Abstract
This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of “here-and-there”. We show that on logic programs equilibrium logic coincides with the inference operation associated with the stable model and answer set semantics of Gelfond and Lifschitz. We thereby obtain a very simple characterisation of answer set semantics as a form of minimal model reasoning in N2, while equilibrium logic itself provides a natural generalisation of this semantics to arbitrary theories. We discuss briefly some consequences and applications of this result.
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References
Akama, S, Constructive Predicate Logic with Strong Negation and Model Theory, Notre Dame J Formal Logic 29 (1988), 18–27.
Akama, S, On the Proof Method for Constructive Falsity, Zeitschr. f. math. Logik und Grundlagen d. Math. 34 (1988), 385–392.
Almukdad, A & Nelson, D, Constructible Falsity and Inexact Predicates, JSL 49 (1984), 231–233.
Alferes, J J & Pereira, L M, Reasoning with Logic Programming LNAI 1111, Springer, 1996.
van Dalen, D, Intuitionistic Logic, in D Gabbay, & F Guenthner (eds), Handbook of Philosophical Logic, Vol. III, Kluwer, Dordrecht, 1986.
Dietrich, J & Herre, H, Outline of Nonmonotonic Model Theory, NTZ Report, Universität Leipzig, 1994.
Gelfond, M, Logic Programming and Reasoning with Incomplete Information, Annals of Mathematics and Artificial Intelligence 12 (1994), 89–116.
Gelfond, M, & Lifschitz, V, The Stable Model Semantics for Logic Programs, in K Bowen & R Kowalski (eds), Proc 5th Int Conf on Logic Programming 2, MIT Press, 1070–1080.
Gelfond, M & Lifschitz, V, Logic Programs with Classical Negation, in D Warren & P Szeredi (eds), Proc ICLP-90, MIT Press, 1990, 579–597.
Gelfond, M, & Lifschitz, V, Classical Negation in Logic Programs and Disjunctive Databases, New Generation Computing (1991), 365–387.
Gödel, K, Zum intuitionistischen Aussagenkalkül, Anzeiger der Akademie der Wissenschaften in Wien 69 65–66; reprinted in em Kurt Gödel, Collected Works, Volume 1, OUP, 1986.
Gurevich, Y, Intuitionistic Logic with Strong Negation, Studia Logica 36 (1977), 49–59.
Heyting, A, Die formalen Regeln der intuitionistischen Logik, Sitz. Berlin 1930, 42–56.
Kracht, M, On Extensions of Intermediate Logics by Strong Negation, Journal of Philosophical Logic, 1997, forthcoming.
Leone, N, Rullo, P, & Scarcello, F, Declarative and Fixpoint Characterizations of Disjunctive Stable Models, in J Lloyd (ed), Proc ILPS '95, MIT Press, 1995.
Lifschitz, V, Minimal Belief and Negation as Failure Artificial Intelligence. 70 (1994), 53–72.
Lifschitz, V & Schwarz, G, Extended Logic Programs as Autoepistemic Theories, in L M Pereira & A Nerode (eds), Logic Programming and Non-Monotonic Reasoning, MIT Press, 1993.
Lin, F & Shoham, Y, A Logic of Knowledge and Justified Assumption, Artificial Intelligence 57 (1992), 271–290.
Lukasiewicz, J, Die Logik und das Grundlagenproblem, in Les Entretiens de Zurich sur les Fondaments et la Methode des Sciences Mathematiques 1938, Zurich, 1941.
Marek, V & Truszczynski, M, Nonmonotonic Logic, Springer Verlag, 1993.
Marek, V & Truszczynski, M, Reflexive Autoepistemic Logic and Logic Programming, in L M Pereira & A Nerode (eds), Logic Programming and Non-Monotonic Reasoning, MIT Press, 1993.
Nelson, D, Constructible Falsity, JSL 14 (1949), 16–26.
Pearce, D, Answer Sets and Nonmonotonic S4, in R Dyckhoff (ed), Extensions of Logic Programming, LNAI 798, Springer Verlag, 1994.
Pearce, D, Nonmonotonicity and Answer Set Inference, to appear in A. Nerode (ed), Logic Programming and Nonmonotonic Reasoning. Proceedings LPNMR 95 LNAI, Springer-Verlag, 1995.
Pearce, D, From Here to There: Stable Negation in Logic Programming, in D Gabbay & H Wansing (eds), What is Negation?, Kluwer, 1997, forthcoming.
Pearce, D, Stable Inference Made Simple, in Proc DGNMR97, Third Dutch/German Workshop on Nonmonotonic Reasoning Techniques and their Applications, MPI Saabrücken, 1997.
Rautenberg, W, Klassische und Nichtklassische Aussagenlogik, Vieweg, Wiesbaden, 1979.
Thomason, R H, A Semantical Study of Constructible Falsity, Zeit. f. math. Logik und Grundlagen der Mathematik 15 (1969), 247–257.
Vorob'ev, N N, A Constructive Propositional Calculus with Strong Negation (in Russian), Doklady Akademii Nauk SSR 85 (1952), 465–468.
Vorob'ev, N N, The Problem of Deducibility in Constructive Propositional Calculus with Strong Negation (in Russian), Doklady Akademii Nauk SSR 85 (1952), 689–692.
Wójcicki, R, Theory of Logical Calculi, Kluwer, Dordrecht, 1988.
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Pearce, D. (1997). A new logical characterisation of stable models and answer sets. In: Dix, J., Pereira, L.M., Przymusinski, T.C. (eds) Non-Monotonic Extensions of Logic Programming. NMELP 1996. Lecture Notes in Computer Science, vol 1216. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023801
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DOI: https://doi.org/10.1007/BFb0023801
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