Alternating refinement relations

  • Rajeev Alur
  • Thomas A. Henzinger
  • Orna Kupferman
  • Moshe Y. Vardi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1466)


Alternating transition systems are a general model for composite systems which allow the study of collaborative as well as adversarial relationships between individual system components. Unlike in labeled transition systems, where each transition corresponds to a possible step of the system (which may involve some or all components), in alternating transition systems, each transition corresponds to a possible move in a game between the components. In this paper, we study refinement relations between alternating transition systems, such as “Does the implementation refine the set A of specification components without constraining the components not in A?” In particular, we generalize the definitions of the simulation and trace containment preorders from labeled transition systems to alternating transition systems. The generalizations are called alternating simulation and alternating trace containment. Unlike existing refinement relations, they allow the refinement of individual components within the context of a composite system description. We show that, like ordinary simulation, alternating simulation can be checked in polynomial time using a fixpoint computation algorithm. While ordinary trace containment is PSPACE-complete, we establish alternating trace containment to be EXPTIME-complete. Finally, we present logical characterizations for the two preorders in terms of ATL, a temporal logic capable of referring to games between system components.


Transition System Temporal Logic Winning Strategy Label Transition System Tree Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [AHK97]
    R. Alur, T.A. Henzinger, and O. Kupferman. Alternating-time temporal logic. In Proc. 38th Symp. on Foundations of Computer Science, pp. 100–109. IEEE Computer Society, 1997. Full version in Compositionality-The Significant Difference. Springer-Verlag Lecture Notes in Computer Science, 1998.Google Scholar
  2. [BGS92]
    J. Balcazar, J. Gabarro, and M. Santha. Deciding bisimilarity is P-complete. Formal Aspects of Computing, 4:638–648, 1992.zbMATHCrossRefGoogle Scholar
  3. [HF89]
    J.Y. Halpern and R. Fagin. Modeling knowledge and action in distributed systems. Distributed Computing, 3:159–179, 1989.zbMATHCrossRefGoogle Scholar
  4. [HHK95]
    M.R. Henzinger, T.A. Henzinger, and P.W. Kopke. Computing simulations on finite and infinite graphs. In Proc. 36rd Symp. on Foundations of Computer Science, pp. 453–462. IEEE Computer Society, 1995.Google Scholar
  5. [Imm81]
    N. Immerman. Number of quantifiers is better than number of tape cells. J. Computer and System Sciences, 22:384–406, 1981.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [KV98]
    O. Kupferman and M.Y. Vardi. Verification of fair transition systems. Chicago J. Theoretical Computer Science, 1998(2).Google Scholar
  7. [Mil71]
    R. Milner. An algebraic definition of simulation between programs. In Proc. 2nd Int. Joint Conf. on Artificial Intelligence, pp. 481–489. British Computer Society, 1971.Google Scholar
  8. [Mil90]
    R. Milner. Operational and algebraic semantics of concurrent processes. In Handbook of Theoretical Computer Science, Vol. B, pp. 1201–1242. Elsevier, 1990.Google Scholar
  9. [MS87]
    D.E. Muller and P.E. Schupp. Alternating automata on infinite trees. Theoretical Computer Science, 54:267–276, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [MS95]
    D.E. Muller and P.E. Schupp. Simulating alternating tree automata by nondeterministic automata: new results and new proofs of theorems of Rabin, McNaughton, and Safra. Theoretical Computer Science, 141:69–107, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [Sha53]
    L.S. Shapley. Stochastic games. In Proc. National Academy of Science, 39:1095–1100, 1953.zbMATHMathSciNetCrossRefGoogle Scholar
  12. [VW86]
    M.Y. Vardi and P. Wolper. Automata-theoretic techniques for modal logics of programs. J. Computer and System Sciences, 32:182–221, 1986.MathSciNetCrossRefGoogle Scholar
  13. [VW94]
    M.Y. Vardi and P. Wolper. Reasoning about infinite computations. Information and Computation, 115:1–37, 1994.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Rajeev Alur
    • 1
  • Thomas A. Henzinger
    • 2
  • Orna Kupferman
    • 2
  • Moshe Y. Vardi
    • 3
  1. 1.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Department of Electrical Engineering and Computer SciencesUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of Computer ScienceRice UniversityHoustonUSA

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