Abstract
Second-order quantifier elimination in the context of classical logic emerged as a powerful technique in many applications, including the correspondence theory, relational databases, deductive and knowledge databases, knowledge representation, commonsense reasoning and approximate reasoning.
In the current paper we generalize the result of [19] by allowing modal operators. This allows us to provide a unifying framework for many applications, that require the use of intensional concepts. Examples of applications of the technique in AI are also provided.
Supported in part by the WITAS project grant under the Wallenberg Foundation, Sweden, KBN grant 8 T11C 009 19 and The College of Economics and Computer Science.
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References
M. Cadoli. Tractable Reasoning in Artificial Intelligence, volume 941 of LNAI. Springer-Verlag, Berlin Heidelberg, 1995.
P. Doherty, J. Kachniarz, and A. Szałas. Meta-queries on deductive databases. Fundamenta Informaticae, 40(1):17–30, 1999.
P. Doherty, J. Kachniarz, and A. Szałas. Using contextually closed queries for local closed-world reasoning in rough knowledge databases. In L. Polkowski and A. Skowron, editors, Rough-Neuro Computing: Techniques for Computing with Words, pages 217–248. Springer-Verlag, 2002.
P. Doherty, W. Łukaszewicz, A. Skowron, and A. Szałas. Combining rough and crisp knowledge in deductive databases. In L. Polkowski and A. Skowron, editors, Rough-Neuro Computing: Techniques for Computing with Words, pages 185–216. Springer-Verlag, 2002.
P. Doherty, W. Łukaszewicz, and A. Szałas. Computing circumscription revisited. Journal of Automated Reasoning, 18(3):297–336, 1997.
P. Doherty, W. Łukaszewicz, and A. Szałas. General domain circumscription and its effective reductions. Fundamenta Informaticae, 36(1):23–55, 1998.
P. Doherty, W. Łukaszewicz, and A. Szałas. Declarative PTIME queries for relational databases using quantifier elimination. Journal of Logic and Computation, 9(5):739–761, 1999.
P. Doherty, W. Łukaszewicz, and A. Szałas. Efficient reasoning using the local closed-world assumption. In A. Cerri and D. Dochev, editors, Proc. 9th Int. Conference AIMSA 2000, volume 1904 of LNAI, pages 49–58. Springer-Verlag, 2000.
P. Doherty, W. Łukaszewicz, and A. Szałas. Computing strongest necessary and weakest suficient conditions of first-order formulas. International Joint Conference on AI (IJCAI’2001), pages 145–151, 2001.
D. M. Gabbay and H. J. Ohlbach. Quantifier elimination in second-order predicate logic. In B. Nebel, C. Rich, and W. Swartout, editors, Principles of Knowledge representation and reasoning, KR 92, pages 425–435. Morgan Kauffman, 1992.
J.W. Garson. Quantification in modal logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 249–307. D. Reidel Pub. Co., 1984.
J. Kachniarz and A. Szałas. On a static approach to verification of integrity constraints in relational databases. In E. Orłowska and A. Szałas, editors, Relational Methods for Computer Science Applications, pages 97–109. Springer Physica-Verlag, 2001.
H. Kautz and B. Selman. Knowledge compilation and theory approximation. Journal of the ACM, 43(2):193–224, 1996.
V. Lifschitz. Computing circumscription. In Proc. 9th IJCAI, pages 229–235, Palo Alto, CA, 1985. Morgan Kaufmann.
V. Lifschitz. Circumscription. In D. M. Gabbay, C. J. Hogger, and J. A. Robinson, editors, Handbook of Artificial Intelligence and Logic Programming, volume 3, pages 297–352. Oxford University Press, 1991.
F. Lin. On strongest necessary and weakest suficient conditions. In A.G. Cohn, F. Giunchiglia, and B. Selman, editors, Proc. 7th International Conf. on Principles of Knowledge Representation and Reasoning, KR2000, pages 167–175, San Francisco, Ca., 2000. Morgan Kaufmann Pub., Inc.
J. McCarthy. Approximate objects and approximate theories. In A.G. Cohn, F. Giunchiglia, and B. Selman, editors, Proc. 7th International Conf. on Principles of Knowledge Representation and Reasoning, KR2000, pages 519–526, San Francisco, Ca., 2000. Morgan Kaufmann Pub., Inc.
A. Nonnengart, H.J. Ohlbach, and A. Szałas. Elimination of predicate quantifiers. In H.J. Ohlbach and U. Reyle, editors, Logic, Language and Reasoning. Essays in Honor of Dov Gabbay, Part I, pages 159–181. Kluwer, 1999.
A. Nonnengart and A. Szałas. A fixpoint approach to second-order quantifier elimination with applications to correspondence theory. In E. Orłowska, editor, Logic at Work: Essays Dedicated to the Memory of Helena Rasiowa, volume 24 of Studies in Fuzziness and Soft Computing, pages 307–328. Springer Physica-Verlag, 1998.
H. Simmons. The monotonous elimination of predicate variables. Journal of Logic and Computation, 4:23–68, 1994. (special issue).
A. Szałas. On the correspondence between modal and classical logic: An automated approach. Journal of Logic and Computation, 3:605–620, 1993.
A. Szałas. On an automated translation of modal proof rules into formulas of the classical logic. Journal of Applied Non-Classical Logics, 4:119–127, 1994.
J. van Benthem. Modal Logic and Classical Logic. Bibliopolis, Naples, 1983.
J. van Benthem. Correspondence theory. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 167–247. D. Reidel Pub. Co., 1984.
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Szalas, A. (2002). Second-Order Quantifier Elimination in Modal Contexts. In: Flesca, S., Greco, S., Ianni, G., Leone, N. (eds) Logics in Artificial Intelligence. JELIA 2002. Lecture Notes in Computer Science(), vol 2424. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45757-7_19
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DOI: https://doi.org/10.1007/3-540-45757-7_19
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