# The completeness of propositional dynamic logic

• Rohit Parikh
Communications
Part of the Lecture Notes in Computer Science book series (LNCS, volume 64)

## Abstract

Propositional modal logic of programs has been introduced by Fischer and Ladner [1], following ideas of Pratt [4]. We shall call it propositional dynamic logic (PDL) following the terminology of Harel, Meyer and Pratt. In the following we prove the completeness of a rather natural set of axioms for this logic and for an extension of it obtained by allowing the inverse operation which converts a program into its inverse.

The following is a brief sketch of the plan of the proof. We introduce two auxiliary notions, that of pseudomodel and that of nonstandard model. Pseudomodels are highly syntactic objects and merely represent partial attempts to spell out a model. Thus an inconsistent formula may have a pseudomodel but every attempt to spell out the complete details of a model corresponding to the pseudomodel will, for an inconsistent formula, run into obstacles. A nonstandard model is like a model but we do not insist that Rα*=(Rα). Rα* is some reflexive transitive relation containing Rα, but not necessarily the smallest.

We shall show that if a formula A is not disprovable from the axioms then it has a series of consistent pseudomodels whose union is a nonstandard model satisfying certain special induction axioms. It is then shown how such a nonstandard model can be converted into a model in the usual sense.

## Keywords

Modal Logic Completeness Theorem Inverse Operation Nonstandard Model Propositional Dynamic 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.

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## References

1. [1]
M.J. Fischer and R.E. Ladner, "Propositional Modal Logic of Programs," Proceedings 9th Annual ACM Symposium on Theory of Computing, Boulder, Colorado, May 1977, 286–294.Google Scholar
2. [2]
D. Harel, A.R. Meyer and V.R. Pratt, "Computability and Completeness in Logics of Programs," Ibid., pp. 261–268.Google Scholar
3. [3]
R. Parikh, "A Completeness Result for PDL," Research report, March 1978.Google Scholar
4. [4]
R. Parikh, "A Decidability Result for a Second Order Process Logic," Research report, May 1978.Google Scholar
5. [5]
V.R. Pratt, "Semantical Considerations in Floyd-Hoare Logic," Proceedings 17th IEEE Symposium on Foundations of Computer Science, 1976.Google Scholar
6. [6]
V.R. Pratt, "The Tableau Method for Propositional Dynamic Logic," Research Report, M.I.T., 11/31/77, revised 12/20/77.Google Scholar
7. [7]
K. Segerberg, "A Completeness Theorem in the Modal Logic of Programs," abstract, Notices of the American Mathematical Society, October 1977, p. A552.Google Scholar

© Springer-Verlag Berlin Heidelberg 1978

## Authors and Affiliations

• Rohit Parikh
• 1
1. 1.Laboratory for Computer Science, M.I.T., and Mathematics DepartmentBoston UniversityUSA