# Extension of PDL and consequence relations

## Abstract

Most of the deduction algorithms are of the form of regular expressions. To investigate some properties of these deductions we introduce an extension of PDL with propositional constants and infinite conjunctions and disjunctions. We treat the formulas of the object logical language for which the deductions are as propositional variables, the sets of formulas — as propositional constants and the deductions — as regular programs. A Hilbert's style axiomatization of this logic is shown to be complete for countable Kripke structures of sets of formulas.

Examples include a logic for the PDL logical consequences, a kind of a nonmonotonic deduction algorithm and some properties of logical consequences.

## Keywords

PDL infinitary logic logical consequence relations deduction methods artificial intelligence monotonicity## Preview

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## References

- [A]H. Andréka. Sharpening the Characterization of the Power of Floyd method, Lecture Notes in Computer Science 148 (1983) 1–26Google Scholar
- [B]J.Barwise. Admissible Sets and Structures, Springer-Verlag, 1975Google Scholar
- [Be]J.van Benthem. Partiality and Nonmonotonicity in Classical Logic, report ISLI, 1984Google Scholar
- [Bu]A. Bundy. The Computer Modelling of Mathematical Reasoning, Academic Press, 1983Google Scholar
- [FL]M. Fischer and R. Ladner. Propositional Dynamic Logic of Regular Programs, J. Comp. Sys. Sci. 18 (1979) 194–211Google Scholar
- [G]D.Gabbay. Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics, D.Reidel, 1976Google Scholar
- [GS]Y. Gurevich and S. Shelah. Interpreting Second-Order Logic in the Monadic Logic of Linear Order, J. Symbolic Logic, 48 (1983) 816–828Google Scholar
- [HPS]D. Harel, A. Pnueli and J. Stavi. Further Results on Propositional Dynamic Logic of Nonregular Programs, Lecture Notes in Computer Science 131 (1982) 124–136Google Scholar
- [K]J.Keisler. Model Theory for Infinitary Logic, North-Holland, 1971Google Scholar
- [Ko]D. Kozen. On Induction vs. ж-Continuity, Lecture Notes in Computer Science 131 (1982) 167–176Google Scholar
- [MP]Z. Manna and A. Pnueli. Verification of Concurrent Programs: Temporal Proof Principles, Lecture Notes in Computer Science 131 (1982) 200–252Google Scholar
- [M]D. Miller. Expansion Tree Proofs and their Conversion to Natural Deduction Proofs, Lecture Notes in Computer Science 170 (1984) 375–393Google Scholar
- [N]I. N'emeti. Nonstandard Dynamic Logic, Lecture Notes in Computer Science 131 (1982) 311–348Google Scholar
- [P]S.Passy. Combinatory PDL, Ph.D. Thesis, July 1984, SofiaGoogle Scholar
- [PT]S. Passy and T. Tinchev. PDL with Data Constants, Inf. Proc. Letters 20 (1985) 35–41Google Scholar
- [Pl]D. Plaisted. Using Examples, Case Analysis, and Dependency Graphs in Theorem Proving, Lecture Notes in Computer Science 170 (1984) 356–374Google Scholar
- [R1]S.Radev. Infinitary Propositional Normal Modal Logic and Programming Languages, Ph.D. Thesis, November 1981, WarsawGoogle Scholar
- [R2]S. Radev. Propositional Logics of Formal Languages, Fundamenta Informaticae VII 4 (1984)Google Scholar
- [S]P. Schroeder-Heister. A Natural Extension of Natural Deduction, J. Symbolic Logic, 49 (1984) 1284–1300Google Scholar
- [T]S. Thomason. Reduction of Second-Order Logic to Modal Logic, Zait. Math. Logic und Grund. Math. 21 (1975) 107–114Google Scholar