Advertisement

Size-Change Termination and Satisfiability for Linear-Time Temporal Logics

  • Martin Lange
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6989)

Abstract

In the automata-theoretic framework, finite-state automata are used as a machine model to capture the operational content of temporal logics. Decision problems like satisfiability, subsumption, equivalence, etc. then translate into questions on automata like emptiness, inclusion, language equivalence, etc. Linear-time temporal logics like LTL, PSL and the linear-time μ-calculus have relatively simple translations into alternating parity automata, and this automaton model is closed under all Boolean operations with very simple constructions. Thus, the typical decision problems for such linear-time temporal logics reduce relatively simply to the emptiness problem for alternating parity automata. In this paper we present a method for decision this emptiness problem without going through intermediate automaton models like nondeterministic ones. The method is a direct adaptation of the size-change termination principle which was orgininally used to decide termination of abstract functional programs.

Keywords

Temporal Logic Linear Temporal Logic Atomic Proposition Boolean Expression Nest Depth 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abdulla, P.A., Chen, Y.-F., Clemente, L., Holík, L., Hong, C.-D., Mayr, R., Vojnar, T.: Simulation subsumption in ramsey-based büchi automata universality and inclusion testing. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 132–147. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Inc. Accellera Organization. Formal semantics of Accellera property specification language (2004), In Appendix B of http://www.eda.org/vfv/docs/PSL-v1.1.pdf
  3. 3.
    Banieqbal, B., Barringer, H.: Temporal logic with fixed points. In: Banieqbal, B., Pnueli, A., Barringer, H. (eds.) Temporal Logic in Specification. LNCS, vol. 398, pp. 62–73. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  4. 4.
    Bustan, D., Fisman, D., Havlicek, J.: Automata constructions for PSL. Technical Report MCS05-04, The Weizmann Institute of Science (2005)Google Scholar
  5. 5.
    Dax, C., Hofmann, M., Lange, M.: A proof system for the linear time μ-calculus. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 274–285. Springer, Heidelberg (2006)Google Scholar
  6. 6.
    Dax, C., Klaedtke, F.: Alternation elimination by complementation (Extended abstract). In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 214–229. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Emerson, E.A.: Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B, ch. 16, pp. 996–1072. Elsevier, MIT Press, New York, USA (1990)Google Scholar
  8. 8.
    Fogarty, S., Vardi, M.Y.: Büchi complementation and size-change termination. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 16–30. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Fogarty, S., Vardi, M.Y.: Efficient büchi universality checking. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 205–220. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Kaivola, R.: Using Automata to Characterise Fixed Point Temporal Logics. PhD thesis, LFCS, Division of Informatics, The University of Edinburgh, Tech. Rep. ECS-LFCS-97-356 (1997)Google Scholar
  11. 11.
    Kupferman, O., Piterman, N., Vardi, M.Y.: Extended temporal logic revisited. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 519–535. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. 12.
    Lange, M.: Linear time logics around PSL: Complexity, expressiveness, and a little bit of succinctness. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 90–104. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Lee, C.S., Jones, N.D., Ben-Amram, A.M.: The size-change principle for program termination. ACM SIGPLAN Notices 36(3), 81–92 (2001)CrossRefzbMATHGoogle Scholar
  14. 14.
    Löding, C., Thomas, W.: Alternating automata and logics over infinite words. In: Watanabe, O., Hagiya, M., Ito, T., van Leeuwen, J., Mosses, P.D. (eds.) TCS 2000. LNCS, vol. 1872, pp. 521–535. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. 15.
    Miyano, S., Hayashi, T.: Alternating finite automata on omega-words. TCS 32(3), 321–330 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Pnueli, A.: The temporal logic of programs. In: Proc. 18th Symp. on Foundations of Computer Science, FOCS 1977, pp. 46–57. IEEE, Providence (1977)Google Scholar
  17. 17.
    Ramsey, F.P.: On a problem of formal logic. Proc. London Mathematical Society, Series 2 30(4), 338–384 (1928)Google Scholar
  18. 18.
    Ramsey, F.P.: On a problem in formal logic. Proc. London Math. Soc. 30(3), 264–286 (1930)CrossRefzbMATHGoogle Scholar
  19. 19.
    Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logics. Journal of the Association for Computing Machinery 32(3), 733–749 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Stirling, C.: Comparing linear and branching time temporal logics. In: Banieqbal, B., Pnueli, A., Barringer, H. (eds.) Temporal Logic in Specification. LNCS, vol. 398, pp. 1–20. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  21. 21.
    Vardi.: Linear vs. branching time: A complexity-theoretic perspective. In: LICS: IEEE Symposium on Logic in Computer Science (1998)Google Scholar
  22. 22.
    Vardi, M.Y.: A temporal fixpoint calculus. In: ACM (ed.) Proc. Conf. on Principles of Programming Languages, POPL 1988, pp. 250–259. ACM Press, New York (1988)Google Scholar
  23. 23.
    Vardi, M.Y.: Alternating automata and program verification. In: van Leeuwen, J. (ed.) Computer Science Today. LNCS, vol. 1000, pp. 471–485. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  24. 24.
    Vardi, M.Y.: An Automata-Theoretic Approach to Linear Temporal Logic. In: Moller, F., Birtwistle, G. (eds.) Logics for Concurrency. LNCS, vol. 1043, pp. 238–266. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  25. 25.
    Vardi, M.Y.: Branching vs. Linear time: Final showdown. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 1–22. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  26. 26.
    Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Wolper, P.: Temporal logic can be more expressive. Information and Control 56, 72–99 (1983)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Martin Lange
    • 1
  1. 1.School of Electr. Eng. and Computer ScienceUniversity of KasselGermany

Personalised recommendations

Cite paper

Buy options