An Incremental Technique for Automata-Based Decision Procedures

  • Gulay Unel
  • David Toman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4603)


Automata-based decision procedures commonly achieve optimal complexity bounds. However, in practice, they are often outperformed by sub-optimal (but more local-search based) techniques, such as tableaux, on many practical reasoning problems. This discrepancy is often the result of automata-style techniques global approach to the problem and the consequent need for constructing an extremely large automaton. This is in particular the case when reasoning in theories consisting of large number of relatively simple formulas, such as descriptions of database schemes, is required. In this paper, we propose techniques that allow us to approach a μ-calculus satisfiability problem in an incremental fashion and without the need for re-computation. In addition, we also propose heuristics that guide the problem partitioning in a way that is likely to reduce the size of the problems that need to be solved.


Description Logic Acceptance Condition Tree Automaton Intersection Automaton Emptiness Problem 
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  1. 1.
    Baader, F., Sattler, U.: An Overview of Tableau Algorithms for Description Logics. Studia Logica 69, 5–40 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Berardi, D., Calvanese, D., De Giacomo, G.: Reasoning on UML Class Diagrams using Description Logic Based Systems. In: Baader, F., Brewka, G., Eiter, T. (eds.) KI 2001. LNCS (LNAI), vol. 2174, Springer, Heidelberg (2001)Google Scholar
  3. 3.
    Bernholtz, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. In: Dill, D.L. (ed.) CAV 1994. LNCS, vol. 818, pp. 142–155. Springer, Heidelberg (1994)Google Scholar
  4. 4.
    Bradfield, J., Stirling, C.: Modal Mu-Calculi, ch. 12 (2006)Google Scholar
  5. 5.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundl. Math. 6, 66–92 (1960)zbMATHCrossRefGoogle Scholar
  6. 6.
    Büchi, J.R.: On a decision method in restricted second-order arithmetic. In: Proc. Int. Congr. for Logic, Methodology and Philosophy of Science 1–11 (1962)Google Scholar
  7. 7.
    Calvanese, D., Lenzerini, M., Nardi, D.: Description logics for conceptual data modeling. In: Chomicki, J., Saake, G. (eds.) Logics for Databases and Information Systems, pp. 229–264. Kluwer, Dordrecht (1998)Google Scholar
  8. 8.
    Demri, S., Sattler, U.: Automata-theoretic decision procedures for information logics. Fundam. Inform. 53(1), 1–22 (2002)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Jutla, C.S., Emerson, E.A.: Tree automata, mu-calculus and determinacy. In: Proceedings of the 32nd annual symposium on Foundations of computer science, pp. 368–377. IEEE Computer Society Press, Los Alamitos (1991)Google Scholar
  10. 10.
    Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–52 (1961)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Henriksen, J.G., Jensen, J.L., Jörgensen, M.E., Klarlund, N., Paige, R., Rauhe, T., Sandholm, A.: MONA: Monadic second-order logic in practice. In: Brinksma, E., Steffen, B., Cleaveland, W.R., Larsen, K.G., Margaria, T. (eds.) TACAS 1995. LNCS, vol. 1019, pp. 89–110. Springer, Heidelberg (1995)Google Scholar
  12. 12.
    Hladik, J., Sattler, U.: A Translation of Looping Alternating Automata into Description Logics. In: Baader, F. (ed.) Automated Deduction – CADE-19. LNCS (LNAI), vol. 2741, pp. 90–105. Springer, Heidelberg (2003)Google Scholar
  13. 13.
    Janin, D., Walukiewicz, I.: Automata for the modal μ-calculus and related results. In: Hájek, P., Wiedermann, J. (eds.) MFCS 1995. LNCS, vol. 969, pp. 552–562. Springer, Heidelberg (1995)Google Scholar
  14. 14.
    Klarlund, N.: MONA & FIDO: The logic-automaton connection in practice. Computer Science Logic, 311–326 (1997)Google Scholar
  15. 15.
    Klarlund, N., Møller, A., Schwartzbach, M.I.: MONA implementation secrets. Int. J. Found. Comput. Sci. 13(4), 571–586 (2002)zbMATHCrossRefGoogle Scholar
  16. 16.
    Kozen, D.: Results on the propositional mu-calculus. Theoretical Computer Science 27, 333–354 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wolper, P., Vardi, M.Y.: Automata theoretic techniques for modal logics of programs (extended abstract). In: STOC 1984: Proceedings of the sixteenth annual ACM symposium on Theory of computing, pp. 446–456. ACM Press, New York (1984)Google Scholar
  18. 18.
    McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Information and Control 9, 521–530 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Vardi, M.Y., Kupferman, O.: Safraless decision procedures. In: Proc. 46th IEEE Symp. on Foundations of Computer Science, pp. 531–540, Pittsburgh (October 2005)Google Scholar
  20. 20.
    Vardi, M.Y., Kupferman, O., Piterman, N.: Safraless compositional synthesis. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 31–44. Springer, Heidelberg (2006)Google Scholar
  21. 21.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141, 1–35 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Safra, S.: On the Complexity of ω-Automata. In: FOCS, pp. 319–327 (1988)Google Scholar
  23. 23.
    Safra, S.: Exponential Determinization for omega-Automata with Strong-Fairness Acceptance Condition (Extended Abstract). In: STOC, pp. 275–282 (1992)Google Scholar
  24. 24.
    Sattler, U., Vardi, M.Y.: The Hybrid μ-Calculus. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 76–91. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  25. 25.
    Thomas, W.: Languages, automata, and logic. In: Handbook of Formal Languages, vol. 3, Springer, Heidelberg (1997)Google Scholar
  26. 26.
    Vardi, M.Y.: What makes Modal Logic so Robustly Decidable. Descriptive Complexity and Finite Models. American Mathematical Society (1997)Google Scholar
  27. 27.
    Vardi, M.Y.: Reasoning about the past with two-way automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  28. 28.
    Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification. In: Proc. LICS, pp. 322–331 (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Gulay Unel
    • 1
  • David Toman
    • 1
  1. 1.D.R.Cheriton School of Computer Science, University of Waterloo 

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