Abstract
The paper presents aset-theoretic translation method for polymodal logics that reduces derivability in a large class of propositional polymodal logics to derivability in a very weak first-order set theory Ω. Unlike most existing translation methods, the one we propose applies to any normal complete finitely axiomatizable polymodal logic, regardless of whether it is first-order complete or an explicit semantics is available. The finite axiomatizability of Ω allows one to implement mechanical proof-search procedures via the deduction theorem. Alternatively, more specialized and efficient techniques can be employed. In the last part of the paper, we briefly discuss the application ofset T-resolution to support automated derivability in (a suitable extension of) Ω.
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References
Aczel, P.:Non-Well-Founded sets, CSLI, Lecture Notes No. 14, 1988.
Benthem, J. van:Modal Logic and Classical Logic, Bibliopolis, Napoli and Atlantic Heights, NJ, 1985.
Benthem, J. van, D'Agostino, G., Montanari, A., and Policriti, A.: Modal Deduction in Second-Order Logic and Set Theory, Research Report in the ILLC-series, ML-95-02, University of Amsterdam, February 1995.
Cantone, D.: A Decision Procedure for a Class of Unquantified Formulae of Set Theory Involving the Powerset and Singleton Operators, PhD Thesis, New York University, 1986.
Cantone, D., Ferro, A., and Omodeo, E. G.:Computable Set Theory. Vol. 1, Oxford University Press, Int. Series of Monographs on Computer Science, 1989.
Cantone, D., Ferro, A., and Schwartz, J. T.: Decision procedures for elementary sublanguages of set theory VI. Multilevel syllogistic extended by the powerset operator,Comm. Pure App. Math. 38(1) (1985), 549–571.
Cantone, D., Omodeo, E., and Policriti, A.: The automation of syllogistic II. Optimization and complexity issues,J. Automated Reasoning 6(2) (1990), 173–187.
D'Agostino, G., Montanari, A., and Policriti, A.: Translating Modal Formulae as Set-Theoretic Terms, Research Report 10/94, Dipartimento di Matematica e Informatica, Università di Udine, May 1994 (also in Logic Colloquium '94).
D'Agostino, G., Montanari, A., and Policriti, A.:Decidability Results for Modal Theorem Proving, in preparation.
Fitting, M.:Proofs Methods for Modal and Intuitionistic Logics, Reidel, Dordrecht, 1983.
Gabbay, D. M. and Ohlbach, H. J.: Quantifier elimination in second-order predicate logic, inProc. 4th Int. Conf. on Principles of Knowledge Representation and Reasoning, KR'92, Morgan Kaufmann, 1992, pp. 425–436.
Hughes, G. F. and Cresswell, M. J.:A Companion to Modal Logic, Methuen, London, 1984.
Kracht, M.: Highway to the Danger Zone,J. Logic and Computation (1994), to appear.
Jech, T.:Set Theory; Pure and Applied Mathematics Series, Academic Press, 1978.
Mendelson, E.:Introduction to Mathematical Logic, 2nd edn; Van Nostrand, New York, 1979.
Nonnengart, A.: First-order modal logic theorem proving and functional simulation, inProc. 13th Int. Joint Conf. on Artificial Intelligence, IJCAI-93, Chambery, France, 1993, pp. 80–85.
Ohlbach, H. J.: Semantic-based translation methods for modal logics,J. Logic and Computation 1(5), 1991.
Ohlbach, H. J.: Translation methods for non-classical logics: An overview;Bull. of the IGLP 1(1) (1993), 69–89.
Parlamento, F. and Policriti, A.: Decision procedures for elementary sublanguages of set theory XIII. Model graphs, reflection and decidability,J. Automated Reasoning 7 (1991), 271–284.
Policriti, A. and Schwartz, J. T.: Theorem Proving I; Research Report 08/92, Università di Udine, July 1992, to appear inJ. Symbolic Computation.
Smoryński, C.:Self-Reference and Modal Logic. Springer, New York, 1985.
Thomason, S. K.: Reduction of tense logic to modal logic I,J. Symbolic Logic 39(3) (1974), 549–551.
Thomason, S. K.: Reduction of tense logic to modal logic II,Theoria 41 (1975), 154–169.
Wansing, H.: Sequent calculi for normal modal propositional logics,J. Logic and Computation 4(2) (1994), 125–142.
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This work has been supported by funds MURST 40 and 60%. The second author was supported by a grant from the Italian Consiglio Nazionale delle Ricerche (CNR). A previous version of this paper has appeared as a research report in the ILLC-series, ML-94-09, University of Amsterdam. A short version is to be presented at STACS '95 in Munich.
On leave at ILLC, Universiteit van Amsterdam, Plantage Muidergracht, 24, 1018 TV Amsterdam, The Netherlands.
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D'Agostino, G., Montanari, A. & Policriti, A. A set-theoretic translation method for polymodal logics. J Autom Reasoning 15, 317–337 (1995). https://doi.org/10.1007/BF00881803
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DOI: https://doi.org/10.1007/BF00881803