Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T10:50:52.844Z Has data issue: false hasContentIssue false
  • English
  • Français

On elementary loops of logic programs

Published online by Cambridge University Press:  24 May 2011

MARTIN GEBSER ,
JOOHYUNG LEE  and
YULIYA LIERLER
MARTIN GEBSER
Affiliation:
Institut für Informatik, Universität Potsdam, Germany (e-mail: [email protected])
JOOHYUNG LEE
Affiliation:
School of Computing, Informatics and Decision Systems Engineering, Arizona State University, Tempe, AZ, USA (e-mail: [email protected])
YULIYA LIERLER
Affiliation:
Department of Computer Science, University of Kentucky, Lexington, KY, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Using the notion of an elementary loop, Gebser and Schaub (2005. Proceedings of the Eighth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'05), 53–65) refined the theorem on loop formulas attributable to Lin and Zhao (2004) by considering loop formulas of elementary loops only. In this paper, we reformulate the definition of an elementary loop, extend it to disjunctive programs, and study several properties of elementary loops, including how maximal elementary loops are related to minimal unfounded sets. The results provide useful insights into the stable model semantics in terms of elementary loops. For a nondisjunctive program, using a graph-theoretic characterization of an elementary loop, we show that the problem of recognizing an elementary loop is tractable. On the other hand, we also show that the corresponding problem is coNP-complete for a disjunctive program. Based on the notion of an elementary loop, we present the class of Head-Elementary-loop-Free (HEF) programs, which strictly generalizes the class of Head-Cycle-Free (HCF) programs attributable to Ben-Eliyahu and Dechter (1994. Annals of Mathematics and Artificial Intelligence 12, 53–87). Like an HCF program, an HEF program can be turned into an equivalent nondisjunctive program in polynomial time by shifting head atoms into the body.

Keywords

stable model semanticsloop formulasunfounded sets
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 11 , Issue 6 , November 2011 , pp. 953 - 988
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Apt, K., Blair, H. and Walker, A. 1988. Towards a theory of declarative knowledge. In Foundations of Deductive Databases and Logic Programming, Minker, J., Ed. Morgan Kaufmann, Massachusetts, USA, 89–148.CrossRefGoogle Scholar
Baral, C. and Gelfond, M. 1994. Logic programming and knowledge representation. Journal of Logic Programming 19, 20, 73–148.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
Chen, Y., Lin, F., Wang, Y. and Zhang, M. 2006. First-order loop formulas for normal logic programs. In Proceedings of the Tenth International Conference on Principles of Knowledge Representation and Reasoning (KR'06), Doherty, P., Mylopoulos, J. and Welty, C., Eds. AAAI Press, Menlo Park, CA, USA, 298–307.Google Scholar
Clark, K. 1978. Negation as failure. In Logic and Data Bases, Gallaire, H. and Minker, J., Eds. Plenum Press, New York, 293–322.CrossRefGoogle Scholar
Drescher, C., Gebser, M., Grote, T., Kaufmann, B., König, A., Ostrowski, M. and Schaub, T. 2008. Conflict-driven disjunctive answer set solving. In Proceedings of the Eleventh International Conference on Principles of Knowledge Representation and Reasoning (KR'08), Brewka, G. and Lang, J., Eds. AAAI Press, Menlo Park, CA, USA, 422–432.Google Scholar
Eiter, T. and Fink, M. 2003. Uniform equivalence of logic programs under the stable model semantics. In Proceedings of the Nineteenth International Conference on Logic Programming (ICLP'03), Palamidessi, C., Ed. Springer-Verlag, New York, USA, 224–238.Google 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
Erdem, E. and Lifschitz, V. 2003. Tight Logic Programs. Theory and Practice of Logic Programming 3, 499–518.CrossRefGoogle Scholar
Fages, F. 1994. Consistency of Clark's completion and existence of stable models. Journal of Methods of Logic in Computer Science 1, 51–60.Google Scholar
Fassetti, F. and Palopoli, L. 2010. On the complexity of identifying head elementary set free programs. Theory and Practice of Logic Programming 10, 1, 113–123.CrossRefGoogle Scholar
Ferraris, P., Lee, J. and Lifschitz, V. 2007. A new perspective on stable models. In Proceedings of the Twentieth International Joint Conference on Artificial Intelligence (IJCAI'07), AAAI Press, 372–379.Google Scholar
Ferraris, P., Lee, J. and Lifschitz, V. 2011. Stable models and circumscription. Artificial Intelligence 175, 236–263.CrossRefGoogle Scholar
Gebser, M., Lee, J. and Lierler, Y. 2006. Elementary sets for logic programs. In Proceedings of the Twenty-first AAAI Conference on Artificial Intelligence (AAAI'06), AAAI Press, Menlo Park, CA, USA.Google Scholar
Gebser, M., Lee, J. and Lierler, Y. 2007. Head-elementary-set-free logic programs. In Procedings of the Ninth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'07), Springer, 149–161.CrossRefGoogle Scholar
Gebser, M. and Schaub, T. 2005. Loops: Relevant or redundant? In Proceedings of the Eighth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'05), Springer, 53–65.CrossRefGoogle 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 Fifth International Symposium on Foundations of Information and Knowledge Systems (FoIKS'08), Hartmann, S. and Kern-Isberner, G., Eds. Springer-Verlag, New York, USA, 24–41.CrossRefGoogle Scholar
Gelfond, M., Lifschitz, V., Przymusińska, H. and Truszczyński, M. 1991. Disjunctive defaults. In Proceedings of the Second International Conference on Principles of Knowledge Representation and Reasoning (KR'91), Allen, J., Fikes, R. and Sandewall, E., Eds. Morgan Kaufmann, 230–237.Google Scholar
Giunchiglia, E., Lierler, Y. and Maratea, M. 2004. SAT-based answer set programming. In Proceedings of the Nineteenth AAAI Conference on Artificial Intelligence (AAAI), AAAI Press/The MIT Press, 61–66.Google Scholar
Inoue, K. and Sakama, C. 1998. Negation as failure in the head. Journal of Logic Programming 35, 39–78.CrossRefGoogle Scholar
Janhunen, T. 2006. Some (in)translatability results for normal logic programs and propositional theories. Journal of Applied Non-Classical Logics 16, 1–2, 35–86.CrossRefGoogle Scholar
Janhunen, T., Niemelä, I., Seipel, D., Simons, P. and You, J.-H. 2006. Unfolding partiality and disjunctions in stable model semantics. ACM Transactions on Computational Logic 7, 1, 1–37.CrossRefGoogle Scholar
Janhunen, T., Oikarinen, E., Tompits, H. and Woltran, S. 2009. Modularity aspects of disjunctive stable models. Journal of Artificial Intelligence Research 35, 813–857.CrossRefGoogle Scholar
Koch, C., Leone, N. and Pfeifer, G. 2003. Enhancing disjunctive logic programming systems by SAT checkers. Artificial Intelligence 151, 177–212.CrossRefGoogle Scholar
Lee, J. 2004. Nondefinite vs. definite causal theories. In Proceedings of the Seventh International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'04), Springer, 141–153.Google Scholar
Lee, J. 2005. A model-theoretic counterpart of loop formulas. In Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence (IJCAI'05). Professional Book Center, Denver CO, USA, 503–508.Google Scholar
Lee, J. and Lifschitz, V. 2003. Loop formulas for disjunctive logic programs. In Proceedings of Nineteenth International Conference on Logic Programming (ICLP'03). 451–465.CrossRefGoogle Scholar
Lee, J. and Lin, F. 2006. Loop formulas for circumscription. Artificial Intelligence 170, 2, 160–185.CrossRefGoogle Scholar
Lee, J. and Meng, Y. 2008. On loop formulas with variables. In Proceedings of the Eleventh International Conference on Knowledge Representation and Reasoning (KR'08), AAAI Press, 444–453.Google Scholar
Lee, J. and Meng, Y. 2009. On reductive semantics of aggregates in answer set programming. In Procedings of the Tenth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'09), Springer, 182–195.CrossRefGoogle 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
Leone, N., Rullo, P. and Scarcello, F. 1997. Disjunctive stable models: Unfounded sets, fixpoint semantics, and computation. Information and Computation 135, 2, 69–112.CrossRefGoogle Scholar
Lierler, Y. 2005. cmodels: SAT-based disjunctive answer set solver. In Proceedings of the Eighth International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR'05), Springer, 447–452.CrossRefGoogle Scholar
Lifschitz, V., Tang, L. R. and Turner, H. 1999. Nested expressions in logic programs. Annals of Mathematics and Artificial Intelligence 25, 369–389.CrossRefGoogle Scholar
Lifschitz, V. and Razborov, A. 2006. Why are there so many loop formulas? ACM Transactions on Computational Logic 7, 2, 261–268.CrossRefGoogle Scholar
Lin, F. and Zhao, J. 2003. On tight logic programs and yet another translation from normal logic programs to propositional logic. In Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence (IJCAI'03), Morgan Kaufmann, 853–858.Google Scholar
Lin, F. and Zhao, Y. 2004. ASSAT: Computing answer sets of a logic program by SAT solvers. Artificial Intelligence 157, 115–137.CrossRefGoogle Scholar
Liu, L. and Truszczynski, M. 2006. Properties and applications of programs with monotone and convex constraints. Journal of Artificial Intelligence Research 27, 299–334.CrossRefGoogle Scholar
Saccá, D. and Zaniolo, C. 1990. Stable models and non-determinism in logic programs with negation. In Proceedings of the Ninth ACM Symposium on Principles of Database Systems (PODS), ACM Press, 205–217.Google Scholar
You, J.-H. and Liu, G. 2008. Loop formulas for logic programs with arbitrary constraint atoms. In Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (AAAI'08), AAAI Press, 584–589.Google Scholar
You, J.-H., Yuan, L.-Y. and Zhang, M. 2003. On the equivalence between answer sets and models of completion for nested logic programs. In Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence (IJCAI'03), Morgan Kaufmann, 859–866.Google Scholar