Bisimulations enjoy numerous applications in the analysis of labeled transition systems. Many of these applications are based on two central observations: first, bisimilar systems satisfy the same branching-time properties; second, bisimilarity can be checked efficiently for finite-state systems. The local character of bisimulation, however, makes it difficult to address liveness concerns. Indeed, the definitions of fair bisimulation that have been proposed in the literature sacrifice locality, and with it, also efficient checkability. We put forward a new definition of fair bisimulation which does not suffer from this drawback.

The bisimilarity of two systems can be viewed in terms of a game played between a protagonist and an adversary. In each step of the infinite bisimulation game, the adversary chooses one system, makes a move, and the protagonist matches it with a move of the other system. Consistent with this game-based view, we call two fair transition systems bisimilar if in the bisimulation game, the infinite path produced in the first system is fair iff the infinite path produced in the second system is fair.

We show that this notion of fair bisimulation enjoys the following properties. First, fairly bisimilar systems satisfy the same formulas of the logics Fair-AFMC (the fair alternation-free μ-calculus) and Fair-CTL*. Therefore, fair bisimulations can serve as property-preserving abstractions for these logics and weaker ones, such as Fair-CTL and LTL. Indeed, Fair-AFMC provides an exact logical characterization of fair bisimilarity. Second, it can be checked in time polynomial in the number of states if two systems are fairly bisimilar. This is in stark contrast to all trace-based equivalences, which are traditionally used for addressing liveness but require exponential time for checking.


Binary Relation Winning Strategy Label Transition System Kripke Structure Query Logic 
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  1. ASB+94._A. Aziz, V. Singhal, F. Balarin, R.K. Brayton, and A.L. Sangiovanni-Vincentelli. Equivalences for fair kripke structures. In ICALP 94: Automata, Languages, and Programming, Lecture Notes in Computer Science 820, pages 364–375. Springer-Verlag, 1994. 300, 303, 304Google Scholar
  2. BCG88.
    M.C. Browne, E.M. Clarke, and O. Grumberg. Characterizing finite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59:115–131, 1988. 303CrossRefzbMATHMathSciNetGoogle Scholar
  3. BKO87.
    J.A. Bergstra, J.W. Klop, and E.R. Olderog. Failures without chaos: a new process semantics for fair abstraction. In Formal Description Techniques III, pages 77–103. Elsevier, 1987. 301Google Scholar
  4. BW90.
    J.C.M. Baeten and W.P. Weijland. Process Algebra. Cambridge University Press, 1990. 301Google Scholar
  5. CES86.
    E.M. Clarke, E.A. Emerson, and A.P. Sistla. Automatic verification of finite-state concurrent systems using temporal-logic specifications. ACM Transactions on Programming Languages and Systems, 8(2):244–263, 1986. 303zbMATHCrossRefGoogle Scholar
  6. CPS93.
    R.J. Cleaveland, J. Parrow, and B. Steffen. The Concurrency Workbench: a semantics-based tool for the verification of finite-state systems. ACM Transactions on Programming Languages and Systems, 15(1):36–72, 1993. 300CrossRefGoogle Scholar
  7. VEB95.
    W. Vogler, E. Brinksma, and A. Rensink. Fair testing. In CONCUR 95: Theories of Concurrency, Lecture Notes in Computer Science 962, pages 313–327. Springer-Verlag, July 1995. 301Google Scholar
  8. GL94.
    O. Grumberg and D.E. Long. Model checking and modular verification. ACM Transactions on Programming Languages and Systems, 16(3):843–871, 1994. 300, 304CrossRefGoogle Scholar
  9. Hen87.
    M.C.B. Hennessy. An algebraic theory of fair asynchronous communicating processes. Theretical Computer Science, 49:121–143, 1987. 301zbMATHCrossRefMathSciNetGoogle Scholar
  10. HK96.
    M. Huth and M. Kwiatkowska. The semantics for fair recursion with divergence. Technical Report CSR-96-4, University of Birmingham, 1996. 301Google Scholar
  11. HKR97.
    T.A. Henzinger, O. Kupferman, and S. K. Rajamani. Fair simulation. In CONCUR 97: Theories of Concurrency, Lecture Notes in Computer Science 1243, pages 273–287. Springer-Verlag, July 1997. 300, 305, 307Google Scholar
  12. KS90.
    P.C. Kanellakis and S.A. Smolka. CCS expressions, finite-state processes, and three problems of equivalence. Information and Computation, 86:43–68, 1990. 300zbMATHCrossRefMathSciNetGoogle Scholar
  13. KV96.
    O. Kupferman and M.Y. Vardi. Verification of fair transition systems. In CAV 96: Computer Aided Verification, Lecture Notes in Computer Science 1102, pages 372–381. Springer-Verlag, 1996. 300, 304Google Scholar
  14. KV98.
    O. Kupferman and M.Y. Vardi. Weak alternating automata and tree automata emptiness. In Proceedings of the 30th ACM Symposium on Theory of Computing, pages 224–233. ACM Press, 1998. 308Google Scholar
  15. Mil89.
    R. Milner. Communication and Concurrency. Prentice-Hall, 1989. 300, 302Google Scholar
  16. NC95.
    V. Natarajan and R. Cleaveland. Divergence and fair testing. In ICALP 95: Automata, Languages, and Programming, Lecture Notes in Computer Science 944, pages 648–659. Springer-Verlag, 1995. 301Google Scholar
  17. Par80.
    D.M.R. Park. Concurrency and Automata on Infinite Sequences. Lecture Notes in Computer Science 104. Springer-Verlag, 1980. 300, 302Google Scholar
  18. PT87.
    R. Paige and R.E. Tarjan. Three partition-refinement algorithms. SIAM Journal of Computing, 16(6):973–989, 1987. 300, 303, 307zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Thomas A. Henzinger
    • 1
  • Sriram K. Rajamani
    • 2
  1. 1.University of California at BerkeleyBerkeley
  2. 2.Microsoft ResearchUSA

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