Abstract
In this paper we show that it is possible to eliminate the “converse” operator from the propositional dynamic logic CPDL (Converse PDL), without compromising the soundness and completeness of inference for it. Specifically we present an encoding of CPDL formulae into PDL that eliminates the converse programs from a CPDL formula, but adds enough information so as not to destroy its original meaning with respect to satisfiability, validity, and logical implication. Notably, the resulting PDL formula is polynomially related to the original one. This fact allows one to build inference procedures for CPDL, by encoding CPDL formulae into PDL, and then running an inference procedure for PDL.
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References
M. Ben-Ari, J. Y. Halpern, and A. Pnueli. Deterministic propositional dynamics logic: finite models, complexity, and completeness, Journal of Computer and System Sciences, 25:402–417, 1982.
P. Blackburn and E. Spaan. A modal perspective on computational complexity of attribute value grammar. Journal of Logic, Language and Information, 2:129–169, 1993.
G. De Giacomo. Decidability of Class-Based Knowledge Representation Formalisms. PhD thesis, Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, 1995.
G. De Giacomo and M. Lenzerini. Boosting the correspondence between description logics and propositional dynamic logics. In Proceedings of the Twelth National Conference on Artificial Intelligence (AAAI-94), pages 205–212, 1994.
G. De Giacomo and M. Lenzerini. What's in an aggregate: foundation for description logics with tuples and set. In Proceedings of the Fourteenth International Joint Conference on Artificial Intelligence (IJCAI-95), pages 801–807, 1995.
M. Fattorosi-Barnaba and F. De Caro. Graded modalities I. Studia Logica, 44:197–221, 1985.
N. J. Fisher and R. E. Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18:194–211, 1979.
N. Friedman and J. Halpern. On the complexity of conditional logics. In Proceedings of the Fourth International Conference on Principles of Knowledge Representation and Reasoning, page 202–213, 1994.
G. Gargov and V. Goranko. Modal logic with names. Journal of Philosphical Logic, 22:607–636, 1993.
J. Halpern and Y. Moses. A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence, 54:319–379, 1992.
D. Harel. Dynamic logic. In Handbook of Philosophical Logic, pages 497–603. D. Reidel Publishing Company, Oxford, 1984.
D. Kozen and J. Tiuryn. Logics of programs. In Handbook of Theoretical Computer Science, pages 790–840. Elsevier Science Publishers, 1990.
S. Passy and T. Tinchev. An essay in combinatory dynamic logic. Information and Computation, 93:263–332, 1991.
V. R. Pratt. A practical decision method for propositional dynamic logic. In Proceedings of the 10th Annual Symposium on Theory of Computing, pages 326–337, 1978.
V.R. Pratt. Models of program logics. In Proceedings of the 20th IEEE Symposium on the Foundations of Computer Science, pages 115–122, 1979.
V. R. Pratt. A near-optimal method for reasoning about action. Journal of Computer and System Sciences, 20:231–255, 1980.
K. Schild. A correspondence theory for terminological logics: preliminary report. In Proceedings of the Twelth International Joint Conference on Artificial Intelligence (IJCAI-91), pages 466–471, 1991.
C. Stirling. Modal and temporal logic. In Handbook of Logic in Computer Science, pages 477–563. Clarendon Press, Oxford, 1992.
R. S. Streett. Propositional dynamic logic of looping and converse is elementary decidable. Information and Control, 54:121–141, 1982.
J. Van Benthem and J. Bergstra. Logic of transition systems. Journal of Logic, Language and Information, 3(4):247–283, 1995.
J. Van benthem, J. Van Eijck, and V. Stebletsova. Modal logic, transition systems and processes. Journal of Logic and Computation, 4(5):811–855, 1994.
W. van der Hoek and M. de Rijke. Counting objects. Journal of Logic and Computation, 5(3):325–345, 1995.
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De Giacomo, G. Eliminating “converse” from converse PDL. J Logic Lang Inf 5, 193–208 (1996). https://doi.org/10.1007/BF00173700
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DOI: https://doi.org/10.1007/BF00173700