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∏2 ∩ Σ2 ≡ AFMC

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Automata, Languages and Programming (ICALP 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2719))

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Abstract

The μ-calculus is an expressive specification language in which modal logic is extended with fixpoint operators, subsuming many dynamic, temporal, and description logics. Formulas of μ-calculus are classified according to their alternation depth, which is the maximal length of a chain of nested alternating least and greatest fixpoint operators. Alternation depth is the major factor in the complexity of μ-calculus model-checking algorithms. A refined classification of μ-calculus formulas distinguishes between formulas in which the outermost fixpoint operator in the nested chain is a least fixpoint operator (Σ i formulas, where i is the alternation depth) and formulas where it is a greatest fixpoint operator (∏ i formulas). The alternation-free μ-calculus (AFMC) consists of μ-calculus formulas with no alternation between least and greatest fixpoint operators. Thus, AFMC is a natural closure of Σ 1 ∪ ∏ 1, which is contained in both Σ 2 and ∏ 2. In this work we show that Σ 2 ∩ ∏ 2 ≡ AFMC. In other words, if we can express a property ξ both as a least fixpoint nested inside a greatest fixpoint and as a greatest fixpoint nested inside a least fixpoint, then we can express ξ also with no alternation between greatest and least fixpoints. Our result refers to μ-calculus over arbitrary Kripke structures. A similar result, for directed μ-calculus formulas interpreted over trees with a fixed finite branching degree, follows from results by Arnold and Niwinski. Their proofs there cannot be easily extended to Kripke structures, and our extension involves symmetric nondeterministic Büchi tree automata, and new constructions for them.

Supported in part by by NSF grant CCR-9988172 and by a research grant from the Center for Pure and Applied Mathematics at the University of California, Berkeley

Supported in part by NSF grants CCR-9988322, CCR-0124077, IIS-9908435, IIS-9978135, and EIA-0086264, by BSF grant 9800096, and by a grant from the Intel Corporation.

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References

  1. J.W. Addision, The theory of hierarchies. Proc. Internat. Congr. Logic, Method. and Philos. Sci. 1960, pages 26–37, Stanford University Press, 1962.

    Google Scholar 

  2. A. Arnold and D. Niwiński. Fixed point characterization of Büchi automata on infinite trees. Information Processing and Cybernetics, 8–9:451–459, 1990.

    Google Scholar 

  3. A. Arnold and D. Niwiński. Fixed point characterization of weak monadic logic definable sets of trees. In Tree Automata and Languages, pages 159–188, Elsevier, 1992.

    Google Scholar 

  4. A. Arnold and D. Niwiński. Rediments of μ-calculus. Elsevier, 2001.

    Google Scholar 

  5. A. Arnold and L. Santocanale, On ambiguous classes in the μ-calculus hierarchy of tree languages, Proc. Workshop on Fixed Points in Computer Science, Warsaw, Poland, 2003.

    Google Scholar 

  6. J. Benthem. Languages in actions: categories, lambdas and dynamic logic. Studies in Logic, 130, 1991.

    Google Scholar 

  7. J.C. Bradfield. The modal μ-calculus alternation hierarchy is strict. TCS, 195(2):133–153, March 1998.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Bloem, K. Ravi, and F. Somenzi. Efficient decision procedures for model checking of linear time logic properties. In Proc. 11th CAV, LNCS 1633, pages 222–235. 1999.

    Google Scholar 

  9. R. Cleaveland and B. Steffen. A linear-time model-checking algorithm for the alternation-free modal μ-calculus. In Proc. 3rd CAV, LNCS 575, pages 48–58, 1991.

    Google Scholar 

  10. E.A. Emerson and C.-L. Lei. Temporal model checking under generalized fairness constraints. In Proc. 18th Hawaii International Conference on System Sciences, 1985.

    Google Scholar 

  11. E.A. Emerson and C.-L. Lei. Efficient model checking in fragments of the propositional μ-calculus. In Proc. 1st LICS, pages 267–278, 1986.

    Google Scholar 

  12. M.J. Fischer and R.E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and Systems Sciences, 18:194–211, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness. W. Freeman and Co., San Francisco, 1979.

    MATH  Google Scholar 

  14. M. Jurdzinski. Small progress measures for solving parity games. In 17th STACS, LNCS 1770, pages 290–301. 2000.

    Chapter  Google Scholar 

  15. D. Janin and I. Walukiewicz. Automata for the modal μ-calculus and related results. In Proc. 20th MFCS, LNCS, pages 552–562. 1995.

    Google Scholar 

  16. R. Kaivola. On modal μ-calculus and Büchi tree automata. IPL, 54:17–22, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Kozen. Results on the propositional μ-calculus. TCS, 27:333–354, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. Kozen and R. Parikh. A decision procedure for the propositional μ-calculus. In Logics of Programs, LNCS 164, pages 313–325, 1984.

    Google Scholar 

  19. O. Kupferman and M.Y. Vardi. Freedom, weakness, and determinism: from linear-time to branching-time. In Proc. 13th LICS, pages 81–92, June 1998.

    Google Scholar 

  20. O. Kupferman and M.Y. Vardi. The weakness of self-complementation. In Proc. 16th STACS, LNCS 1563, pages 455–466. 1999.

    Google Scholar 

  21. O. Kupferman and M.Y. Vardi. On clopen specifications. In Proc. 8th LPAR, LNCS 2250, pages 24–38. 2001.

    Google Scholar 

  22. O. Kupferman, M.Y. Vardi, and P. Wolper. An automata-theoretic approach to branching-time model checking. Journal of the ACM, 47(2):312–360, March 2000.

    Article  MATH  MathSciNet  Google Scholar 

  23. K.L. McMillan. Symbolic Model Checking. Kluwer Academic Publishers, 1993.

    Google Scholar 

  24. S. Miyano and T. Hayashi. Alternating finite automata on ω-words. TCS, 32:321–330, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  25. A.W. Mostowski. Regular expressions for infinite trees and a standard form of automata. In Computation Theory, LNCS 208, pages 157–168. 1984.

    Google Scholar 

  26. D.E. Muller and P.E. Schupp. Alternating automata on infinite trees. TCS, 54:267–276, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  27. D.E. Muller, A. Saoudi, and P.E. Schupp. Alternating automata, the weak monadic theory of the tree and its complexity. In Proc. 13th ICALP, LNCS 226, 1986.

    Google Scholar 

  28. D. Niwiński. On fixed point clones. In Proc. 13th ICALP, LNCS 226, pages 464–473. 1986.

    Google Scholar 

  29. M.O. Rabin. Weakly definable relations and special automata. In Proc. Symp. Math. Logic and Foundations of Set Theory, pages 1–23, 1970.

    Google Scholar 

  30. H. Rogers, Theory of recursive functions and effective computability. McGraw-Hill, 1967.

    Google Scholar 

  31. R.S. Street and E.A. Emerson. An elementary decision procedure for the μ-calculus. In Proc. 11th ICALP, LNCS 172, pages 465–472, 1984.

    Google Scholar 

  32. M. Takahashi. The greatest fixed-points and rational ω-tree languages. TCS 44, pp. 259–274, 1986.

    Article  MATH  Google Scholar 

  33. M.Y. Vardi and P. Wolper. An automata-theoretic approach to automatic program verification. In Proc. 1st LICS, pages 332–344, 1986.

    Google Scholar 

  34. I. Walukiewicz. Private communication, 2003.

    Google Scholar 

  35. T. Wilke. CTL+ is exponentially more succinct than CTL. In Proc. 19th FST& TCS, LNCS 1738, pages 110–121, 1999.

    Google Scholar 

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Author information

Authors and Affiliations

  1. School of Engineering and Computer Science, Hebrew University, Jerusalem, 91904, Israel

    Orna Kupferman

  2. Department of Computer Science, Rice University, Houston, TX, 77251-1892, USA

    Moshe Y. Vardi

Authors
  1. Orna Kupferman
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  2. Moshe Y. Vardi
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Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Jos C. M. Baeten

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive, Atlanta, GA, 30332-0205, USA

    Jan Karel Lenstra

  3. Department of Information Technology, Uppsala University, P.O. Box 337, 75105, Uppsala, Sweden

    Joachim Parrow

  4. Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    Gerhard J. Woeginger

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Kupferman, O., Vardi, M.Y. (2003). ∏2 ∩ Σ2 ≡ AFMC. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_55

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  • DOI: https://doi.org/10.1007/3-540-45061-0_55

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