Formal Methods in System Design

, Volume 39, Issue 3, pp 261–296 | Cite as

Antichains and compositional algorithms for LTL synthesis

  • Emmanuel Filiot
  • Naiyong Jin
  • Jean-François Raskin
Article

Abstract

In this paper, we present new monolithic and compositional algorithms to solve the LTL realizability problem. Those new algorithms are based on a reduction of the LTL realizability problem to a game whose winning condition is defined by a universal automaton on infinite words with a k-co-Büchi acceptance condition. This acceptance condition asks that runs visit at most k accepting states, so it implicitly defines a safety game. To obtain efficient algorithms from this construction, we need several additional ingredients. First, we study the structure of the underlying automata constructions, and we show that there exists a partial order that structures the state space of the underlying safety game. This partial order can be used to define an efficient antichain algorithm. Second, we show that the algorithm can be implemented in an incremental way by considering increasing values of k in the acceptance condition. Finally, we show that for large LTL formulas that are written as conjunctions of smaller formulas, we can solve the problem compositionally by first computing winning strategies for each conjunct that appears in the large formula. We report on the behavior of those algorithms on several benchmarks. We show that the compositional algorithms are able to handle LTL formulas that are several pages long.

Keywords

LTL realizability and synthesis Automata on infinite words Compositional algorithms Antichain algorithms 

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References

  1. 1.
    Abadi M, Lamport L, Wolper P (1989) Realizable and unrealizable specifications of reactive systems. In: Proceedings of 16th international colloquium on automata, languages, and programming (ICALP). Lecture notes in computer science, vol 372. Springer, Berlin, pp 1–17 CrossRefGoogle Scholar
  2. 2.
    Alur R, Henzinger TA, Kupferman O, Vardi MY (1998) Alternating refinement relations. In: Proceedings of the 9th international conference on concurrency theory (CONCUR). Lecture notes in computer science, vol 1466. Springer, Berlin, pp 163–178 Google Scholar
  3. 3.
    Bloem R, Galler S, Jobstmann B, Piterman N, Pnueli A, Weiglhofer M (2007) Specify compile, run: hardware from psl. Electron Notes Theor Comput Sci 190(4):3–16 CrossRefGoogle Scholar
  4. 4.
    Boker U, Kupferman O (2009) Co-ing büchi made tight and useful. In: Proceedings of the 24th IEEE annual symposium on logic in computer science (LICS). IEEE Computer Society, New York, pp 245–254 Google Scholar
  5. 5.
    Cassez F, David A, Fleury E, Larsen KG, Lime D (2005) Efficient on-the-fly algorithms for the analysis of timed games. In: Proceedings of the 16th international conference on concurrency theory (CONCUR). Lecture notes in computer science, vol 3653. Springer, Berlin, pp 66–80 Google Scholar
  6. 6.
    De Wulf M, Doyen L, Henzinger TA, Raskin JF (2006) Antichains: a new algorithm for checking universality of finite automata. In: Proceedings of the 18th international conference on computer aided verification (CAV). Lecture notes in computer science, vol 4144. Springer, Berlin, pp 17–30 CrossRefGoogle Scholar
  7. 7.
    Doyen L, Raskin JF (2007) Improved algorithms for the automata-based approach to model-checking. In: Proceedings of the 13th international conference on tools and algorithms for the construction and analysis of systems (TACAS). Lecture notes in computer science, vol 4424. Springer, Berlin, pp 451–465 CrossRefGoogle Scholar
  8. 8.
    Doyen L, Raskin JF (2010) Antichain algorithms for finite automata. In: Proceedings of the 16th international conference on tools and algorithms for the construction and analysis of systems (TACAS). Lecture notes in computer science, vol 6015. Springer, Berlin, pp 2–22 CrossRefGoogle Scholar
  9. 9.
    Ehlers R (2010) Symbolic bounded synthesis. In: Proceedings of the 22nd international conference on computer aided verification (CAV). Lecture notes in computer science, vol 6174. Springer, Berlin, pp 365–379 CrossRefGoogle Scholar
  10. 10.
    Filiot E, Jin N, Raskin JF (2009) An antichain algorithm for LTL realizability. In: Proceedings of the 21st international conference on computer aided verification (CAV). Lecture notes in computer science, vol 5643. Springer, Berlin, pp 263–277 CrossRefGoogle Scholar
  11. 11.
    Filiot E, Jin N, Raskin JF (2010) Compositional algorithms for LTL synthesis. In: Proceedings of the 8th international symposium on automated technology for verification and analysis (ATVA). Lecture notes in computer science, vol 6252. Springer, Berlin, pp 122–127 Google Scholar
  12. 12.
    Grädel E, Thomas W, Wilke T (2002) Automata, logics, and infinite games: a guide to current research. Lecture notes in computer science, vol 2500. Springer, Berlin CrossRefMATHGoogle Scholar
  13. 13.
    Greimel K, Bloem R, Jobstmann B, Vardi MY (2008) Open implication. In: Proceedings of the 35th international colloqium on automata, languages and programming (ICALP). Lecture notes in computer science, vol 5126. Springer, Berlin, pp 361–372 CrossRefGoogle Scholar
  14. 14.
  15. 15.
    Jobstmann B, Bloem R (2006) Optimizations for LTL synthesis. In: Proceedings of the 6th international conference on formal methods in computer aided design (FMCAD). IEEE Computer Society, New York, pp 117–124 Google Scholar
  16. 16.
    Kuijper W, van de Pol J (2009) Compositional control synthesis for partially observable systems. In: Proceedings of the 20th international conference on concurrency theory (CONCUR). Lecture notes in computer science, vol 5710. Springer, Berlin, pp 431–447 Google Scholar
  17. 17.
    Kupferman O, Piterman N, Vardi MY (2006) Safraless compositional synthesis. In: Proceedings of the 18th international conference on computer aided verification (CAV). Lecture notes in computer science, vol 4144. Springer, Berlin, pp 31–44 CrossRefGoogle Scholar
  18. 18.
    Kupferman O, Vardi MY (2001) On bounded specifications. In: Proceedings of the 8th international conference on logic for programming, artificial intelligence, and reasoning (LPAR). Lecture notes in computer science, vol 2250. Springer, Berlin, pp 24–38 CrossRefGoogle Scholar
  19. 19.
    Kupferman O, Vardi MY (2005) Safraless decision procedures. In: Proceedings of the IEEE symposium on foundations of computer science (FOCS). IEEE Computer Society, New York, pp 531–542 Google Scholar
  20. 20.
    Liu X, Smolka SA (1998) Simple linear-time algorithms for minimal fixed points. In: Proceedings of the 25th international colloqium on automata, languages and programming (ICALP). Lecture notes in computer science, vol 1443. Springer, Berlin, pp 53–66 CrossRefGoogle Scholar
  21. 21.
    Martin D (1975) Borel determinacy. Ann Math 102:363–371 CrossRefMATHGoogle Scholar
  22. 22.
    Piterman N (2007) From nondeterministic büchi and streett automata to deterministic parity automata. Log Methods Comput Sci 3(3) Google Scholar
  23. 23.
    Piterman N, Pnueli A, Sa’ar Y (2006) Synthesis of reactive(1) designs. In: Proceedings of the 7th international conference on verification, model checking, and abstract interpretation (VMCAI). Lecture notes in computer science, vol 3855. Springer, Berlin, pp 364–380 CrossRefGoogle Scholar
  24. 24.
    Pnueli A, Rosner R (1989) On the synthesis of a reactive module. In: Proceedings of the ACM symposium on principles of programming languages (POPL). ACM, New York, pp 179–190 Google Scholar
  25. 25.
    Raskin JF, Chatterjee K, Doyen L, Henzinger TA (2007) Algorithms for omega-regular games with imperfect information. Log Methods Comput Sci 3(3) Google Scholar
  26. 26.
    Rosner R (1992) Modular synthesis of reactive systems. PhD dissertation, Weizmann Institute of Science Google Scholar
  27. 27.
    Safra S (1988) On the complexity of ω automata. In: Proceedings of the IEEE symposium on foundations of computer science (FOCS). IEEE Computer Society, New York, pp 319–327 Google Scholar
  28. 28.
    Schewe S, Finkbeiner B (2007) Bounded synthesis. In: Proceedings of the 5th international symposium on automated technology for verification and analysis (ATVA). Lecture notes in computer science, vol 4762. Springer, Berlin, pp 474–488 CrossRefGoogle Scholar
  29. 29.
    Sohail S, Somenzi F (2009) Safety first: a two-stage algorithm for LTL games. In: Proceedings of the 9th international conference on formal methods in computer aided design (FMCAD). IEEE Computer Society, New York, pp 77–84 Google Scholar
  30. 30.
    Somenzi F, Bloem R (2000) Efficient büchi automata from LTL formulae. In: Proceedings of the 12th international conference on computer aided verification (CAV). Lecture notes in computer science, vol 1855. Springer, Berlin, pp 248–263 CrossRefGoogle Scholar
  31. 31.
    Thomas W (2008) Church’s problem and a tour through automata theory. In: Pillars of Computer Science. Lecture notes in computer science, vol 4800. Springer, Berlin, pp 635–655 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Emmanuel Filiot
    • 1
  • Naiyong Jin
    • 1
  • Jean-François Raskin
    • 1
  1. 1.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium

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