Abstract
The fragment ∃ ∀ SO(ID) of second order logic extended with inductive definitions is expressive, and many interesting problems, such as conformant planning, can be naturally expressed as finite domain satisfiability problems of this logic. Such satisfiability problems are computationally hard (\(\Sigma^P_2\)). In this paper, we develop an approximate, sound but incomplete method for solving such problems that transforms a ∃ ∀ SO(ID) to a ∃ SO(ID) problem. The finite domain satisfiability problem for the latter language is in NP and can be handled by several existing solvers. We show that this provides an effective method for solving practically useful problems, such as common examples of conformant planning. We also propose a more complete translation to ∃ SO(FP), existential SO extended with nested inductive and coinductive definitions.
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References
Baral, C.: Knowledge representation, reasoning and declarative problem solving. Cambridge university press, Cambridge (2003)
Baral, C., Gelfond, M., Kosheleva, O.: Expanding queries to incomplete databases by interpolating general logic programs. J. Log. Program. 35(3), 195–230 (1998)
Denecker, M.: The well-founded semantics is the principle of inductive definition. In: Dix, J., Fariñas del Cerro, L., Furbach, U. (eds.) JELIA 1998. LNCS (LNAI), vol. 1489, pp. 1–16. Springer, Heidelberg (1998)
Denecker, M., Cortés-Calabuig, A., Bruynooghe, M., Arieli, O.: Towards a logical reconstruction of a theory for locally closed databases. ACM Transactions on Database Systems (2010) (accepted)
Denecker, M., Ternovska, E.: A logic of nonmonotone inductive definitions. ACM Trans. Comput. Log. 9(2) (2008)
Denecker, M., Vennekens, J., Bond, S., Gebser, M., Truszczynski, M.: The second answer set programming competition. In: Erdem, E., Lin, F., Schaub, T. (eds.) LPNMR 2009. LNCS, vol. 5753, pp. 637–654. Springer, Heidelberg (2009)
Doherty, P., Magnusson, M., Szalas, A.: Approximate databases: a support tool for approximate reasoning. Journal of Applied Non-Classical Logics 16(1-2), 87–118 (2006)
Immerman, N.: Descriptive Complexity. Springer, Heidelberg (1998)
Mariën, M., Wittocx, J., Denecker, M.: The IDP framework for declarative problem solving. In: Search and Logic: Answer Set Programming and SAT, pp. 19–34 (2006)
Mitchell, D.G., Ternovska, E.: A framework for representing and solving np search problems. In: AAAI, pp. 430–435 (2005)
Ping, H., De Cat, B., Denecker, M.: Fo(fd): Extending classical logic with rule-based fixpoint definitions. In: International Conference on Logic Programming, ICLP 2010 (2010)
Son, T.C., Tu, P.H., Gelfond, M., Ricardo Morales, A.: An approximation of action theories of and its application to conformant planning. In: Baral, C., Greco, G., Leone, N., Terracina, G. (eds.) LPNMR 2005. LNCS (LNAI), vol. 3662, pp. 172–184. Springer, Heidelberg (2005)
Stickel, M.E.: A prolog technology theorem prover: Implementation by an extended prolog compiler. J. Autom. Reasoning 4(4), 353–380 (1988)
Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Slisenko, A.O. (ed.) Studies in Constructive Mathematics and Mathematical Logic II. Seminars in Mathematics: Steklov Mathematical Institute, vol. 8, pp. 115–125. Consultants Bureau, New York (1968)
Van Gelder, A.: The alternating fixpoint of logic programs with negation. Journal of Computer and System Sciences 47(1), 185–221 (1993)
Wittocx, J.: Finite Domain and Symbolic Inference Methods for Extensions of First-Order Logic. PhD thesis, K.U.Leuven (May 2010)
Wittocx, J., Mariën, M., Denecker, M.: Approximate reasoning in first-order logic theories. In: KR, pp. 103–112 (2008)
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Vlaeminck, H., Wittocx, J., Vennekens, J., Denecker, M., Bruynooghe, M. (2010). An Approximative Inference Method for Solving ∃ ∀SO Satisfiability Problems. In: Janhunen, T., Niemelä, I. (eds) Logics in Artificial Intelligence. JELIA 2010. Lecture Notes in Computer Science(), vol 6341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15675-5_28
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DOI: https://doi.org/10.1007/978-3-642-15675-5_28
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