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A decision procedure for combinations of propositional temporal logic and other specialized theories

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Abstract

We present a decision procedure for formulae of discrete linear time propositional temporal logic whose propositional part may include assertions in a specialized theory. The combined decision procedure may be viewed as an extension of known decision procedures for quantifier-free theories to theories including temporal logic connectives. A combined decision procedure given by Pratt restricted to linear time temporal logic, runs in polynomial space relative to an oracle for the underlying theory. Our procedure differs from this one in that it can handle assertions containing arbitrary mixtures of global variables, whose values cannot change with time, and state variables, whose values can change with time. This new procedure can also handle assertions containing functions and predicates whose interpretations do not change with time. However, it requires the computation of least and greatest fixed points and has a worse asymptotic running time than that of Pratt. This new procedure has been implemented and seems efficient enough to be practical on simple formulae, although an upper bound derived for the worst case running time is triple exponential in the length of the formula. The techniques used appear to apply to logics other than temporal logic which have decision procedures based on tableaux.

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References

  1. Ben-Ari, M., Manna, Z., and Pneuli, A., ‘The temporal logic of branching time’,Proceedings of the 8th ACM Symposium on Principles of Programming Languages, 1981, pp. 164–176.

  2. EmersonE. and ClarkeE. M., ‘Using branching time temporal logic to synthesize synchronization skeletons’,Science of Computer Programming 2 241–266 (1982).

    Article  MATH  Google Scholar 

  3. Emerson, E. and Sistla, A., ‘Deciding full branching time logic’,Information and Control 61 (1984).

  4. FlonL. and SuzukiN., ‘The total correctness of parallel programs’,SIAM J. Comput. 10, 227–246 (1981).

    Article  MATH  MathSciNet  Google Scholar 

  5. HalpernJ. and ReifJ., ‘The propositional dynamic logic of deterministic, well-structured programs’,Theoret. Comput. Sci. 27, 127–166 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  6. Manna, Z. and Pneuli, A., ‘Verification of concurrent programs: the temporal framework, inThe Correctness Problem in Computer Science (eds R. Boyer and J. S. Moore), Academic Press, pp. 215–273 (1981).

  7. Manna, Z. and Pneuli, A., ‘Verification of concurrent programs, part II: temporal proof principles’, Department of Computer Science, Stanford University technical report STAN-CS-81-843 (1981).

  8. Manna, Z. and Wolper, P., ‘Synthesis of communicating processes from temporal logic specifications’,ACM Transactions on Programming Languages and Systems 68–93 (1984).

  9. NelsonG. and OppenD., ‘Simplification by cooperating decision procedures’,ACM Transactions on Programming Languages and Systems 1, 245–257 (1979).

    Article  MATH  Google Scholar 

  10. NelsonG. and OppenD., ‘Fast decision procedures based on congruence closure’,JACM 27, 356–364 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  11. Plaisted, D., ‘A low level language for obtaining decision procedures for classes of temporal logics’, inProceedings of the Workshop on Logics of Programs, (eds. E. Clarke and D. Kozen), pp. 403–420 (1983).

  12. Pneuli, A., ‘The temporal logic of programs’,Proceedings of the Eighteenth Annual Symposium on Foundations of Computer Science, IEEE, pp. 46–57 (1977).

  13. Pratt, V., ‘Program logic without binding is decidable’,Proceedings of the Eighth ACM Symposium on Principles of Programming Languages, pp. 159–163 (1981).

  14. Schwartz, R., Melliar-Smith, P. M., and Vogt, F., ‘An interval logic for higher level temporal reasoning: language definition and examples’, Computer Science Laboratory, SRI International technical report CSL-138 (1983).

  15. Shostak, R., ‘Deciding combinations of theories’,Proceedings of the Sixth Conference on Automated Deduction, pp. 209–222 (1982).

  16. SistlaA. P. and ClarkeE. M., ‘The complexity of propositional linear temporal logics’,JACM 32, 733–749 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  17. SmullyanR.,First Order Logic. Springer-Verlag, New York (1968).

    Google Scholar 

  18. Wolper, P., ‘Temporal logic can be more expressive’,22nd Annual Symposium on Foundations of Computer Science, Nashville, Tennessee, 1981, pp. 340–348.

  19. Wolper, P., ‘Synthesis of communicating processes from temporal logic specifications’, PhD thesis, Stanford University (1982).

  20. YasuharaA.,Recursive Function Theory and Logic, Academic Press, New York (1971).

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of North Carolina at Chapel Hill, 27514, Chapel Hill, NC, U.S.A.

    David A. Plaisted

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  1. David A. Plaisted
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Additional information

Most of this work was done while the author was at SRI International, Menlo Park, California, and at the University of Illinois, Urbana, Illinois, U.S.A.

This work was partially supported by the National Science Foundation under grants MCS 81-09831 and MCS 83-07755.

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Plaisted, D.A. A decision procedure for combinations of propositional temporal logic and other specialized theories. J Autom Reasoning 2, 171–190 (1986). https://doi.org/10.1007/BF02432150

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