An automata-theoretic decision procedure for Future Interval Logic

  • Y. S. Ramakrishna
  • L. K. Dillon
  • L. E. Moser
  • P. M. Melliar-Smith
  • G. Kutty
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 652)


Graphical Interval Logic (GIL) is a temporal logic in which all reasoning is done by means of diagrammatic formulæ. It is a discrete linear-time modal logic in which the basic temporal modality is the interval. Future Interval Logic (FIL) provides the logical foundation for GIL. In this paper we present an automata-theoretic decision procedure for FIL with complexity DTIME\((2^{O(n^k )} )\), where n is the size of the formula and k is the depth of interval nesting. For formulæ with bounded depth but length unbounded, the satisfiability problem for FIL is shown to be PSPACE-complete. We believe that this is the first result giving a direct decision procedure of elementary complexity for an interval logic. We also show that the logic is transparent to finite stuttering over the class of ω-sequences, a feature that is useful for composition and refinement.


Temporal Logic Decision Procedure World Model Propositional Dynamic Logic Interval Logic 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Aaby A A, Narayana K T, Propositional Temporal Interval Logic is PSPACE-Complete, Proc. 9th CADE, 1988, pp 218–237.Google Scholar
  2. [2]
    Alur R, Techniques for Automatic Verification of Real-Time Systems, PhD Dissertation, Dept of Computer Science, Stanford University, August 1991.Google Scholar
  3. [3]
    Barringer H, Kuiper R, Pnueli A, Now You May Compose Temporal Logic Specifications, Proc. 16th ACM STOC, Washington DC, April 1984, pp 51–63.Google Scholar
  4. [4]
    Barringer H, Kuiper R, Pnueli A, A Really Abstract Concurrent Model and Its Temporal Logic, Proc. 18th ACM POPL, St. Petersburg Beach, January 1986, pp 173–183.Google Scholar
  5. [5]
    Browne M C, Clarke E M, Grümberg O, Reasoning about Networks with Many Identical Finite State Processes, Information and Computation, Vol 81, No 1, April 1989, pp 13–31.CrossRefGoogle Scholar
  6. [6]
    Büchi J R, On a Decision Method in Restricted Second Order Arithmetic, Logic, Methodology and Philosophy of Science, Proc. 1960 Congress, Stanford University Press, Stanford, pp 1–11.Google Scholar
  7. [7]
    Büchi J R, The Monadic Second Order Theory of All Countable Ordinals, in G H Müller and D Siefkes, Decidable Theories II, Lecture Notes in Mathematics, 328, Springer-Verlag, 1973, pp 1–126.Google Scholar
  8. [8]
    Clarke E M, Grümberg O, Avoiding the State Explosion Problem in Temporal Logic Model Checking Algorithms, Proc. 6th ACM PODC, Vancouver, August 1987, pp 294–303.Google Scholar
  9. [9]
    Dillon L K, Kutty G, Moser L E, MelliarSmith P M, Ramakrishna Y S, Graphical Specifications for Concurrent Software Systems, Proc. 14th ICSE, Melbourne, May 1992, pp 214–224.Google Scholar
  10. [10]
    Fischer M J, Ladner R E, Propositional Dynamic Logic of Regular Programs, J. Computer and System Sciences, Vol 18, 1979, pp 194–211.CrossRefGoogle Scholar
  11. [11]
    Griswold V J, Core Algorithms for Autonomous Monitoring of Distributed Systems, Proc. ACM/ONR Workshop on Parallel and Distributed Debugging, May 1991, pp 23–34.Google Scholar
  12. [12]
    Halpern J, Manna Z, Moszkowski B, A Hardware Semantics Based on Temporal Intervals, Proc. 10th ICALP, Barcelona, 1983, pp 278–291.Google Scholar
  13. [13]
    Halpern J, Shoham Y, A Propositional Modal Logic of Time Intervals, J. ACM, Vol 38, No 4, October 1991, pp 935–962.CrossRefGoogle Scholar
  14. [14]
    Harel D, Peleg D, Process Logic with Regular Formulas, Theoretical Computer Science, Vol 38, 1985, pp 307–322.CrossRefGoogle Scholar
  15. [15]
    Kutty G, Ramakrishna Y S, Dillon L K, Moser L E, Melliar-Smith P M, Specification of a Communication Protocol in Graphical Interval Logic, Proc. IEE Conf. Information Engineering, Singapore, December 1991, pp 432–441.Google Scholar
  16. [16]
    Lamport L, What Good is Temporal Logic, Proc. IFIP Congress, Paris, 1983, pp 657–668.Google Scholar
  17. [17]
    Lamport L, Specifying Concurrent Program Modules, ACM TOPLAS, Vol 5, No 2, 1983, pp 190–222.CrossRefGoogle Scholar
  18. [18]
    Lamport L, The Temporal Logic of Actions, Tech Rep 79, DEC Systems Research Center, Palo Alto, December 1991.Google Scholar
  19. [19]
    Manna Z, Pnueli A, Verification of Concurrent Programs: The Temporal Framework, In R S Boyer and J S Moore, editors, The Correctness Problem in Computer Science, Academic Press, London, 1982, pp 215–273.Google Scholar
  20. [20]
    Narayana K T, Aaby A A, Specification of Real-Time Systems in Real-Time Interval Logic, Proc. IEEE RTSS, December 1988, pp 86–95.Google Scholar
  21. [21]
    Newman M H A, On Theories with a Combinatorial Definition of “Equivalence”, Annals of Mathematics, Vol 43, No 2, April 1942, pp 223–243.Google Scholar
  22. [22]
    Plaisted D, A Low Level Language for Obtaining Decision Procedures for Classes of Temporal Logics, in Schwartz et al, An Interval Logic for Higher-Level Temporal Reasoning, NASA Contractor Report 172262, September 1983, pp 65–82.Google Scholar
  23. [23]
    Rabin M O, Decidability of Second Order Theories and Automata on Infinite Trees, Trans. AMS, Vol 141, 1969, pp 1–35.Google Scholar
  24. [24]
    Ramakrishna Y S, Melliar-Smith P M, Moser L E, Dillon L K, Kutty G, Interval Logic is Non-Elementarily Decidable, unpublished manuscript, February 1992.Google Scholar
  25. [25]
    Razouk R R, Gorlick M M, A Real-Time Interval Logic For Reasoning About Executions of Real-Time Programs, Proc. 3rd Symp. Software Testing, Analysis and Verification, Key West, SIGSOFT Software Engineering Notes, Vol 114, No 3, December 1989, pp 10–19.Google Scholar
  26. [26]
    Rosner R, Pnueli A, A Choppy Logic, Proc. IEEE LICS, July 1986, pp 306–313.Google Scholar
  27. [27]
    Savitch W J, Relationship between Non-Deterministic and Deterministic Tape Complexities, J. Computer and System Sciences, Vol 4, No 2, pp 177–192.Google Scholar
  28. [28]
    Schwartz R L, Melliar-Smith P M, Vogt F, An Interval Logic for Higher-Level Temporal Reasoning, Proc. ACM PODC, Montreal, Canada, August 1983, pp. 173–186.Google Scholar
  29. [29]
    Vardi M, Wolper P, Automata-Theoretic Techniques for Modal Logics of Programs, J. Computer and System Sciences, Vol 32, No 2, April 1986, pp 183–210.CrossRefGoogle Scholar
  30. [30]
    Wolper P, On the Relation of Programs and Computations to Models of Temporal Logic, Proc. Conf. Temporal Logic in Specification, Altrincham, April 1987, LNCS 398, Springer-Verlag, pp 75–123.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Y. S. Ramakrishna
    • 1
  • L. K. Dillon
    • 1
  • L. E. Moser
    • 1
  • P. M. Melliar-Smith
    • 1
  • G. Kutty
    • 1
  1. 1.Departments of Electrical and Computer Engineering and of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA

Personalised recommendations