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Complexity of super-coherence problems in ASP*

Published online by Cambridge University Press:  09 August 2013

MARIO ALVIANO ,
WOLFGANG FABER  and
STEFAN WOLTRAN
MARIO ALVIANO
Affiliation:
University of Calabria, Rende (CS) 87036, Italy (e-mail: alviano@mat.unical.it, faber@mat.unical.it)
WOLFGANG FABER
Affiliation:
University of Calabria, Rende (CS) 87036, Italy (e-mail: alviano@mat.unical.it, faber@mat.unical.it)
STEFAN WOLTRAN
Affiliation:
Vienna University of Technology, Vienna 1040, Austria (e-mail: woltran@dbai.tuwien.ac.at)
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Abstract

Adapting techniques from database theory in order to optimize Answer Set Programming (ASP) systems, and in particular the grounding components of ASP systems, is an important topic in ASP. In recent years, the Magic Set method has received some interest in this setting, and a variant of it, called Dynamic Magic Set, has been proposed for ASP. However, this technique has a caveat, because it is not correct (in the sense of being query-equivalent) for all ASP programs. In a recent work, a large fragment of ASP programs, referred to as super-coherent programs, has been identified, for which Dynamic Magic Set is correct. The fragment contains all programs which possess at least one answer set, no matter which set of facts is added to them. Two open question remained: How complex is it to determine whether a given program is super-coherent? Does the restriction to super-coherent programs limit the problems that can be solved? Especially the first question turned out to be quite difficult to answer precisely. In this paper, we formally prove that deciding whether a propositional program is super-coherent is Π3 P -complete in the disjunctive case, while it is Π2 P -complete for normal programs. The hardness proofs are the difficult part in this endeavor: We proceed by characterizing the reductions by the models and reduct models which the ASP programs should have, and then provide instantiations that meet the given specifications. Concerning the second question, we show that all relevant ASP reasoning tasks can be transformed into tasks over super-coherent programs, although this transformation is more of theoretical than practical interest.

Keywords

Answer Set Programming (ASP)complexity analysiscoherenceuniform equivalencefoundations
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 14 , Issue 3 , May 2014 , pp. 339 - 361
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

*

Preliminary versions of this paper have been presented at the International Conference on Logic Programming (ICLP) workshop on Answer Set Programming and Other Computing Paradigms (ASPOCP) and at the Convegno Italiano di Logica Computazionale (CILC).

References

Alviano, M. and Faber, W. 2010. Dynamic magic sets for super-consistent answer set programs. In Proceedings of the 3rd Workshop on Answer Set Programming and Other Computing Paradigms (ASPOCP 2010), M. Balduccini and S. Woltran, Eds.Google Scholar
Alviano, M. and Faber, W. 2011. Dynamic magic sets and super-coherent answer set programs. AI Communications 24, 2, 125–145.CrossRefGoogle Scholar
Alviano, M., Faber, W., Greco, G. and Leone, N. 2012. Magic sets for disjunctive datalog programs. Artificial Intelligence (Elsevier) 187–188, 156–192.CrossRefGoogle Scholar
Apt, K. R., Blair, H. A. and Walker, A. 1988. Towards a Theory of Declarative Knowledge. In Foundations of Deductive Databases and Logic Programming, Minker, J., Ed. Morgan Kaufmann, Washington, DC, 89–148.CrossRefGoogle Scholar
Bancilhon, F., Maier, D., Sagiv, Y. and Ullman, J. D. 1986. Magic sets and other strange ways to implement logic programs. In Proceedings of the Fifth ACM SIGACT-SIGMOD Symposium on Principles of Database Systems, Cambridge, MA, 1–15.Google Scholar
Baral, C. 2003. Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Beeri, C. and Ramakrishnan, R. 1991. On the power of magic. Journal of Logic Programming 10, 255–259.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
Cumbo, C., Faber, W., Greco, G. and Leone, N. 2004. Enhancing the magic-set method for disjunctive datalog programs. In Proceedings of the the 20th International Conference on Logic Programming – ICLP'04. LNCS, vol. 3132. Springer-Verlag, Berlin, Germany, 371–385.Google 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 2008), Brewka, G. and Lang, J., Eds. AAAI Press, Sydney, Australia, 422–432.Google Scholar
Dung, P. M. 1992. On the relations between stable and well-founded semantics of logic programs. Theoretical Computer Science 105, 1, 7–25.CrossRefGoogle Scholar
Eiter, T., Fink, M., Tompits, H. and Woltran, S. 2004. On eliminating disjunctions in stable logic programming. In Proceedings of the 9th International Conference on Principles of Knowledge Representation and Reasoning (KR 2004). AAAI Press, Palo Alto, CA, 447–458.Google Scholar
Eiter, T., Fink, M. and Woltran, S. 2007. Semantical characterizations and complexity of equivalences in stable logic 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 (September), 364–418.CrossRefGoogle Scholar
Eiter, T., Tompits, H. and Woltran, S. 2005. On solution correspondences in answer set programming. In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI'05), Kaelbling, L. P. and Saffiotti, A., Eds. Professional Book Center, Denver CO, 97–102.Google Scholar
Gelfond, M. and Lifschitz, V. 1991. Classical negation in logic programs and disjunctive databases. New Generation Computing 9, 365–385.CrossRefGoogle Scholar
Gottlob, G., Leone, N. and Veith, H. 1999. Succinctness as a source of expression complexity. Annals of Pure and Applied Logic 97, 1–3, 231–260.CrossRefGoogle Scholar
Greco, S. 2003. Binding propagation techniques for the optimization of bound disjunctive queries. IEEE Transactions on Knowledge and Data Engineering 15, 2 (March/April), 368–385.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 (January), 1–37.CrossRefGoogle Scholar
Leone, N., Gottlob, G., Rosati, R., Eiter, T., Faber, W., Fink, M., Greco, G., Ianni, G., Kałka, E., Lembo, D., Lenzerini, M., Lio, V., Nowicki, B., Ruzzi, M., Staniszkis, W. and Terracina, G. 2005. The INFOMIX system for advanced integration of incomplete and inconsistent data. In Proceedings of the 24th ACM SIGMOD International Conference on Management of Data (SIGMOD 2005). ACM Press, Baltimore, MD, 915–917.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 (July), 499–562.CrossRefGoogle Scholar
Lierler, Y. 2005. Disjunctive answer set programming via satisfiability. In Proceedings of Logic Programming and Nonmonotonic Reasoning – 8th International Conference (LPNMR'05), Diamante, Italy, September 2005, Baral, C., Greco, G., Leone, N. and Terracina, G., Eds. LNCS, vol. 3662. Springer-Verlag, New York, NY, 447–451.Google Scholar
Lifschitz, V. and Turner, H. 1994. Splitting a logic program. In Proceedings of the 11th International Conference on Logic Programming (ICLP'94), Santa Margherita Ligure, Italy, Van Hentenryck, P., Ed. MIT Press, Cambridge, MA, 23–37.Google Scholar
Manna, M., Ruffolo, M., Oro, E., Alviano, M. and Leone, N. 2012. The HiLeX system for semantic information extraction. Transactions on Large-Scale Data and Knowledge-Centered Systems, 91–125.Google Scholar
Manna, M., Scarcello, F. and Leone, N. 2011. On the complexity of regular-grammars with integer attributes. Journal of Computer and System Sciences 77, 2, 393–421.CrossRefGoogle Scholar
Oetsch, J., Tompits, H. and Woltran, S. 2007. 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'07). AAAI Press, Palo Alto, CA, 458–464.Google Scholar
Papadimitriou, C. H. and Yannakakis, M. 1997. Tie-breaking semantics and structural totality. Journal of Computer and System Sciences 54, 1, 48–60.CrossRefGoogle Scholar
Ricca, F., Alviano, M., Dimasi, A., Grasso, G., Ielpa, S. M., Iiritano, S., Manna, M., and Leone, N. 2010. A logic-based system for e-tourism. Fundamenta Informaticae 105, 1–2, 35–55.CrossRefGoogle Scholar
Ricca, F., Grasso, G., Alviano, M., Manna, M., Lio, V., Iiritano, S. and Leone, N. 2012. Team-building with answer set programming in the Gioia-Tauro seaport. Theory and Practice of Logic Programming (Cambridge University Press) 12, 3, 361–381.CrossRefGoogle Scholar
Ullman, J. D. 1989. Principles of Database and Knowledge Base Systems. Computer Science Press, Los Alamitos, CA.Google Scholar
You, J.-H. and Yuan, L. Y. 1994. A three-valued semantics for deductive databases and logic programs. Journal of Computer and System Sciences 49, 2, 334–361.CrossRefGoogle Scholar