Abstract
Standard and nonstandard models of Propositional Dynamic Logic differ in their interpretation of loops. In Standard models, a loop is interpreted as the Kleene closure of the interpretation of its loop body; in nonstandard (Loop Invariant) models, a loop is interpreted as a program which preserves invariant assertions over the loop body.
In this paper we show that both interpretations are adequate to represent loops in PDL. We demonstrate this in two ways: First we note that Standard and Loop Invariant models are distinct but not distinguishable within PDL. Second, we show that the class of Loop Invariant models is complete with respect to the Segerberg axiomatization of PDL. Since completeness of the class of Loop Invariant models implies completeness of the class of Standard models, Standard models are also complete with respect to this axiomatization.
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References
Berman, F. “Syntactic and Semantic Structure in Propositional Dynamic Logic,” Ph. D. Dissertation. Seattle: University of Washington, 1979.
Berman, F. “A Completeness Technique forD-Axiomatizable Semantics,” Proceedings of the 11th Annual ACM Symposium on the Theory of Computing. Atlanta, Ga., May 1979.
Fischer, M. J., and R. E. Ladner. “Propositional Dynamic Logic of Regular Programs,” JCSS, vol. 18, no. 2, April 1979.
Harel, D.,First-Order Dynamic Logic, Lecture Notes in Computer Science no. 68, New York/Berlin: Springer-Verlag, 1979.
Kozen, D. “A Representation Theorem for Models of*-Free PDL.” Proceedings of the 7th Colloquium on Automata, Languages and Programming, Noordwijkerhout, the Netherlands, July 1980.
Parikh, R. “A Completeness Result for Propositional Dynamic Logic,” Symposium on the Mathematical Foundations of Computer Science. Zakopane, Poland, 1978.
Pratt, V. “Dynamic Algebras and the Nature of Induction,” Proceedings of the 12th Annual ACM Symposium on the Theory of Computing. Los Angeles, California, April 1980.
Segerberg, K. “A Completeness Theorem in the Modal Logic of Programs,” Preliminary Report, Notices of the AMS, 24:6, A-552, Oct. 1977.
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Berman, F. Semantics of looping programs in Propositional Dynamic Logic. Math. Systems Theory 15, 285–294 (1981). https://doi.org/10.1007/BF01786985
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DOI: https://doi.org/10.1007/BF01786985