Advertisement

Logic of Programs 1985: Logics of Programs pp 359-372 | Cite as

Fixpoints and program looping: Reductions from the Propositional mu-calculus into Propositional Dynamic Logics of Looping

  • Robert S. Streett
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 193)

Abstract

The propositional mu-calculus is a propositional logic of programs which incorporates a least fixpoint operator and subsumes the Propositional Dynamic Logic of Fischer and Ladner, the infinite looping constructs of Streett and Sherman, and the Game Logic of Parikh. The propositional mu-calculus is strictly stronger in expressive power than any of these other logics. However, the mu-calculus satisfiability problem is polynomially reducible to the satisfiability problem for Propositional Dynamic Logic with Streett's looping construct. This result shows the connection between fixpoints and program looping, provides an alternative decision procedure for the mu-calculus, and rules out the possibility that the complexities of the two logics could be separated by new upper and lower bound results. We also give reductions from several weaker mu-calculi (including those investigated by Kozen, Vardi and Wolper) into variants of Propositional Dynamic Logic with weaker looping constructs (including Sherman's looping construct). The deterministic exponential time upper bounds obtained for these mu-calculi may also be obtained via these reductions.

Keywords

Modal Logic Regular Expression Choice Function Finite Automaton Infinite Chain 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. de Bakker, J., and de Roever, W. P. (1973), A Calculus for Recursive Program Schemes, First International Colloquium on Automata, Languages, and Programming, 167–196.Google Scholar
  2. de Roever, W. P. (1974), Recursive Program Schemes: Semantics and Proof Theory, Ph.D. thesis, Free University, Amsterdam.Google Scholar
  3. Ehrenfeucht, A., and Zeiger, P. (1976), Complexity Measures for Regular Expressions, Journal of Computer System Science12, 134–146.Google Scholar
  4. Emerson, A. E., and Clarke, E. C. (1980), Characterizing Correctness Properties of Parallel Programs using Fixpoints, Seventh International Colloquium on Automata, Languages and Programming, 169–181.Google Scholar
  5. Fischer, M. J., and Ladner, R. E. (1979), Propositional Dynamic Logic of Regular Programs, Journal of Computer System Science18, 194–211.Google Scholar
  6. Harel, D. (1979), First Order Dynamic Logic, Springer-Verlag Lecture Notes in Computer Science68.Google Scholar
  7. Harel, D., and Sherman, R. (1984), Looping versus Repeating in Dynamic Logic, Information and Control55, 175–192, 1984.Google Scholar
  8. Hitchcock, P., and Park, D. M. R. (1973), Induction Rules and Termination Proofs, First International Colloquium on Automata, Languages, and Programming, 225–251.Google Scholar
  9. Hossley, R., and Rackoff, C. W. (1972), The Emptiness Problem for Automata on Infinite Trees, Thirteenth IEEE Symposium on Switching and Automata Theory, 121–124.Google Scholar
  10. Kfoury, A. J., and Park, D. M. R. (1975), On Termination of Program Schemes, Information and Control29, 243–251.Google Scholar
  11. Kozen, D. (1982), Results on the Propositional Mu-Calculus, Ninth International Colloquium on Automata, Languages, and Programming, 348–359.Google Scholar
  12. Kozen, D., and Parikh, R. J. (1983), A Decision Procedure for the Propositional MuCalculus, Second Workshop on Logics of Programs.Google Scholar
  13. Kripke, S. A. (1963), Semantical Considerations on Modal Logics, Acta Philosophica Fennica.Google Scholar
  14. McNaughton, R. (1966), Testing and Generating Infinite Sequences by a Finite Automaton, Information and Control9, 521–530.Google Scholar
  15. Meyer, A. R. (1974), Weak Monadic Second Order Theory of Successor is not Elementary Recursive, Boston Logic Colloquium, Springer-Verlag Lecture Notes in Mathematics453.Google Scholar
  16. Niwinski, D. (1984), The Propositional Mu-Calculus is More Expressive than the Propositional Dynamic Logic of Looping, unpublished manuscript.Google Scholar
  17. Parikh, R. J. (1979), A Decidability Result for a Second Order Process Logic, Nineteenth IEEE Symposium on the Foundations of Computer Science, 177–183.Google Scholar
  18. Parikh, R. J. (1983a), Cake Cutting, Dynamic Logic, Games, and Fairness, Second Workshop on Logics of Programs.Google Scholar
  19. Parikh, R. J. (1983b), Propositional Game Logic, Twenty-third IEEE Symposium on the Foundations of Computer Science.Google Scholar
  20. Park, D. M. R. (1970), Fixpoint Induction and Proof of Program Semantics, Machine Intelligence5, Edinburgh University Press.Google Scholar
  21. Park, D. M. R. (1976), Finiteness is Mu-Ineffable, Theoretical Computer Science3, 173–181.Google Scholar
  22. Pnueli, A., and Sherman, R. (1983),Propositional Dynamic Logic of Looping Flowcharts, Technical Report, Weizmann Institute of Science.Google Scholar
  23. Pratt, V. R. (1976), Semantical Considerations on Floyd-Hoare Logic, Seventeenth IEEE Symposium on Foundations of Computer Science, 109–121.Google Scholar
  24. Pratt, V. R. (1982), A Decidable Mu-Calculus: Preliminary Report, Twenty-second IEEE Symposium on the Foundations of Computer Science, 421–427.Google Scholar
  25. Rabin, M. O. (1969), Decidability of Second Order Theories and Automata on Infinite Trees, Transactions of the American Mathematical Society141, 1–35.Google Scholar
  26. Sherman, R. (1984), Variants of Propositional Dynamic Logic, Ph.D. thesis, Weizmann Institute of Science.Google Scholar
  27. Streett, R. S. (1980), A Propositional Dynamic Logic for Reasoning About Program Divergence, M.S. thesis, Massachusetts Institute of Technology.Google Scholar
  28. Streett, R. S. (1981), Propositional Dynamic Logic of Looping and Converse, MIT LCS Technical Report TR-263.Google Scholar
  29. Streett, R. S. (1982), Propositional Dynamic Logic of Looping and Converse is Elemantarily Decidable, Information and Control54, 121–141.Google Scholar
  30. Streett, R. S., and Emerson, E. A. (1984), The Propositional Mu-Calculus is Elementary, Eleventh International Colloquium on Automata, Languages, and Programming, Springer-Verlag Lecture Notes in Computer Science172, 465–472.Google Scholar
  31. Streett, R. S., and Emerson, E. A. (1985), An Automata Theoretic Decision Procedure for the Propositional Mu-Calculus, in preparation.Google Scholar
  32. Vardi, M. Y., and Stockmeyer, L. (1984), Improved Upper and Lower Bounds for Modal Logics of Programs, unpublished paper.Google Scholar
  33. Vardi, M., and Wolper, P (1984), Automata Theoretic Techniques for Modal Logics of Programs, Sixteenth ACM Symposium on the Theory of Computing.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Robert S. Streett
    • 1
  1. 1.Computer Science DepartmentBoston UniversityBoston

Personalised recommendations