Andrei Voronkov
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Abstract
We present a bottom-up decision procedure for propositional modal logic K based on the inverse method. The procedure is based on the “inverted” version of a sequent calculus. To restrict the search space, we prove a number of redundancy criteria for derivations in the sequent calculus. We introduce a new technique of proving redundancy criteria, based on the analysis of tableau-based derivations in K. Moreover, another new technique is based on so-called traces. A new search with a strong notion of subsumption. This technique is based on so-called traces. A new formalization of the inverse method in the form of a path calculus considerably simplifies all proofs as compared to the previously published presentations of the inverse method. Experimental results demonstrate that our method is competitive with many state-of-the-art implementations of K.
- BAADER,F.AND HOLLUNDER, B. 1991. A terminological knowledge representation system with complete inference algorithms. In PDK'91, H. Boley and M. Richter, Eds. Lecture Notes in Artificial Intelligence, vol. 567. Springer Verlag, 67-86.]] Google Scholar
- BACHMAIR,L.AND GANZINGER, H. 2000. Resolution theorem proving. In Handbook of Automated Reasoning, A. Robinson and A. Voronkov, Eds. Elsevier Science and MIT Press. To appear.]]Google Scholar
- BALSIGER, P., HEUERDING, A., AND SCHWENDIMANN, S. 1998. Logics Workbench 1.0. In Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX'98, H. de Swart, Ed. Lecture Notes in Artificial Intelligence, vol. 1397. Springer Verlag, Oisterwijk, The Netherlands, 35.]] Google Scholar
- CALVANESE, D., DE GIACOMO, G., LENZERINI, M., AND NARDI, D. 2000. Reasoning in expressive description logics. In Handbook of Automated Reasoning, A. Robinson and A. Voronkov, Eds. Elsevier Science and MIT Press. To appear.]] Google Scholar
- CATACH, L. 1991. Tableaux: a general theorem prover for modal logics. Journal of Automated Reasoning 7, 4, 489-510.]]Google Scholar
- DEGTYAREV,A.AND VORONKOV, A. 1996. Equality elimination for the tableau method. In Design and Implementation of Symbolic Computation Systems. International Symposium, DISCO'96, J. Calmet and C. Limongelli, Eds. Lecture Notes in Computer Science, vol. 1128. Springer Verlag, Karlsruhe, Germany, 46-60.]] Google Scholar
- FITTING, M. 1983. Proof methods for modal and intuitionistic logics. Synthese Library, vol. 169. Reidel Publ. Comp.]]Google Scholar
- GENTZEN, G. 1934. Untersuchungen uber das logische Schlieben. Mathematical Zeitschrift 39, 176-210, 405-431. Translated as {Gentzen 1969}.]]Google Scholar
- GENTZEN, G. 1969. Investigations into logical deduction. In The Collected Papers of Gerhard Gentzen, M. Szabo, Ed. North Holland, Amsterdam, 68-131. Originally appeared as {Gentzen 1934}.]]Google Scholar
- GIUNCHIGLIA, E., GIUNCHIGLIA,F.,AND TACCHELLA, A. 1999. SAT-based decision procedures for classical modal logics. Journal of Automated Reasoning. To appear in the Special Issues: Satisfiability at the start of the year 2000 (SAT 2000).]] Google Scholar
- GIUNCHIGLIA, E., GIUNCHIGLIA, F., SEBASTIANI, R., AND TACCHELLA, A. 1998. More evaluation of decision procedures for modal logics. In Principles of Knowledge Representation and Reasoning: Proceedings of the Sixth International Conference (KR'98), A. Cohn, L. Schubert, and S. Shapiro, Eds. Morgan Kaufmann, San Francisco, CA, 626-635.]]Google Scholar
- GIUNCHIGLIA,F.AND SEBASTIANI, R. 1996a. Building decision procedures for modal logics from propositional decision procedures: Case study of modal K. In CADE-13, M. McRobbie and J. Slaney, Eds. Lecture Notes in Artificial Intelligence, vol. 1104. Springer Verlag, 583-597.]] Google Scholar
- GIUNCHIGLIA,F.AND SEBASTIANI, R. 1996b. A SAT-based decision procedure for ALC .InKR'96. 304-314.]]Google Scholar
- GOBLE, L. 1974. Gentzen systems for modal logic. Notre Dame J. of Formal Logic 15, 455-461.]]Google Scholar
- HAARSLEV, V., M OLLER, R., AND TURHAN, A.-Y. 1998. Implementing an ALCRP(D) ABox reasonerprogress report. In Proc. DL-98 International Description Logic Workshop. Trento, Italy.]]Google Scholar
- HORROCKS, I. 1997. Optimizing tableaux decision procedures for modal logic. Ph.D. thesis, University of Manchester.]]Google Scholar
- HORROCKS, I. 1998. Using an expressive description logic: FaCT or fiction? In Principles of Knowledge Representation and Reasoning: Proceedings of the Sixth International Conference (KR'98), A. Cohn, L. Schubert, and S. Shapiro, Eds. Morgan Kaufmann, San Francisco, CA, 636-647.]]Google Scholar
- HORROCKS,I.AND PATEL-SCHNEIDER, P. 1998. FaCT and DLP. In Automated Reasoning with Ana-lytic Tableaux and Related Methods, International Conference, TABLEAUX'98, H. de Swart, Ed. Lecture Notes in Artificial Intelligence, vol. 1397. Springer Verlag, Oisterwijk, The Netherlands, 27-30.]] Google Scholar
- HORROCKS, I., PATEL-SCHNEIDER,P.,AND SEBASTIANI, R. 2000. An analysis of empirical testing for modal decision procedures. Logic Journal of the IGPL 8, 3, 293-323.]]Google Scholar
- HUSTADT,U.AND SCHMIDT, R. 1997. On evaluating decision procedures for modal logic. In IJCAI-97. Vol. 1. 202-207.]]Google Scholar
- MASLOV, S. 1971. Proof-search strategies for methods of the resolution. In Machine Intelligence, B. Meltzer and D. Michie, Eds. Vol. 6. American Elsevier, 77-90.]]Google Scholar
- MINTS, G. 1993. Resolution calculus for the first order linear logic. Journal of Logic, Language and Information 2,58-93.]]Google Scholar
- MINTS, G. 1994. Resolution strategies for the intuitionistic logic. In Constraint Programming. NATO ASI Series F. Springer Verlag, 289-311.]]Google Scholar
- MINTS, G., DEGTYAREV, A., AND VORONKOV, A. 2000. The inverse method. In Handbook of Automated Reasoning, A. Robinson and A. Voronkov, Eds. Elsevier Science and MIT Press. To appear.]]Google Scholar
- MINTS, G., OREVKOV,V.,AND TAMMET, T. 1996. Transfer of sequent calculus strategies to resolution for S4. In Proof Theory of Modal Logic. Studies in Pure and Applied Logic. Kluwer Academic Publishers.]]Google Scholar
- NIEWENHUIS,R.AND RUBIO, A. 2000. Paramodulation-based theorem proving. In Handbook of Automated Reasoning, A. Robinson and A. Voronkov, Eds. Elsevier Science and MIT Press. To appear.]]Google Scholar
- PATEL-SCHNEIDER, P. 1998. DLP system description. In Collected Papers from the International Description Logic Workshop (DL'98), E. Franconi, G. Giacomo, R. MacGregor, W. Nutt, C. Welty, and F. Sebastiani, Eds. CEUR-WS, vol. 11. 87-89.]]Google Scholar
- SCHMIDT, R. 1998. Resolution is a decision procedure for many propositional modal logics. In Advances in Modal Logic, I, M. Kracht, M. de Rijke, H. Wansing, and M. Zakharyaschev, Eds. Lecture Notes, vol. 87. CSLI Publications, Stanford, 189-208.]]Google Scholar
- TACCHELLA, A. 1999. *SAT system description. In Proceedings of the 1999 International Description Logic Workshop (DL'99), P. Lambrix, A. Borgida, M. Lenzerini, R. M oller, and P. Patel-Schneider, Eds. CEUR-WS, vol. 22. 142-144.]]Google Scholar
- TAMMET, T. 1994. Proof strategies in linear logic. Journal of Automated Reasoning 12, 3, 273-304.]]Google Scholar
- TAMMET, T. 1996. A resolution theorem prover for intuitionistic logic. In CADE-13, M. McRobbie and J. Slaney, Eds. Lecture Notes in Artificial Intelligence, vol. 1104. 2-16.]] Google Scholar
- VORONKOV, A. 1985. A proof search method (in Russian). Vychislitelnye Sistemy 107, 109-123.]]Google Scholar
- VORONKOV, A. 1990. A proof-search method for the first order logic. In COLOG'88, P. Martin-L of and G. Mintz, Eds. Lecture Notes in Computer Science, vol. 417. Springer Verlag, 327-340.]] Google Scholar
- VORONKOV, A. 1992. Theorem proving in non-standard logics based on the inverse method. In 11th International Conference on Automated Deduction, D. Kapur, Ed. Lecture Notes in Artificial Intelligence, vol. 607. Springer Verlag, 648-662.]] Google Scholar
- VORONKOV, A. 1999. K K : a theorem prover for k.InAutomated Deduction-CADE-16. 16th International Conference on Automated Deduction, H. Ganzinger, Ed. Lecture Notes in Artificial Intelligence, vol. 1632. Springer Verlag, Trento, Italy, 383-387.]] Google Scholar
- VORONKOV, A. 2000. How to decide K using K .InPrinciples of Knowledge Representation and Reasoning (KR'2000), A. Cohn, F. Giunchiglia, and B. Selman, Eds. 198-209.]]Google Scholar
- WALLEN, L. 1990. Automated Deduction in Nonclassical Logics. The MIT Press.]] Google Scholar
- WEIDENBACH, C., GAEDE,B.,AND ROCK, G. 1996. SPASS & FLOTTER. version 0.42. In CADE-13, M. McRobbie and J. Slaney, Eds. Lecture Notes in Artificial Intelligence, vol. 1104. Springer Verlag, 141-145.]] Google Scholar
Index Terms
- How to optimize proof-search in modal logics: new methods of proving redundancy criteria for sequent calculi
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