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A Hierarchical Completeness Proof for Propositional Temporal Logic

  • Ben Moszkowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2772)

Abstract

We present a new proof of axiomatic completeness for Proposition Temporal Logic (PTL) for discrete, linear time for both finite and infinite time (without past-time). This makes use of a natural hierarchy of logics and notions and is an interesting alternative to the proofs in the literature based on tableaux, filtration, game theory and other methods. In particular we exploit the deductive completeness of a sublogic in which the only temporal operator is O (“next”). This yields a proof which is in certain respects more direct and higher-level than previous ones. The presentation also reveals unexpected fundamental links to a natural and preexisting framework for interval-based reasoning and fixpoints of temporal operators.

Keywords

Modal Logic Temporal Logic State Sequence Completeness Proof Axiom System 
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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References

  1. [1]
    B. Banieqbal and H. Barringer. A study of an extended temporal logic and a temporal fixed point calculus. Technical Report UMCS-86-10-2, Dept. of Computer Science, University of Manchester, England, Oct. 1986. (Revised June 1987).Google Scholar
  2. [2]
    M. Ben-Ari, Z. Manna, and A. Pnueli. The temporal logic of branching time. In Eighth ACM Symposium on Principles of Programming Languages, pages 164-176. ACM, JAN 1981.Google Scholar
  3. [3]
    M. Ben-Ari, Z. Manna, and A. Pnueli. The temporal logic of branching time. Acta Informatica, 20(3):207–226, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    R. E. Bryant. Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computers, C-35(8), 1986.Google Scholar
  5. [5]
    B. F. Chellas. Modal Logic: An Introduction. Cambridge University Press, Cambridge, England, 1980.CrossRefzbMATHGoogle Scholar
  6. [6]
    E. A. Emerson. Temporal and modal logic. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics, chapter 16, pages 995-1072. Elsevier/MIT Press, Amsterdam, 1990.Google Scholar
  7. [7]
    T. French. A proof of the completeness of PLTL. Available as http://www.cs.uwa.edu.au/~tim/papers/pltlcomp.ps, 2000.Google Scholar
  8. [8]
    D. Gabbay, A. Pnueli, S. Shelah, and J. Stavi. On the temporal analysis of fairness. In Seventh Annual ACM Symposium on Principles of Programming Languages, pages 163-173, 1980.Google Scholar
  9. [9]
    R. Goldblatt. Logics of Time and Computation, volume 7 of CSLI Lecture Notes. CLSI/SRI International, 333 Ravenswood Av., Menlo Park, CA 94025, 1987.Google Scholar
  10. [10]
    J. Halpern, Z. Manna, and B. Moszkowski. A hardware semantics based on temporal intervals. In J. Diaz, editor, Proceedings of the 10-th I nternati onal Colloquium on Automata, Languages and Programming, volume 154 of LNCS, pages 278-291, Berlin, 1983. Springer-Verlag.Google Scholar
  11. [11]
    G. E. Hughes and M. J. Cresswell. A New Introduction to Modal Logic. Routledge, London, 1996.CrossRefzbMATHGoogle Scholar
  12. [12]
    Y. Kesten and A. Pnueli. A complete proof system for QPTL. In Proc. 10th IEEE Symp. on Logic in Computer Science, pages 2-12. IEEE Computer Society Press, 1995.Google Scholar
  13. [13]
    D. Kozen and R. Parikh. An elementary proof of the completeness of PDL. Theor. Comp. Sci., 14:113–118, 1981.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    F. Kroger. Temporal Logic of Programs, volume 8 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1987.Google Scholar
  15. [15]
    M. Lange and C. Stirling. Focus games for satisfiability and completeness of temporal logic. In Proc. 16th Annual IEEE Symp. on Logic in Computer Science, LICS’01, pages 357-365, Boston, MA, USA, June 2001. IEEE Computer Society Press.Google Scholar
  16. [16]
    O. Lichtenstein and A. Pnueli. Propositional temporal logics: Decidability and completeness. Logic Journal of the IGPL, 8(1):55–85, 2000. Available at http://www3.oup.co.uk/igpl/Volume_08/Issue_01/#Lichtenstein.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    B. Moszkowski. Reasoning about Digital Circuits. PhD thesis, Department of Computer Science, Stanford University, 1983. Technical report STAN-CS-83-970.Google Scholar
  18. [18]
    B. Moszkowski. A temporal logic for multilevel reasoning about hardware. Computer, 18:10–19, 1985.CrossRefGoogle Scholar
  19. [19]
    B. Moszkowski. Executing Temporal Logic Programs. Cambridge University Press, Cambridge, England, 1986.Google Scholar
  20. [20]
    B. Moszkowski. Some very compositional temporal properties. In E.-R. Olderog, editor, Programming Concepts, Methods and Calculi, volume A-56 of IFIP Transactions, pages 307-326. IFIP, Elsevier Science B.V. (North-Holland), 1994.Google Scholar
  21. [21]
    B. Moszkowski. Using temporal fixpoints to compositionally reason about liveness. In He Jifeng, J. Cooke, and P. Wallis, editors, BCS-FACS 7th Refinement Workshop, electronic Workshops in Computing, London, 1996. BCS-FACS, Springer-Verlag and British Computer Society.Google Scholar
  22. [22]
    B. Moszkowski. Compositional reasoning using Interval Temporal Logic and Tempura. In W.-P. de Roever, H. Langmaack, and A. Pnueli, editors, Compositionality: The Significant Difference, volume 1536 of LNCS, pages 439-464, Berlin, 1998. Springer-Verlag.Google Scholar
  23. [23]
    B. Moszkowski. An automata-theoretic completeness proof for Interval Temporal Logic (extended abstract). In U. Montanari, J. Rolim, and E. Welzl, editors, Proceedings of the 27th International Colloquium on Automata, Languages and Programming (ICALP 2000), volume 1853 of LNCS, pages 223-234, Geneva, Switzerland, July 2000. Springer-Verlag.Google Scholar
  24. [24]
    B. Moszkowski. A complete axiomatization of Interval Temporal Logic with infinite time (extended abstract). In Proc. of the 15th Annual IEEE Symposium on Logic in Computer Science (LICS 2000), pages 242-251. IEEE Computer Society Press, June 2000.Google Scholar
  25. [25]
    B. Moszkowski. A hierarchical completeness proof for interval temporal logic with finite time. To appear in Proc. of the Workshop on Interval Temporal Logics and Duration Calculi (part of 15th European Summer School in Logic Language and Information (ESSLLI-2003)), Vienna, August 25–29, 2003.Google Scholar
  26. [26]
    B. Paech. Gentzen-systems for propositional temporal logics. In E. Borger, H. K. Büning, and M. M. Richter, editors, Proceedings of the 2nd Workshop on Computer Science Logic, Duisburg (FRG), volume 385 of LNCS, pages 240-253. Springer-Verlag, Oct. 1988.Google Scholar
  27. [27]
    A. Pnueli. The temporal logic of programs. In Proceedings of the 18th Sym posium on the Foundation of Computer Science, pages 46-57. ACM, 1977.Google Scholar
  28. [28]
    V. R. Pratt. Process logic. In Sixth Annual ACM Symposium on Principles of Programming Languages, pages 93-100, 1979.Google Scholar
  29. [29]
    N. Rescher and A. Urquhart. Temporal Logic. Springer-Verlag, New York, 1971.CrossRefzbMATHGoogle Scholar
  30. [30]
    R. Rosner and A. Pnueli. A choppy logic. In First Annual IEEE Symposium on Logic in Computer Science, pages 306-313. IEEE Computer Society Press, June 1986.Google Scholar
  31. [31]
    S. Safra. On the complexity of ω-automata. In Proc. 29th Ann. IEEE Symp. on the Foundations of Computer Science (FOCS 1988)), pages 319-327, White Plains, New York, Oct. 1988. IEEE Computer Society Press.Google Scholar
  32. [32]
    G. H. von Wright. An Essay in Modal Logic. North Holland Publishing Co., Amsterdam, 1951.zbMATHGoogle Scholar
  33. [33]
    P. Wolper. Temporal logic can be more expressive. In Proc. 22nd Annual Symposium on Foundations of Computer Science (FOCS), pages 340-348, Nashville, Tennessee, Oct. 1981. IEEE Computer Society.Google Scholar
  34. [34]
    P. L. Wolper. Temporal logic can be more expressive. Information and Control, 56(1–2):72–99, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [35]
    P. L. Wolper. The tableau method for temporal logic: An overview. Logique et Analyse, 110–111:119–136, 1985.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Ben Moszkowski
    • 1
  1. 1.Software Technology Research Laboratory, Hawthorn BuildingDe Montfort UniversityThe Gateway, LeicesterGreat Britain

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