notes in Computer Science Vol. 3907 (eds Murlak, F. & Sankowski, P.) 206–218 (Springer, 2011). https://doi.org/10.1007/978-3-642-22993-0_21 . (PMID: 10.1007/978-3-642-22993-0_21)
López-Barrientos, J. D., Jiménez-Lizárraga, M. & Escobedo-Trujillo, B. A. On the discrete-time minimum principle in multiple-mode systems. Cybern. Syst.[SPACE] https://doi.org/10.1080/01969722.2023.2175492 (2023). (PMID: 10.1080/01969722.2023.2175492)
Zhao, W., Li, R., Liu, W., Dong, W. & Liu, Z. Probabilistic synthesis against GR(1) winning condition. Front. Comput. Sci. 16, 162203. https://doi.org/10.1007/s11704-020-0076-z (2022). (PMID: 10.1007/s11704-020-0076-z)
Entry Date(s): Date Created: 20250117 Latest Revision: 20250121
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PubMed Central ID: PMC11742064
DOI: 10.1038/s41598-025-85589-9
PMID: 39824925
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  Data: A synthesis method for zero-sum mean-payoff asynchronous probabilistic games.
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  Data: <searchLink fieldCode="AU" term="%22Zhao+W%22">Zhao W</searchLink>; The Department of Computer Engineering, Jiangsu University of Technology, Changzhou, 213001, China. zhaowei618@jsut.edu.cn.<br /><searchLink fieldCode="AU" term="%22Liu+W%22">Liu W</searchLink>; The National University of Defense Technology, Changsha, 410073, China.<br /><searchLink fieldCode="AU" term="%22Liu+Z%22">Liu Z</searchLink>; Southwest University, Chongqing, 400715, China. zhimingliu88@swu.edu.cn.<br /><searchLink fieldCode="AU" term="%22Wang+T%22">Wang T</searchLink>; The College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, China.
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  Data: The traditional synthesis problem aims to automatically construct a reactive system (if it exists) satisfying a given Linear Temporal Logic (LTL) specifications, and is often referred to as a qualitative problem. There is also a class of synthesis problems aiming at quantitative properties, such as mean-payoff values, and this type of problem is called a quantitative problem. For the two types of synthesis problems, the research on the former has been relatively mature, and the latter also has received huge amounts of attention. System designers prefer to synthesize systems that satisfy resource constraints. To this end, this paper focuses on the reactive synthesis problem of combining quantitative and qualitative objectives. First, zero-sum mean-payoff asynchronous probabilistic games are proposed, where the system aims at the expected mean payoff in a probabilistic environment while satisfying an LTL winning condition against an adversarial environment. Then, the case of taking the wider class of Generalized Reactivity(1) (GR(1)) formula as an LTL winning condition is studied, that is, the synthesis problem of the expected mean payoffs is studied for the system with the probability of winning. Next, two symbolic algorithms running in polynomial time are proposed to calculate the expected mean payoffs, and both algorithms adopt uniform random strategies. Combining the probability of system winning, the expected mean payoffs of the system when it has the probability of winning is calculated. Finally, our two algorithms are implemented, and their convergence and volatility are demonstrated through experiments.<br /> (© 2025. The Author(s).)
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  Data: Declarations. Competing interests: The authors declare no competing interests.
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  Label: References
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  Data: Henzinger, T. A. Quantitative reactive models. In Model Driven Engineering Languages and Systems—15th International Conference, MODELS 2012, Innsbruck, Austria, September 30–October 5, 2012. Proceedings. Lecture Notes in Computer Science Vol. 7590 (eds France, R. B. et al.) 1–2 (Springer, 2012). https://doi.org/10.1007/978-3-642-33666-9_1 . (PMID: <searchLink fieldCode="PM" term="%2210%2E1007%2F978-3-642-33666-9%26#95%3B1%22">10.1007/978-3-642-33666-9_1)</searchLink><br />Hunter, P., Pauly, A., Pérez, G. A. & Raskin, J. Mean-payoff games with partial observation. Theor. Comput. Sci. 735, 82–110. https://doi.org/10.1016/j.tcs.2017.03.038 (2018). (PMID: <searchLink fieldCode="PM" term="%2210%2E1016%2Fj%2Etcs%2E2017%2E03%2E038%22">10.1016/j.tcs.2017.03.038)</searchLink><br />Chatterjee, K., Doyen, L., Gimbert, H. & Oualhadj, Y. Perfect-information stochastic mean-payoff parity games. In Foundations of Software Science and Computation Structures—17th International Conference, FOSSACS 2014, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2014, Grenoble, France, April 5–13, 2014, Proceedings. Lecture Notes in Computer Science Vol. 8412 (ed. Muscholl, A.) 210–225 (Springer, 2014). https://doi.org/10.1007/978-3-642-54830-7_14 .<br />Velner, Y. et al. The complexity of multi-mean-payoff and multi-energy games. Inf. Comput. 241, 177–196. https://doi.org/10.1016/j.ic.2015.03.001 (2015). (PMID: <searchLink fieldCode="PM" term="%2210%2E1016%2Fj%2Eic%2E2015%2E03%2E001%22">10.1016/j.ic.2015.03.001)</searchLink><br />Brim, L., Chaloupka, J., Doyen, L., Gentilini, R. & Raskin, J. Faster algorithms for mean-payoff games. Formal Methods Syst. Des. 38, 97–118. https://doi.org/10.1007/s10703-010-0105-x (2011). (PMID: <searchLink fieldCode="PM" term="%2210%2E1007%2Fs10703-010-0105-x%22">10.1007/s10703-010-0105-x)</searchLink><br />Björklund, H., Sandberg, S. & Vorobyov, S. G. Memoryless determinacy of parity and mean payoff games: a simple proof. Theor. Comput. Sci. 310, 365–378. https://doi.org/10.1016/S0304-3975(03)00427-4 (2004). (PMID: <searchLink fieldCode="PM" term="%2210%2E1016%2FS0304-3975%2803%2900427-4%22">10.1016/S0304-3975(03)00427-4)</searchLink><br />Zwick, U. & Paterson, M. The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158, 343–359. https://doi.org/10.1016/0304-3975(95)00188-3 (1996). (PMID: <searchLink fieldCode="PM" term="%2210%2E1016%2F0304-3975%2895%2900188-3%22">10.1016/0304-3975(95)00188-3)</searchLink><br />Bruyère, V., Filiot, E., Randour, M. & Raskin, J. Meet your expectations with guarantees: Beyond worst-case synthesis in quantitative games. Inf. Comput. 254, 259–295. https://doi.org/10.1016/j.ic.2016.10.011 (2017). (PMID: <searchLink fieldCode="PM" term="%2210%2E1016%2Fj%2Eic%2E2016%2E10%2E011%22">10.1016/j.ic.2016.10.011)</searchLink><br />Church, A. Logic, arithmetic, and automata. J. Symb. Logic 29, 210 (1964).<br />Pnueli, A. & Rosner, R. On the synthesis of an asynchronous reactive module. In Automata, Languages and Programming, 16th International Colloquium, ICALP89, Stresa, Italy, July 11–15, 1989, Proceedings. Lecture Notes in Computer Science Vol. 372 (eds Ausiello, G. et al.) 652–671 (Springer, 1989). https://doi.org/10.1007/BFb0035790 . (PMID: <searchLink fieldCode="PM" term="%2210%2E1007%2FBFb0035790%22">10.1007/BFb0035790)</searchLink><br />Wallmeier, N., Hütten, P. & Thomas, W. Symbolic synthesis of finite-state controllers for request-response specifications. In Implementation and Application of Automata, 8th International Conference, CIAA 2003, Santa Barbara, California, USA, July 16–18, 2003, Proceedings. Lecture Notes in Computer Science Vol. 2759 (eds Ibarra, O. H. & Dang, Z.) 11–22 (Springer, 2003). https://doi.org/10.1007/3-540-45089-0_3 . (PMID: <searchLink fieldCode="PM" term="%2210%2E1007%2F3-540-45089-0%26#95%3B3%22">10.1007/3-540-45089-0_3)</searchLink><br />Alur, R. & Torre, S. L. Deterministic generators and games for ltl fragments. ACM Trans. Comput. Log. 5, 1–25. https://doi.org/10.1145/963927.963928 (2004). (PMID: <searchLink fieldCode="PM" term="%2210%2E1145%2F963927%2E963928%22">10.1145/963927.963928)</searchLink><br />Harding, A., Ryan, M. & Schobbens, P. A new algorithm for strategy synthesis in LTL games. In Tools and Algorithms for the Construction and Analysis of Systems, 11th International Conference, TACAS 2005, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2005, Edinburgh, UK, April 4–8, 2005, Proceedings. Lecture Notes in Computer Science Vol. 3440 (eds Halbwachs, N. & Zuck, L. D.) 477–492 (Springer, 2005). https://doi.org/10.1007/978-3-540-31980-1_31 .<br />Jobstmann, B., Griesmayer, A. & Bloem, R. Program repair as a game. In Computer Aided Verification, 17th International Conference, CAV 2005, Edinburgh, Scotland, UK, July 6–10, 2005, Proceedings. Lecture Notes in Computer Science Vol. 3576 (eds Etessami, K. & Rajamani, S. K.) 226–238 (Springer, 2005). https://doi.org/10.1007/11513988_23 . (PMID: <searchLink fieldCode="PM" term="%2210%2E1007%2F11513988%26#95%3B23%22">10.1007/11513988_23)</searchLink><br />Bloem, R., Jobstmann, B., Piterman, N., Pnueli, A. & Sa’ar, Y. Synthesis of reactive(1) designs. J. Comput. Syst. Sci. 78, 911–938. https://doi.org/10.1016/j.jcss.2011.08.007 (2012). (PMID: <searchLink fieldCode="PM" term="%2210%2E1016%2Fj%2Ejcss%2E2011%2E08%2E007%22">10.1016/j.jcss.2011.08.007)</searchLink><br />Ehrenfeucht, A. & Mycielski, J. Positional strategies for mean payoff games. Int. J. Game Theory 8, 109–113 (1979). (PMID: <searchLink fieldCode="PM" term="%2210%2E1007%2FBF01768705%22">10.1007/BF01768705)</searchLink><br />Ummels, M. & Wojtczak, D. The complexity of Nash equilibria in limit-average games. In CONCUR 2011—Concurrency Theory—22nd International Conference, CONCUR 2011, Aachen, Germany, September 6–9, 2011. Proceedings. Lecture Notes in Computer Science Vol. 6901 (eds Katoen, J. & König, B.) 482–496 (Springer, 2011). https://doi.org/10.1007/978-3-642-23217-6_32 . (PMID: <searchLink fieldCode="PM" term="%2210%2E1007%2F978-3-642-23217-6%26#95%3B32%22">10.1007/978-3-642-23217-6_32)</searchLink><br />Bouyer, P., Brenguier, R. & Markey, N. Nash equilibria for reachability objectives in multi-player timed games. In CONCUR 2010—Concurrency Theory, 21st International Conference, CONCUR 2010, Paris, France, August 31–September 3, 2010. Proceedings. Lecture Notes in Computer Science Vol. 6269 (eds Gastin, P. & Laroussinie, F.) 192–206 (Springer, 2010). https://doi.org/10.1007/978-3-642-15375-4_14 . (PMID: <searchLink fieldCode="PM" term="%2210%2E1007%2F978-3-642-15375-4%26#95%3B14%22">10.1007/978-3-642-15375-4_14)</searchLink><br />Chatterjee, K., Henzinger, T. A. & Jurdzinski, M. Mean-payoff parity games. In 20th IEEE Symposium on Logic in Computer Science (LICS 2005), 26-29 June 2005, Chicago, IL, USA, Proceedings 178–187 (IEEE Computer Society, 2005). https://doi.org/10.1109/LICS.2005.26.<br />Chatterjee, K., Komárková, Z. & Kretínský, J. Unifying two views on multiple mean-payoff objectives in Markov decision processes. In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, July 6–10, 2015 244–256 (IEEE Computer Society, 2015). https://doi.org/10.1109/LICS.2015.32.<br />Almagor, S., Kupferman, O. & Velner, Y. 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