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Characterising equilibrium logic and nested logic programs: Reductions and complexity1,2

Published online by Cambridge University Press:  31 July 2009

DAVID PEARCE ,
HANS TOMPITS  and
STEFAN WOLTRAN
DAVID PEARCE
Affiliation:
Universidad Politécnica de Madrid, Departamento de Inteligencia Artificial, Grupo CLIP E-28660 Boadilla del Monte, Madrid, Spain (e-mail: david.pearce@upm.es)
HANS TOMPITS
Affiliation:
Technische Universität Wien, Institut für Informationssysteme 184/3, Arbeitsbereich Wissensbasierte Systeme, Favoritenstrasse 9-11, A-1040 Vienna, Austria (e-mail: tompits@kr.tuwien.ac.at)
STEFAN WOLTRAN
Affiliation:
Technische Universität Wien, Institut für Informationssysteme 184/2, Arbeitsbereich Datenbanken und Artificial Intelligence, Favoritenstrasse 9-11, A-1040 Vienna, Austria (e-mail: woltran@dbai.tuwien.ac.at)
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Abstract

Equilibrium logic is an approach to non-monotonic reasoning that extends the stable-model and answer-set semantics for logic programs. In particular, it includes the general case of nested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present polynomial reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs into quantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. Thus, quantified propositional logic is a fragment of second-order logic, and its formulas are usually referred to as quantified Boolean formulas (QBFs). We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some QBF such that the latter is valid precisely in case the former holds. The basic tasks we deal with here are the consistency problem, brave reasoning and skeptical reasoning. Additionally, we also provide encodings for testing equivalence of theories or programs under different notions of equivalence, viz. ordinary, strong and uniform equivalence. For all considered reasoning tasks, we analyse their computational complexity and give strict complexity bounds. Hereby, our encodings yield upper bounds in a direct manner. Besides this useful feature, our approach has the following benefits: First, our encodings yield a uniform axiomatisation for a variety of problems in a common language. Second, extant solvers for QBFs can be used as back-end inference engines to realise implementations of the encoded task in a rapid prototyping manner. Third, our axiomatisations also allow us to straightforwardly relate equilibrium logic with circumscription.

Keywords

answer-set programmingequilibrium logicreduction techniquesquantified Boolean formulascomputational complexityequivalence testing

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Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 9 , Issue 5 , September 2009 , pp. 565 - 616
Copyright
Copyright © Cambridge University Press 2009

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References

Aguado, F., Cabalar, P., Pérez, G. and Vidal, C. 2008. Strongly equivalent temporal logic programs. In Proceedings of the 11th European Conference on Logics in Artificial Intelligence (JELIA 2008), Hölldobler, S., Lutz, C., and Wansing, H., Eds. LNCS, vol. 5293. Springer, Berlin, 8–20.Google Scholar
Arieli, O. and Denecker, M. 2003. Reducing preferential paraconsistent reasoning to classical entailment. Journal of Logic and Computation 13 (4), 557–580.CrossRefGoogle Scholar
Baral, C. 2003. Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Ben-Eliyahu, R. and Dechter, R. 1994. Propositional semantics for disjunctive logic programs. Annals of Mathematics and Artificial Intelligence 12, 53–87.CrossRefGoogle Scholar
Besnard, P., Schaub, T., Tompits, H. and Woltran, S. 2005. Representing paraconsistent reasoning via quantified propositional logic. In Inconsistency Tolerance, Bertossi, L., Hunter, A. and Schaub, T., Eds. LNCS, vol. 3300. Springer, Berlin, 84–118.CrossRefGoogle Scholar
Biere, A. 2005. Resolve and expand. In Proceedings of the 7th International Conference on Theory and Applications of Satisfiability Testing (SAT 2004), Hoos, H. H. and Mitchell, D. G., Eds. LNCS, vol. 3542. Springer, Berlin, 59–70.CrossRefGoogle Scholar
Brewka, G. 2002. Logic programming with ordered disjunction. In Proceedings of the 18th National Conference on Artificial Intelligence (AAAI 2002). AAAI Press, Menlo Park, CA, 100–105.Google Scholar
Brewka, G., Niemelä, I. and Syrjänen, T. 2004. Logic programs with ordered disjunctions. Computational Intelligence 20 (2), 335–357.CrossRefGoogle Scholar
Cabalar, P. and Ferraris, P. 2007. Propositional theories are strongly equivalent to logic programs. Theory and Practice of Logic Programming 7, 6, 745–759.CrossRefGoogle Scholar
Cabalar, P., Pearce, D. and Valverde, A. 2005. Reducing propositional theories in equilibrium logic to logic programs. In Proceedings of the 12th Portuguese Conference on Artificial Intelligence (EPIA 2005), Bento, C., Cardoso, A. and Dias, G., Eds. LNCS, vol. 3808. Springer, Berlin, 4–17.Google Scholar
Cabalar, P., Pearce, D. and Valverde, A. 2007. Minimal logic programs. In Proceedings of the 23rd International Conference on Logic Programming (ICLP 2007), Dahl, V. and Niemelä, I., Eds. LNCS, vol. 4670. Springer, Berlin, 104–118.Google Scholar
Cabalar, P. and Pérez Vega, G. 2007. Temporal equilibrium logic: A first approach. In Proceedings of the 11th International Conference on Computer Aided Systems Theory (EUROCAST 2007), Moreno-Díaz, R., Pichler, F. and Quesada-Arencibia, A., Eds. LNCS, vol. 4739. Springer, Berlin, 241–248.CrossRefGoogle Scholar
Chen, J. 1993. Minimal knowledge + negation as failure = only knowing (sometimes). In Proceedings of the 2nd International Workshop on Logic Programming and Nonmonotonic Reasoning (LPNMR '93), Pereira, L. M. and Nerode, A., Eds. MIT Press, Cambridge, MA, 132–150.Google Scholar
Chen, Y., Lin, F. and Li, L. 2005. SELP - A system for studying strong equivalence between logic programs. In Proceedings of the 8th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2005), Baral, C., Greco, G., Leone, N. and Terracina, G., Eds. LNCS, vol. 3662. Springer, Berlin, 442–446.CrossRefGoogle Scholar
Church, A. 1956. Introduction to Mathematical Logic, Volume I. Princeton University Press, Princeton, NJ.Google Scholar
Clark, K. L. 1978. Negation as failure. In Logic and Data Bases, Gallaire, H. and Minker, J., Eds. Plenum Press, New York, NY, 127–138.Google Scholar
de Bruijn, J., Eiter, T., Polleres, A. and Tompits, H. 2007. Embedding non-ground logic programs into autoepistemic logic for knowledge-base combination. In Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI 2007), Veloso, M., Ed. AAAI Press, Menlo Park, CA, 304–309.Google Scholar
de Jongh, D. and Hendriks, L. 2003. Characterizations of strongly equivalent logic programs in intermediate logics Theory and Practice of Logic Programming 3 (3), 259–270.CrossRefGoogle Scholar
Delgrande, J., Schaub, T., Tompits, H. and Woltran, S. 2004. On computing solutions to belief change scenarios. Journal of Logic and Computation 14 (6), 801–826.CrossRefGoogle Scholar
Dix, J., Gottlob, G. and Marek, V. 1996. Reducing disjunctive to non-disjunctive semantics by shift-operations. Fundamenta Informaticae XXVIII, (1/2), 87–100.CrossRefGoogle Scholar
Egly, U., Eiter, T., Tompits, H. and Woltran, S. 2000. Solving advanced reasoning tasks using quantified Boolean formulas. In Proceedings of the 17th National Conference on Artificial Intelligence (AAAI 2000). AAAI Press/MIT Press, Menlo Park, CA, 417–422.Google Scholar
Egly, U., Seidl, M., Tompits, H., Woltran, S. and Zolda, M. 2004. Comparing different prenexing strategies for quantified Boolean formulas. In Proceedings of the 6th International Conference on Theory and Applications of Satisfiability Testing (SAT 2003). Selected Revised Papers, Giunchiglia, E. and Tacchella, A., Eds. LNCS, vol. 2919. Springer, Berlin, 214–228.Google Scholar
Egly, U., Seidl, M. and Woltran, S. 2006. A solver for QBFs in nonprenex form. In Proceedings of the 17th European Conference on Artificial Intelligence (ECAI 2006), Brewka, G., Coradeschi, S., Perini, A., and Traverso, P., Eds. IOS Press, Amsterdam, 477–481.Google Scholar
Eiter, T., Faber, W. and Traxler, P. 2005a. Testing strong equivalence of nonmonotonic datalog programs – implementation and examples. In Proceedings of the 8th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2005), Baral, C., Greco, G., Leone, N. and Terracina, G., Eds. LNCS, vol. 3662. Springer, Berlin, 437–441.CrossRefGoogle Scholar
Eiter, T. and Fink, M. 2003. Uniform equivalence of logic programs under the stable model semantics. In Proceedings of the 19th International Conference on Logic Programming (ICLP 2003), Palamidessi, C., Ed. LNCS, vol. 2916. Springer, Berlin, 224–238.Google Scholar
Eiter, T., Fink, M., Tompits, H. and Woltran, S. 2004a. On eliminating disjunctions in stable logic programming. In Proceedings of the 9th International Conference on Principles of Knowledge Representation and Reasoning (KR 2004), Dubois, D., Welty, C. A. and Williams, M.-A., Eds. AAAI Press, Menlo Park, CA, 447–458.Google Scholar
Eiter, T., Fink, M., Tompits, H. and Woltran, S. 2004b. Simplifying logic programs under uniform and strong equivalence. In Proceedings of the 7th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2004), Lifschitz, V. and Niemelä, I., Eds. LNCS, vol. 2923. Springer, Berlin, 87–99.Google Scholar
Eiter, T., Fink, M. and Woltran, S. 2007. Semantical characterizations and complexity of equivalences in answer set programming. ACM Transactions on Computational Logic 8 (3), 1–53.CrossRefGoogle Scholar
Eiter, T. and Gottlob, G. 1995. On the computational cost of disjunctive logic programming: Propositional case. Annals of Mathematics and Artificial Intelligence 15 (3–4), 289–323.CrossRefGoogle Scholar
Eiter, T., Gottlob, G. and Mannila, H. 1997. Disjunctive datalog. ACM Transactions on Database Systems 22 (3), 364–418.CrossRefGoogle Scholar
Eiter, T., Klotz, V., Tompits, H. and Woltran, S. 2002. Modal nonmonotonic logics revisited: Efficient encodings for the basic reasoning tasks. In Proceedings of the 11th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2002), Egly, U. and Fermüller, C., Eds. LNCS, vol. 2381. Springer, Berlin, 100–114.CrossRefGoogle Scholar
Eiter, T., Tompits, H. and Woltran, S. 2005b. On solution correspondences in answer set programming. In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005), Kaelbling, L. Pack and Saffiotti, A., Eds. Professional Book Center, Denver, CO, 97–102.Google Scholar
Erdem, E. and Lifschitz, V. 2001. Fages' theorem for programs with nested expressions. In Proceedings of the 18th International Conference on Logic Programming (ICLP 2001), Codognet, P., Ed. LNCS, vol. 2237. Springer, Berlin, 242–254.Google Scholar
Erdem, E. and Lifschitz, V. 2003. Tight logic programs. Theory and Practice of Logic Programming 3 (4–5), 499–518.CrossRefGoogle Scholar
Faber, W. and Konczak, K. 2005. Strong equivalence for logic programs with preferences. In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005), Kaelbling, L. Pack and Saffiotti, A., Eds. Professional Book Center, Denver, CO, 430–435.Google Scholar
Faber, W., Tompits, H. and Woltran, S. 2008. Notions of strong equivalence for logic programs with ordered disjunction. In Proceedings of the 11th International Conference on Principles of Knowledge Representation and Reasoning (KR 2008), Brewka, G. and Lang, J., Eds. AAAI Press, Menlo Park, CA, 433–443.Google Scholar
Fages, F. 1994. Consistency of Clark's completion and existence of stable models. Methods of Logic in Computer Science 1, 51–60.Google Scholar
Ferraris, P. 2005. Answer sets for propositional theories. In Proceedings of the 8th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2005), Baral, C., Greco, G., Leone, N. and Terracina, G., Eds. LNCS, vol. 3662. Springer, Berlin, 119–131.CrossRefGoogle Scholar
Ferraris, P., Lee, J. and Lifschitz, V. 2006. A generalization of the Lin-Zhao theorem. Annals of Mathematics and Artificial Intelligence 47, 1–2, 79–101.CrossRefGoogle Scholar
Ferraris, P., Lee, J., and Lifschitz, V. 2007. A new perspective on stable models. In Proceedings of the 20th International Joint Conference on Artificial Intelligence (IJCAI 2007), Veloso, M., Ed. AAAI Press, Menlo Park, CA, 372–379.Google Scholar
Fink, M. 2008. Equivalences in answer-set programming by countermodels in the logic of here-and-there. In Proceedings of the 24th International Conference on Logic Programming (ICLP 2008), de la Banda, M. G. and Pontelli, E., Eds. LNCS, vol. 5366. Springer, Berlin, 99–113.Google Scholar
Garey, M. R. and Johnson, D. S. 1979. Computers and Intractability. W. H. Freeman, New York, NY.Google Scholar
Gebser, M., Lee, J. and Lierler, Y. 2006. Elementary sets for logic programs. In Proceedings of the 21st National Conference on Artificial Intelligence (AAAI 2006). AAAI Press, Menlo Park, CA, 244–249.Google Scholar
Gebser, M., Schaub, T., Tompits, H. and Woltran, S. 2008. Alternative characterizations for program equivalence under answer-set semantics based on unfounded sets. In Proceedings of the 5th International Symposium on Foundations of Information and Knowledge Systems (FoIKS 2008), Hartmann, S. and Kern-Isberner, G., Eds. LNCS, vol. 4932. Springer, Berlin, 24–41.Google Scholar
Gelfond, M. 1987. On stratified autoepistemic theories. In Proceedings of 6th National Conference on Artificial Intelligence (AAAI '87), Forbus, K. and Shrobe, H. E., Eds. AAAI Press/MIT Press, Menlo Park, CA, 207–211.Google Scholar
Gelfond, M. and Lifschitz, V. 1991. Classical negation in logic programs and disjunctive databases. New Generation Computing 9 (3–4), 365–385.CrossRefGoogle Scholar
Gelfond, M., Lifschitz, V., Przymusinska, H. and Schwarz, G. 1994. Autoepistemic logic and introspective circumscription. In Proceedings of the 5th Conference on Theoretical Aspects of Reasoning about Knowledge (TARK '94), Fagin, R., Ed. Morgan Kaufmann, San Mateo, CA, 197–207.CrossRefGoogle Scholar
Gelfond, M., Lifschitz, V., Przymusinska, H. and Truszczyński, M. 1991. Disjunctive defaults. In Proceedings of the 2nd Conference on Principles of Knowledge Representation and Reasoning (KR '91), Allen, J., Fikes, R. and Sandewall, B., Eds. Morgan Kaufmann, San Mateo, CA, 230–237.Google Scholar
Gelfond, M., Przymusinska, H. and Przymusinski, T. C. 1989. On the relationship between circumscription and negation as failure. Artificial Intelligence 38 (1), 75–94.CrossRefGoogle Scholar
Gelfond, M., Przymusinska, H. and Przymusinski, T. C. 1990. On the relationship between CWA, minimal model, and minimal Herbrand model semantics. International Journal of Intelligent Systems 5, 549–564.CrossRefGoogle Scholar
Giunchiglia, E., Narizzano, M. and Tacchella, A. 2003. Backjumping for quantified Boolean logic satisfiability. Artificial Intelligence 145, 99–120.CrossRefGoogle Scholar
Gödel, K. 1932. Zum intuitionistischen Aussagenkalkül. Anzeiger der Akademie der Wissenschaften in Wien, 65–66.Google Scholar
Gurevich, Y. 1977. Intuitionistic logic with strong negation. Studia Logica 36 (1–2), 49–59.CrossRefGoogle Scholar
Heyting, A. 1930. Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, 42–56. Reprint in Logik-Texte: Kommentierte Auswahl zur Geschichte der Modernen Logik, Akademie-Verlag, Berlin, 1986.Google Scholar
Inoue, K. and Sakama, C. 1998. Negation as failure in the head. Journal of Logic Programming 35, 1, 39–78.CrossRefGoogle Scholar
Inoue, K. and Sakama, C. 2004. Equivalence of logic programs under updates. In Proceedings of the 9th European Conference on Logics in Artificial Intelligence (JELIA 2004), Alferes, J. J. and Leite, J. A., Eds. LNCS, vol. 3229. Springer, Berlin, 174–186.Google Scholar
Janhunen, T. 2001. On the effect of default negation on the expressiveness of disjunctive rules. In Proceedings of the 6th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2001), Eiter, T., Faber, W. and Truszczyński, M., Eds. LNCS, vol. 2173. Springer, Berlin, 93–106.Google Scholar
Janhunen, T. 2004. Representing normal programs with clauses. In Proceedings of the 16th European Conference on Artificial Intelligence (ECAI 2004), de Mántaras, R. L. and Saitta, L., Eds. IOS Press, Amsterdam, 358–362.Google Scholar
Janhunen, T. and Oikarinen, E. 2002. Testing the equivalence of logic programs under stable model semantics. In Proceedings of the 8th European Conference on Logics in Artificial Intelligence (JELIA 2002), Flesca, S., Greco, S., Leone, N. and Ianni, G., Eds. LNCS, vol. 2424. Springer, Berlin, 493–504.Google Scholar
Kowalski, V. 1968. The calculus of the weak “law of excluded middle”. Mathematics of the USSR 8, 648–658.Google Scholar
Lee, J. 2005. A model-theoretic counterpart of loop formulas. In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI 2005), Kaelbling, L. Pack and Saffiotti, A., Eds. Professional Book Center, Denver, CO, 503–508.Google Scholar
Lee, J. and Lifschitz, V. 2003. Loop formulas for disjunctive logic programs. In Proceedings of the 19th International Conference on Logic Programming (ICLP 2003), Palamidessi, C., Ed. LNCS, vol. 2916. Springer, Berlin, 451–465.Google Scholar
Lee, J., Lifschitz, V. and Palla, R. 2008. A reductive semantics for counting and choice in answer set programming. In Proceedings of the 23rd National Conference on Artificial Intelligence (AAAI 2008), Fox, D. and Gomes, C. P., Eds. AAAI Press, Menlo Park, CA, 472–479.Google Scholar
Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S. and Scarcello, F. 2006. The DLV system for knowledge representation and reasoning. ACM Transactions on Computational Logic 7 (3), 499–562.CrossRefGoogle Scholar
Leśniewski, S. 1929. Grundzüge eines neuen Systems der Grundlagen der Mathematik. Fundamenta Mathematica 14, 1–81.CrossRefGoogle Scholar
Letz, R. 2002. Lemma and model caching in decision procedures for quantified Boolean formulas. In Proceedings of the 11th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2002), Egly, U. and Fermüller, C., Eds. LNCS, vol. 2381. Springer, Berlin, 160–175.CrossRefGoogle Scholar
Lierler, Y. 2005. Disjunctive answer set programming via satisfiability. In Proceedings of the 8th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2005), Baral, C., Greco, G., Leone, N. and Terracina, G., Eds. LNCS, vol. 3662. Springer, Berlin, 447–451.CrossRefGoogle Scholar
Lifschitz, V. 1989. Between circumscription and autoepistemic logic. In Proceedings of the 1st International Conference on Principles of Knowledge Representation and Reasoning (KR '89), Brachman, R., Levesque, H. and Reiter, R., Eds. Morgan Kaufmann, San Mateo, CA, 235–244.Google Scholar
Lifschitz, V. 1994. Circumscription. In Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3: Nonmonotonic Reasoning and Uncertain Reasoning, Gabbay, D. M., Hogger, C. J. and Robinson, J. A., Eds. Clarendon Press, Oxford, 297–352.CrossRefGoogle Scholar
Lifschitz, V., Pearce, D. and Valverde, A. 2001. Strongly equivalent logic programs. ACM Transactions on Computational Logic 2 (4), 526–541.CrossRefGoogle Scholar
Lifschitz, V., Pearce, D. and Valverde, A. 2007. A characterization of strong equivalence for logic programs with variables. In Proceedings of the 9th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2007), Baral, C., Brewka, G. and Schlipf, J. S., Eds. LNCS, vol. 4483. Springer, Berlin, 188–200.CrossRefGoogle Scholar
Lifschitz, V. and Schwarz, G. 1993. Extended logic programs as autoepistemic theories. In Proceedings of the 2nd International Workshop on Logic Programming and Nonmonotonic Reasoning (LPNMR '93), Pereira, L. M. and Nerode, A., Eds. MIT Press, Cambridge, MA, 101–114.Google Scholar
Lifschitz, V., Tang, L. and Turner, H. 1999. Nested expressions in logic programs. Annals of Mathematics and Artificial Intelligence 25 (3–4), 369–389.CrossRefGoogle Scholar
Lifschitz, V. and Turner, H. 1994. Splitting a logic program. In Proceedings of the 11th International Conference on Logic Programming (ICLP '94). MIT Press, Cambridge, MA, 23–38.Google Scholar
Lin, F. 1991. A Study of Nonmonotonic Reasoning. Ph.D. thesis, Stanford University, California.Google Scholar
Lin, F. 2002. Reducing strong equivalence of logic programs to entailment in classical propositional logic. In Proceedings of the 8th International Conference on Principles of Knowledge Representation and Reasoning (KR 2002), Fensel, D., Giunchiglia, F., McGuinness, D., and Williams, M.-A., Eds. Morgan Kaufmann, San Mateo, CA, 170–176.Google Scholar
Lin, F. and Zhao, Y. 2002. ASSAT: Computing answer sets of a logic program by SAT solvers. In Proceedings of the 18th National Conference on Artificial Intelligence (AAAI 2002). AAAI Press/MIT Press, Menlo Park, CA, 112–117.Google Scholar
Linke, T., Tompits, H. and Woltran, S. 2004. On acyclic and head-cycle free nested logic programs. In Proceedings of the 20th International Conference on Logic Programming (ICLP 2004), Demoen, B. and Lifschitz, V., Eds. LNCS, vol. 3132. Springer, Berlin, 225–239.Google Scholar
Liu, L. and Truszczyński, M. 2005. Properties of programs with monotone and convex constraints. In Proceedings of the 20th National Conference on Artificial Intelligence (AAAI 2005), Veloso, M. and Kambhampati, S., Eds. AAAI Press, Menlo Park, CA, 701–706.Google Scholar
Łukasiewicz, J. and Tarski, A. 1930. Untersuchungen über den Aussagenkalkül. Comptes Rendus Séances Société des Sciences et Lettres Varsovie 23, Cl. III, 30–50.Google Scholar
Maher, M. J. 1988. Equivalence of logic programs. In Foundations of Deductive Databases and Logic Programming, Minker, J., Ed. Morgan Kaufmann, San Mateo, CA, 627–658.CrossRefGoogle Scholar
Marek, V. W. and Truszczyński, M. 1993. Reflexive autoepistemic logic and logic programming. In Proceedings of the 2nd International Workshop on Logic Programming and Nonmonotonic Reasoning (LPNMR '93), Pereira, L. M. and Nerode, A., Eds. MIT Press, Cambridge, MA, 115–131.Google Scholar
McCarthy, J. 1980. Circumscription – a form of nonmonotonic reasoning. Artificial Intelligence 13, 27–39.CrossRefGoogle Scholar
McCluskey, E. J. 1956. Minimization of boolean functions. Bell System Technical Journal 35, 1417–1444.CrossRefGoogle Scholar
Meyer, A. R. and Stockmeyer, L. J. 1973. Word problems requiring exponential time. In ACM Symposium on Theory of Computing (STOC '73). ACM Press, New York, NY, 1–9.Google Scholar
Moore, R. C. 1985. Semantical considerations on nonmonotonic logic. Artificial Intelligence 25, 75–94.CrossRefGoogle Scholar
Mundici, D. 1987. Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science 52 (1–2), 145–153.CrossRefGoogle Scholar
Nelson, D. 1949. Constructible falsity. Journal of Symbolic Logic 14 (2), 16–26.CrossRefGoogle Scholar
Oetsch, J., Seidl, M., Tompits, H. and Woltran, S. 2006a. cc⊤: A correspondence-checking tool for logic programs under the answer-set semantics. In Proceedings of the 10th European Conference on Logics in Artificial Intelligence (JELIA 2006), Fisher, M., van der Hoek, W., Konev, B. and Lisitsa, A., Eds. LNCS, vol. 4160. Springer, Berlin, 502–505.Google Scholar
Oetsch, J., Seidl, M., Tompits, H. and Woltran, S. 2006b. cc⊤: A tool for checking advanced correspondence problems in answer-set programming. In Proceedings of the 15th International Conference on Computing (CIC 2006), Gelbukh, A. and Guerra, S. S., Eds. IEEE Computer Society Press, Los Alamitos, CA, 3–10.CrossRefGoogle Scholar
Oetsch, J., Seidl, M., Tompits, H. and Woltran, S. 2007a. An extension of the system cc⊤ for testing relativised uniform equivalence under answer-set projection. In Proceedings of the 16th International Conference on Computing (CIC 2007) Gelbukh, A. and Guerra, S. S., Eds. Instituto Politécnico Nacional, Mexico City.Google Scholar
Oetsch, J. and Tompits, H. 2008. Program correspondence under the answer-set semantics: The non-ground case. In Proceedings of the 24th International Conference on Logic Programming (ICLP 2008), de la Banda, M. G. and Pontelli, E., Eds. LNCS, vol. 5366. Springer, Berlin, 591–605.Google Scholar
Oetsch, J., Tompits, H. and Woltran, S. 2007b. Facts do not cease to exist because they are ignored: Relativised uniform equivalence with answer-set projection. In Proceedings of the 22nd National Conference on Artificial Intelligence (AAAI 2007). AAAI Press, Menlo Park, CA, 458–464.Google Scholar
Oikarinen, E. and Janhunen, T. 2004. Verifying the equivalence of logic programs in the disjunctive case. In Proceedings of the 7th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2004), Lifschitz, V. and Niemelä, I., Eds. LNCS, vol. 2923. Springer, Berlin, 180–193.Google Scholar
Oikarinen, E. and Janhunen, T. 2006. Modular equivalence for normal logic programs. In Proceedings of the 11th International Workshop on Nonmonotonic Reasoning (NMR 2006), Dix, J. and Hunter, A., Eds. University of Clausthal, Department of Informatics, Technical Report, IfI-06-04, 10–18.Google Scholar
Osorio, M., Navarro, J. A. and Arrazola, J. 2005. Safe beliefs for propositional theories. Annals of Pure and Applied Logic 134 (1), 63–82.CrossRefGoogle Scholar
Papadimitriou, C. 1994. Computational Complexity. Addison-Wesley, Reading, MA.Google Scholar
Pearce, D. 1997. A new logical characterisation of stable models and answer sets. In Non-Monotonic Extensions of Logic Programming, Dix, J., Pereira, L. and Przymusinski, T., Eds. LNCS, vol. 1216. Springer, Berlin, 57–70.CrossRefGoogle Scholar
Pearce, D. 1999. From here to there: Stable negation in logic programming. In What is Negation?, Gabbay, D. and Wansing, H., Eds. Kluwer, Dordrecht, 161–181.CrossRefGoogle Scholar
Pearce, D. 2004. Simplifying logic programs under answer set semantics. In Proceedings of the 20th International Conference on Logic Programming (ICLP 2004), Demoen, B. and Lifschitz, V., Eds. LNCS, vol. 3132. Springer, Berlin, 210–224.Google Scholar
Pearce, D. 2006. Equilibrium logic. Annals of Mathematics and Artificial Intelligence 47, 3–41.CrossRefGoogle Scholar
Pearce, D., de Guzmán, I. and Valverde, A. 2000a. Computing equilibrium models using signed formulas. In Proceedings of the 1st International Conference on Computational Logic (CL 2000), Lloyd, J., Dahl, V., Furbach, U., Kerber, M., Lau, K.-K., Palamidessi, C., Pereira, L. M., Sagiv, Y. and Stuckey, P. J., Eds. LNCS, vol. 1861. Springer, Berlin, 688–702.Google Scholar
Pearce, D., de Guzmán, I. and Valverde, A. 2000b. A tableau calculus for equilibrium entailment. In Proceedings of the 9th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2000), Dyckhoff, R., Ed. LNCS, vol. 1847. Springer, Berlin, 352–367.CrossRefGoogle Scholar
Pearce, D., Sarsakov, V., Schaub, T., Tompits, H. and Woltran, S. 2002. A polynomial translation of logic programs with nested expressions into disjunctive logic programs: Preliminary report. In Proceedings of the 19th International Conference on Logic Programming (ICLP 2002), Stuckey, P., Ed. LNCS, vol. 2401. Springer, Berlin, 405–420.Google Scholar
Pearce, D., Tompits, H. and Woltran, S. 2001. Encodings for equilibrium logic and logic programs with nested expressions. In Proceedings of the 10th Portuguese Conference on Artificial Intelligence (EPIA 2001), Brazdil, P. and Jorge, A. Eds. LNCS, vol. 2258. Springer, Berlin, 306–320.Google Scholar
Pearce, D. and Valverde, A. 2004a. Synonymous theories in answer set programming and equilibrium logic. In Proceedings 16th European Conference on Artificial Intelligence (ECAI 2004). de Mántaras, R. L. and Saitta, L., Eds. IOS Press, Amsterdam, 388–392.Google Scholar
Pearce, D. and Valverde, A. 2004b. Uniform equivalence for equilibrium logic and logic programs. In Proceedings of the 7th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2004), Lifschitz, V. and Niemelä, I., Eds. LNCS, vol. 2923. Springer, Berlin, 194–206.Google Scholar
Pearce, D. and Valverde, A. 2005. A first order nonmonotonic extension of constructive logic. Studia Logica 80 (2–3), 321–346.CrossRefGoogle Scholar
Pearce, D. and Valverde, A. 2008. Quantified equilibrium logic and foundations for answer set programs. In Proceedings of the 24th International Conference on Logic Programming (ICLP 2008), de la Banda, M. G. and Pontelli, E., Eds. LNCS, vol. 5366. Springer, Berlin, 546–560.Google Scholar
Pearce, D. and Wagner, G. 1991. Logic programming with strong negation. In Workshop on Extensions of Logic Programming, Proceedings, Schroeder-Heiste, P., Ed. LNAI, vol. 475. Springer, Berlin, 311–326.CrossRefGoogle Scholar
Perlis, D. 1988. Autocircumscription. Artificial Intelligence 36, 223–236.CrossRefGoogle Scholar
Przymusinski, T. C. 1991. Stable semantics for disjunctive programs. New Generation Computing 9 (3–4), 401–424.CrossRefGoogle Scholar
Pührer, J., Tompits, H. and Woltran, S. 2008. Elimination of disjunction and negation in answer-set programs under hyperequivalence. In Proceedings of the 24th International Conference on Logic Programming (ICLP 2008), de la Banda, M. G. and Pontelli, E., Eds. LNCS, vol. 5366. Springer, Berlin, 561–575.Google Scholar
Quine, W. V. O. 1952. The problem of simplifying truth functions. American Mathematical Monthly 59, 521–531.CrossRefGoogle Scholar
Rintanen, J. 1999. Constructing conditional plans by a theorem prover. Journal of Artificial Intelligence Research 10, 323–352.CrossRefGoogle Scholar
Russell, B. 1906. The theory of implication. American Journal of Mathematics 28 (2), 159–202.CrossRefGoogle Scholar
Sagiv, Y. 1988. Optimising datalog programs. In Foundations of Deductive Databases and Logic Programming, Minker, J., Ed. Morgan Kaufmann, San Mateo, CA, 659–698.CrossRefGoogle Scholar
Sarsakov, V., Schaub, T., Tompits, H. and Woltran, S. 2004. nlp: A compiler for nested logic programming. In Proceedings of the 7th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2004), Lifschitz, V. and Niemelä, I., Eds. LNCS, vol. 2923. Springer, Berlin, 361–364.Google Scholar
Simons, P., Niemelä, I. and Soininen, T. 2002. Extending and implementing the stable model semantics. Artificial Intelligence 138, 181–234.CrossRefGoogle Scholar
Srzednicki, J. and Stachniak, Z., Eds. 1998. Lesniewski's Systems Protothetic. Kluwer, Dordrecht.CrossRefGoogle Scholar
Stockmeyer, L. J. 1976. The polynomial-time hierarchy. Theoretical Computer Science 3 (1), 1–22.CrossRefGoogle Scholar
Tompits, H. 2003. Expressing default abduction problems as quantified Boolean formulas. AI Communications 16 (2), 89–105.Google Scholar
Tompits, H. and Woltran, S. 2005. Towards implementations for advanced equivalence checking in answer-set programming. In Proceedings of the 21st International Conference on Logic Programming (ICLP 2005), Gabbrielli, M. and Gupta, G., Eds. LNCS, vol. 3668. Springer, Berlin, 189–203.Google Scholar
Truszczyński, M. and Woltran, S. 2008. Hyperequivalence of logic programs with respect to supported models. In Proceedings of the 23rd National Conference on Artificial Intelligence (AAAI 2008), Fox, D. and Gomes, C. P., Eds. AAAI Press, Menlo Park, CA, 560–565.Google Scholar
Turner, H. 2001. Strong equivalence for logic programs and default theories (made easy). In Proceedings of the 6th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2001), Eiter, T., Faber, W. and Truszczyński, M., Eds. LNCS, vol. 2173. Springer, Berlin, 81–92.Google Scholar
Turner, H. 2003. Strong equivalence made easy: Nested expressions and weight constraints. Theory and Practice of Logic Programming 3 (4–5), 609–622.CrossRefGoogle Scholar
Valverde, A. 2004. tabeql: A tableau based suite for equilibrium logic. In Proceedings of the 9th European Conference on Logics in Artificial Intelligence (JELIA 2004), Alferes, J. J. and Leite, J. A., Eds. LNCS, vol. 3229. Springer, Berlin, 734–737.Google Scholar
van Dalen, D. 1986. Intuitionistic logic. In Handbook of Philosophical Logic, Volume III: Alternatives to Classical Logic, Gabbay, D. and Guenthner, F., Eds. Synthese Library, vol. 166. D. Reidel Publishing Co., Dordrecht, Chapter III.4, 225–339.CrossRefGoogle Scholar
van Gelder, A., Ross, K. and Schlipf, J. 1991. The well-founded semantics for general logic programs. Journal of the ACM 38 (3), 620–650.CrossRefGoogle Scholar
Vorob'ev, N. 1952. A constructive propositional calculus with strong negation (in Russian). Doklady Akademii Nauk SSR 85, 689–692.Google Scholar
Whitehead, A. N. and Russell, B. 1910–1913. Principia Mathematica. vol. 1–3. Cambridge University Press, Cambridge.Google Scholar
Woltran, S. 2003. Quantified Boolean Formulas – From Theory to Practice. Ph.D. thesis, Technische Universität Wien, Institut für Informationssysteme.Google Scholar
Woltran, S. 2004. Characterizations for relativized notions of equivalence in answer set programming. In Proceedings of the 9th European Conference on Logics in Artificial Intelligence (JELIA 2004), Alferes, J. J. and Leite, J. A., Eds. LNCS, vol. 3229. Springer, Berlin, 161–173.Google Scholar
Woltran, S. 2005. Answer set programming: Model applications and proofs-of-concept. Tech. Rep. WP5, Working Group on Answer Set Programming (WASP, IST-FET-2001-37004). Available at http://www.kr.tuwien.ac.at/research/projects/WASP/.Google Scholar
Woltran, S. 2008. A common view on strong, uniform, and other notions of equivalence in answer-set programming. Theory and Practice of Logic Programming 8 (2), 217–234.CrossRefGoogle Scholar
Wrathall, C. 1976. Complete sets and the polynomial-time hierarchy. Theoretical Computer Science 3 (1), 23–33.CrossRefGoogle Scholar
You, J.-H., Yuan, L.-Y. and Mingyi, Z. 2003. On the equivalence between answer sets and models of completion for nested logic programs. In Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI 2003), Gottlob, G. and Walsh, T., Eds. Morgan Kaufmann, San Mateo, CA, 859–865.Google Scholar
Yuan, L.-Y. 1994. Autoepistemic logic of first order and its expressive power. Journal of Automated Reasoning 13 (1), 69–82.CrossRefGoogle Scholar