Abstract
Following our recent development of a compositional model checking algorithm for Markov decision processes, we present a compositional framework for solving mean payoff games (MPGs). The framework is derived from category theory: MPGs (extended with open ends) get composed in so-called string diagrams and thus organized in a monoidal category; their solution is then expressed as a functor, whose preservation properties embody compositionality. The key question to compositionality is how to enrich the semantic domain; the categorical framework gives an informed guidance in solving the question by singling out the algebraic structure required in the extended semantic domain. We implemented our compositional solution in Haskell; depending on benchmarks, it can outperform an existing algorithm by an order of magnitude.
We thank to Massimo Benerecetti, Daniele Dell’Erba, and Fabio Mogavero for kindly sharing their implementation. We thank the reviewers for the previous versions of the paper for their useful comments. K.W., C.E., and I.H. are supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603) and the ASPIRE grant No. JPMJAP2301, JST. K.W. is supported by the JST grants No. JPMJFS2136 and JPMJAX23CU.
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it is available at https://github.com/Kazuuuuuki/compMPG.
References
Alur, R., Stanford, C., Watson, C.: A robust theory of series parallel graphs. Proc. ACM Program. Lang. 7(POPL), 1058–1088 (2023). https://doi.org/10.1145/3571230DOI
Benerecetti, M., Dell’Erba, D., Mogavero, F.: Solving mean-payoff games via quasi dominions. In: TACAS 2020. LNCS, vol. 12079, pp. 289–306. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45237-7_18
Benerecetti, M., Dell’Erba, D., Mogavero, F.: Solving parity games via priority promotion. Formal Methods Syst. Des. 52(2), 193–226 (2018)
Bonchi, F., Holland, J., Piedeleu, R., Sobocinski, P., Zanasi, F.: Diagrammatic algebra: from linear to concurrent systems. Proc. ACM Program. Lang. 3(POPL), 25:1–25:28 (2019). https://doi.org/10.1145/3290338
Brim, L., Chaloupka, J., Doyen, L., Gentilini, R., Raskin, J.: Faster algorithms for mean-payoff games. Formal Methods Syst. Des. 38(2), 97–118 (2011). https://doi.org/10.1007/s10703-010-0105-x
Chakrabarti, A., de Alfaro, L., Henzinger, T.A., Stoelinga, M.: Resource Interfaces. In: Alur, R., Lee, I. (eds.) EMSOFT 2003. LNCS, vol. 2855, pp. 117–133. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45212-6_9
Clarke, E.M., Long, D.E., McMillan, K.L.: Compositional model checking. In: Proceedings of the Fourth Annual Symposium on Logic in Computer Science (LICS ’89), Pacific Grove, California, USA, June 5-8, 1989. pp. 353–362. IEEE Computer Society (1989). https://doi.org/10.1109/LICS.1989.39190
Comin, C., Posenato, R., Rizzi, R.: Hyper temporal networks - A tractable generalization of simple temporal networks and its relation to mean payoff games. Constraints An Int. J. 22(2), 152–190 (2017). https://doi.org/10.1007/s10601-016-9243-0
Comin, C., Rizzi, R.: Dynamic consistency of conditional simple temporal networks via mean payoff games: A singly-exponential time dc-checking. In: Grandi, F., Lange, M., Lomuscio, A. (eds.) 22nd International Symposium on Temporal Representation and Reasoning, TIME 2015, Kassel, Germany, September 23-25, 2015. pp. 19–28. IEEE Computer Society (2015). https://doi.org/10.1109/TIME.2015.18
Cruttwell, G.S.: Normed spaces and the change of base for enriched categories. Ph.D. thesis, Dalhousie University (2008)
Duffin, R.J.: Topology of series-parallel networks. J. Math. Anal. Appl. 10(2), 303–318 (1965)
Ehrenfeucht, A., Mycielski, J.: Positional strategies for mean payoff games. Internat. J. Game Theory 8(2), 109–113 (1979)
Eilenberg, S., Kelly, G.M.: Closed categories. In: Eilenberg, S., Harrison, D.K., MacLane, S., Röhrl, H. (eds.) Proceedings of the Conference on Categorical Algebra, pp. 421–562. Springer Berlin Heidelberg, Berlin, Heidelberg (1966)
Eppstein, D.: Parallel recognition of series-parallel graphs. Inf. Comput. 98(1), 41–55 (1992)
Friedmann, O., Lange, M.: Solving parity games in practice. In: Liu, Z., Ravn, A.P. (eds.) ATVA 2009. LNCS, vol. 5799, pp. 182–196. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04761-9_15
Grellois, C., Melliès, P.A.: Finitary semantics of linear logic and higher-order model-checking. In: International Symposium on Mathematical Foundations of Computer Science, pp. 256–268. Springer (2015)
Heunen, C., Vicary, J.: Categories for Quantum Theory: an introduction. Oxford University Press (2019)
Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Cambridge Philos. Soc. 119(3), 447–468 (1996)
Jurdziński, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-46541-3_24
Kelly, G., Laplaza, M.: Coherence for compact closed categories. J. Pure Appl. Algebra 19, 193–213 (1980). https://doi.org/10.1016/0022-4049(80)90101-2. https://www.sciencedirect.com/science/article/pii/0022404980901012
Khovanov, M.: A functor-valued invariant of tangles. Algebraic Geometric Topol. 2(2), 665–741 (2002)
Kwiatkowska, M.Z., Norman, G., Parker, D., Qu, H.: Compositional probabilistic verification through multi-objective model checking. Inf. Comput. 232, 38–65 (2013). https://doi.org/10.1016/j.ic.2013.10.001
Mac Lane, S.: Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer-Verlag, New York (1978)
Moggi, E.: Notions of computation and monads. Inf. Comput. 93(1), 55–92 (1991). https://doi.org/10.1016/0890-5401(91)90052-4
Piedeleu, R., Kartsaklis, D., Coecke, B., Sadrzadeh, M.: Open system categorical quantum semantics in natural language processing. In: Moss, L.S., Sobocinski, P. (eds.) 6th Conference on Algebra and Coalgebra in Computer Science, CALCO 2015, June 24-26, 2015, Nijmegen, The Netherlands. LIPIcs, vol. 35, pp. 270–289. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015). https://doi.org/10.4230/LIPIcs.CALCO.2015.270
Rathke, J., Sobocinski, P., Stephens, O.: Compositional reachability in Petri nets. In: Ouaknine, J., Potapov, I., Worrell, J. (eds.) Reachability Problems - 8th International Workshop, RP 2014, Oxford, UK, September 22-24, 2014. Proceedings. Lecture Notes in Computer Science, vol. 8762, pp. 230–243. Springer (2014). https://doi.org/10.1007/978-3-319-11439-2_18
Stephens, O.: Compositional specification and reachability checking of net systems. Ph.D. thesis, University of Southampton, UK (2015), http://eprints.soton.ac.uk/385201/
Tomita, T., Ueno, A., Shimakawa, M., Hagihara, S., Yonezaki, N.: Safraless LTL synthesis considering maximal realizability. Acta Informatica 54(7), 655–692 (2017). https://doi.org/10.1007/s00236-016-0280-3
Tsukada, T., Ong, C.L.: Compositional higher-order model checking via \(\omega \)-regular games over Böhm trees. In: Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS ’14, Vienna, Austria, July 14 - 18, 2014. pp. 78:1–78:10. ACM (2014)
Watanabe, K., Eberhart, C., Asada, K., Hasuo, I.: A compositional approach to parity games. In: Sokolova, A. (ed.) Proceedings 37th Conference on Mathematical Foundations of Programming Semantics, MFPS 2021, Hybrid: Salzburg, Austria and Online, 30th August - 2nd September, 2021. EPTCS, vol. 351, pp. 278–295 (2021) https://doi.org/10.4204/EPTCS.351.17
Watanabe, K., Eberhart, C., Asada, K., Hasuo, I.: Compositional probabilistic model checking with string diagrams of MDPs. In: Enea, C., Lal, A. (eds.) Computer Aided Verification - 35th International Conference, CAV 2023, Paris, France, July 17-22, 2023, Proceedings, Part III. Lecture Notes in Computer Science, vol. 13966, pp. 40–61. Springer (2023). https://doi.org/10.1007/978-3-031-37709-9_3
Watanabe, K., Eberhart, C., Asada, K., Hasuo, I.: Compositional solution of mean payoff games by string diagrams. CoRR abs/2307.08034 (2023)
Watanabe, K., van der Vegt, M., Hasuo, I., Rot, J., Junges, S.: Pareto curves for compositionally model checking string diagrams of MDPs. In: TACAS (2). LNCS, vol. 14571, pp. 279–298. Springer (2024)
Watanabe, K., van der Vegt, M., Junges, S., Hasuo, I.: Compositional value iteration with pareto caching. In: CAV (3). LNCS, vol. 14683, pp. 467–491. Springer (2024)
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To Joost-Pieter Katoen on his 60th birthday. This paper comes from our recent endeavor to use the language of category theory for a general account of, and efficient algorithms for, model checking. In this endeavor, the beautiful theory of probabilistic model checking developed by Joost-Pieter and his coauthors has been the main source of inspiration. Comparing the theory with the conventional (non-probabilistic) model checking, we have learned what theoretical constructs are essential and thus amenable to categorical abstraction. We have also been thrilled to work with the gang of (former and present) students of Joost-Pieter—they are not only excellent researchers but also have open and fun-loving minds. We thus dedicate this paper to Joost-Pieter’s heritage, in scientific work and in people.
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Watanabe, K., Eberhart, C., Asada, K., Hasuo, I. (2025). Compositional Solution of Mean Payoff Games by String Diagrams. In: Jansen, N., et al. Principles of Verification: Cycling the Probabilistic Landscape . Lecture Notes in Computer Science, vol 15262. Springer, Cham. https://doi.org/10.1007/978-3-031-75778-5_20
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