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Formal Methods in System Design

, Volume 52, Issue 2, pp 193–226 | Cite as

Solving parity games via priority promotion

  • Massimo Benerecetti
  • Daniele Dell’Erba
  • Fabio MogaveroEmail author
Article

Abstract

We consider parity games, a special form of two-player infinite-duration games on numerically labeled graphs, whose winning condition requires that the maximal value of a label occurring infinitely often during a play be of some specific parity. The problem of identifying the corresponding winning regions has a rather intriguing status from a complexity theoretic viewpoint, since it belongs to the class \({\textsc {UPTime}} \cap {\textsc {CoUPTime}}\), and still open is the question whether it can be solved in polynomial time. Parity games also have great practical interest, as they arise in many fields of theoretical computer science, most notably logic, automata theory, and formal verification. In this paper, we propose a new algorithm for the solution of this decision problem, based on the idea of promoting vertexes to higher priorities during the search for winning regions. The proposed approach has nice computational properties, exhibiting the best space complexity among the currently known solutions. Experimental results on both random games and benchmark families show that the technique is also very effective in practice.

Keywords

Parity games Infinite-duration games on graphs Algorithmic complexity Formal methods 

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References

  1. 1.
    Agrawal M, Kayal N, Saxena N (2004) PRIMES is in P. Ann Math 160(2):781–793MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alur R, Henzinger TA, Kupferman O (2002) Alternating-time temporal logic. J ACM 49(5):672–713MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Apt K, Grädel E (2011) Lectures in game theory for computer scientists. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  4. 4.
    Benerecetti M, Dell’Erba D, Mogavero F (2016) Solving parity games via priority promotion. In Computer aided verification’16, LNCS 9780 (Part II). Springer, New York, pp 270–290Google Scholar
  5. 5.
    Benerecetti M, Mogavero F, Murano A (2013) Substructure temporal logic. In: Logic in computer science’13. IEEE Computer Society, pp 368–377Google Scholar
  6. 6.
    Benerecetti M, Mogavero F, Murano A (2015) Reasoning about substructures and games. Trans Comput Log 16(3):25:1–25:46MathSciNetzbMATHGoogle Scholar
  7. 7.
    Berwanger D, Dawar A, Hunter P, Kreutzer S (2006) DAG-width and parity games. In: Symposium on theoretical aspects of computer science’06, LNCS 3884. Springer, New York, pp 524–536Google Scholar
  8. 8.
    Berwanger D, Grädel E (2001) Games and model checking for guarded logics. In: Logic for programming artificial intelligence and reasoning’01, LNCS 2250. Springer, New York, pp 70–84Google Scholar
  9. 9.
    Berwanger D, Grädel E (2004) Fixed-point logics and solitaire games. Theor Comput Sci 37(6):675–694MathSciNetzbMATHGoogle Scholar
  10. 10.
    Berwanger D, Grädel E, Kaiser L, Rabinovich R (2012) Entanglement and the complexity of directed graphs. Theor Comput Sci 463:2–25MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Calude CS, Jain S, Khoussainov B, Li W, Stephan F (2017) Deciding parity games in quasipolynomial time. In: Symposium on theory of computing’17. Association for Computing Machinery, pp 252–263Google Scholar
  12. 12.
    Chatterjee K, Doyen L (2012) Energy parity games. Theor Comput Sci 458:49–60MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chatterjee K, Doyen L, Henzinger TA, Raskin J-F (2010) Generalized mean-payoff and energy games. In Foundations of software technology and theoretical computer science’10, LIPIcs 8. Leibniz-Zentrum fuer Informatik, pp 505–516Google Scholar
  14. 14.
    Chatterjee K, Henzinger TA, Horn F (2010) Finitary winning in omega-regular games. Trans Comput Log 11(1):1:1–1:26zbMATHGoogle Scholar
  15. 15.
    Chatterjee K, Henzinger TA, Jurdziński M (2005) Mean-payoff parity games. In: Logic in computer science’05. IEEE Computer Society, pp 178–187Google Scholar
  16. 16.
    Chatterjee K, Henzinger TA, Piterman N (2010) Strategy logic. Inf Comput 208(6):677–693MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Condon A (1992) The complexity of stochastic games. Inf Comput 96(2):203–224MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ehrenfeucht A, Mycielski J (1979) Positional strategies for mean payoff games. Int J Game Theory 8(2):109–113MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Emerson EA, Jutla CS (1991) Tree automata, mu-calculus, and determinacy. In: Foundation of computer science’91. IEEE Computer Society, pp 368–377Google Scholar
  20. 20.
    Emerson EA, Jutla CS, Sistla AP (1993) On model checking for the mu-calculus and its fragments. In: Computer aided verification’93, LNCS 697. Springer, New York, pp 385–396Google Scholar
  21. 21.
    Emerson EA, Jutla CS, Sistla AP (2001) On model checking for the \(\mu \)-calculus and its fragments. Theor Comput Sci 258(1–2):491–522MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Emerson EA, Lei C-L (1986) Temporal reasoning under generalized fairness constraints. In: Symposium on theoretical aspects of computer science’86, LNCS 210. Springer, New York, pp 267–278Google Scholar
  23. 23.
    Fearnley J (2010) Non-oblivious strategy improvement. In: Logic for programming artificial intelligence and reasoning’10, LNCS 6355. Springer, New York, pp 212–230Google Scholar
  24. 24.
    Fearnley J, Jain S, Schewe S, Stephan F, Wojtczak D (2017) An ordered approach to solving parity games in quasi polynomial time and quasi linear space. In: SPIN symposium on model checking of software’2017. Association for Computing Machinery, pp 112–121Google Scholar
  25. 25.
    Fearnley J, Lachish O (2011) Parity games on graphs with medium tree-width. In: Mathematical foundations of computer science’11, LNCS 6907. Springer, New York, pp 303–314Google Scholar
  26. 26.
    Fearnley J, Schewe S (2012) Time and parallelizability results for parity games with bounded treewidth. In: International colloquium on automata, languages, and programming’12, LNCS 7392. Springer, pp 189–200Google Scholar
  27. 27.
    Fellows MR, Koblitz N (1992) Self-witnessing polynomial-time complexity and prime factorization. In: Conference on structure in complexity theory’92. IEEE Computer Society, pp 107–110Google Scholar
  28. 28.
    Fellows MR, Koblitz N (1992) Self-witnessing polynomial-time complexity and prime factorization. Des Codes Crypt 2(3):231–235MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Fijalkow N, Zimmermann M (2012) Cost-parity and cost-streett games. In: Foundations of software technology and theoretical computer science’12, LIPIcs 18. Leibniz-Zentrum fuer Informatik, pp 124–135Google Scholar
  30. 30.
    Fijalkow N, Zimmermann M (2014) Cost-parity and cost-streett games. Log Methods Comput Sci 10(2):1–29MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Friedmann O, Lange M (2009) Solving parity games in practice. In: Automated technology for verification and analysis’09, LNCS 5799. Springer, pp 182–196Google Scholar
  32. 32.
    Grädel E, Thomas W, Wilke T (2002) Automata, logics, and infinite games: a guide to current research. LNCS 2500. Springer, New YorkGoogle Scholar
  33. 33.
    Gurvich VA, Karzanov AV, Khachivan LG (1990) Cyclic games and an algorithm to find minimax cycle means in directed graphs. USSR Comput Math Math Phys 28(5):85–91CrossRefzbMATHGoogle Scholar
  34. 34.
    Jurdziński M (1998) Deciding the winner in parity games is in UP \(\cap \) co-UP. Inf Process Lett 68(3):119–124MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Jurdziński M (2000) Small progress measures for solving parity games. In: Symposium on theoretical aspects of computer science’00, LNCS 1770. Springer, pp 290–301Google Scholar
  36. 36.
    Jurdziński M, Lazic R (2017) Succinct progress measures for solving parity games. In: Logic in computer science’17. Association for Computing Machinery. Accepted for publication, pp 1–9Google Scholar
  37. 37.
    Jurdziński M, Paterson M, Zwick U (2006) A deterministic subexponential algorithm for solving parity games. In: Symposium on discrete algorithms’06. Society for Industrial and Applied Mathematics, pp 117–123Google Scholar
  38. 38.
    Jurdziński M, Paterson M, Zwick U (2008) A Deterministic Subexponential Algorithm for Solving Parity Games. SIAM J Comput 38(4):1519–1532MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Klarlund N, Kozen D (1991) Rabin measures and their applications to fairness and automata theory. In: Logic in computer science’91. IEEE Computer Society, pp 256–265Google Scholar
  40. 40.
    Kupferman O, Vardi MY (1998) Weak alternating automata and tree automata emptiness. In: Symposium on theory of computing’98. Association for Computing Machinery, pp 224–233Google Scholar
  41. 41.
    Martin AD (1975) Borel determinacy. Ann Math 102(2):363–371MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Martin AD (1985) A purely inductive proof of Borel determinacy. In: Symposia in pure mathematics’82, recursion theory. American Mathematical Society and Association for Symbolic Logic, pp 303–308Google Scholar
  43. 43.
    McNaughton R (1993) Infinite games played on finite graphs. Ann Pure Appl Log 65:149–184MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Mogavero F, Murano A, Perelli G, Vardi MY (2012) What makes ATL* decidable? A decidable fragment of strategy logic. In: Concurrency theory’12, LNCS 7454. Springer, Berlin, pp 193–208Google Scholar
  45. 45.
    Mogavero F, Murano A, Perelli G, Vardi MY (2014) Reasoning about strategies: on the model-checking problem. Trans Comput Log 15(4):341–3442MathSciNetzbMATHGoogle Scholar
  46. 46.
    Mogavero F, Murano A, Sorrentino L (2013) On promptness in parity games. In: Logic for programming artificial intelligence and reasoning’13, LNCS 8312. Springer, New York, pp 601–618Google Scholar
  47. 47.
    Mogavero F, Murano A, Vardi MY (2010) Reasoning about strategies. In: Foundations of software technology and theoretical computer science’10, LIPIcs 8. Leibniz-Zentrum fuer Informatik, pp 133–144Google Scholar
  48. 48.
    Mostowski AW (1984) Regular expressions for infinite trees and a standard form of automata. In: Symposium on computation theory’84, LNCS 208. Springer, New York, pp 157–168Google Scholar
  49. 49.
    Mostowski AW (1991) Games with forbidden positions. Technical report, University of Gdańsk, Gdańsk, PolandGoogle Scholar
  50. 50.
    Obdrzálek J (2003) Fast mu-calculus model checking when tree-width is bounded. In: Computer aided verification’03, LNCS 2725. Springer, New York, pp 80–92Google Scholar
  51. 51.
    Obdrzálek J (2007) Clique-width and parity games. In: Computer science logic’07, LNCS 4646. Springer, New York, pp 54–68Google Scholar
  52. 52.
    Schewe S (2007) Solving parity games in big steps. In: Foundations of software technology and theoretical computer science’07, LNCS 4855. Springer, New York, pp 449–460Google Scholar
  53. 53.
    Schewe S (2008) An optimal strategy improvement algorithm for solving parity and payoff games. In: Computer science logic’08, LNCS 5213. Springer, New York, pp 369–384Google Scholar
  54. 54.
    Schewe S (2008) ATL* satisfiability is 2EXPTIME-complete. In: International colloquium on automata, languages, and programming’08, LNCS 5126. Springer, New York, pp 373–385Google Scholar
  55. 55.
    Schewe S, Finkbeiner B (2006) Satisfiability and finite model property for the alternating-time \(\mu \)-calculus. In: Computer science logic’06, LNCS 6247. Springer, New York, pp 591–605Google Scholar
  56. 56.
    Vöge J, Jurdziński M (2000) A discrete strategy improvement algorithm for solving parity games. In: Computer aided verification’00, LNCS 1855. Springer, New York, pp 202–215Google Scholar
  57. 57.
    Wilke T (2001) Alternating tree automata, parity games, and modal \(\mu \)-calculus. Bull Belg Math Soc 8(2):359–391MathSciNetzbMATHGoogle Scholar
  58. 58.
    Zielonka W (1998) Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor Comput Sci 200(1–2):135–183MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Zwick U, Paterson M (1996) The complexity of mean payoff games on graphs. Theor Comput Sci 158(1–2):343–359MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Università degli Studi di Napoli Federico INaplesItaly
  2. 2.University of OxfordOxfordUK

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