Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T07:23:57.625Z Has data issue: false hasContentIssue false
  • English
  • Français

Extended ASP Tableaux and rule redundancy in normal logicprograms1

Published online by Cambridge University Press:  01 November 2008

MATTI JÄRVISALO  and
EMILIA OIKARINEN
MATTI JÄRVISALO
Affiliation:
Helsinki University of Technology (TKK), Department of Information and Computer Science, P.O. Box 5400, FI-02015 TKK, Finland (e-mail: [email protected], [email protected])
EMILIA OIKARINEN
Affiliation:
Helsinki University of Technology (TKK), Department of Information and Computer Science, P.O. Box 5400, FI-02015 TKK, Finland (e-mail: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce an extended tableau calculus for answer set programming (ASP). Theproof system is based on the ASP tableaux defined in the work by Gebser andSchaub (Tableau calculi for answer set programming. In Proceedings ofthe 22nd International Conference on Logic Programming (ICLP 2006),S. Etalle and M. Truszczynski, Eds. Lecture Notes in Computer Science, vol.4079. Springer, 11–25) with an added extension rule. We investigatethe power of Extended ASP Tableaux both theoretically and empirically. We studythe relationship of Extended ASP Tableaux with the Extended Resolution proofsystem defined by Tseitin for sets of clauses, and separate Extended ASPTableaux from ASP Tableaux by giving a polynomial-length proof for a family ofnormal logic programs {Φn} for which ASP Tableaux has exponential-length minimal proofs withrespect to n. Additionally, Extended ASP Tableaux implyinteresting insight into the effect of program simplification on the lengths ofproofs in ASP. Closely related to Extended ASP Tableaux, we empiricallyinvestigate the effect of redundant rules on the efficiency of ASP solving.

Keywords

Answer set programmingtableau methodextension ruleproof complexityproblem structure
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 8 , Issue 5-6 , November 2008 , pp. 691 - 716
Copyright
Copyright © Cambridge University Press 2008

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

Anger, C., Gebser, M., Janhunen, T. and Schaub, T. 2006. What's a head without a body? 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, the Netherlands, 769–770.Google Scholar
Anger, C., Gebser, M., Linke, T., Neumann, A. and Schaub, T. 2005. The nomore++ approach to answer set solving. In Proceedings of the 12th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2005), Sutcliffe, G. and Voronkov, A., Eds. Lecture Notes in Computer Science, vol. 3835. Springer, Heidelberg, Germany, 95–109.Google Scholar
Baral, C. 2003. Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
Beame, P., Kautz, H. and Sabharwal, A. 2004. Towards understanding and harnessing the potential of clause learning. Journal of Artificial Intelligence Research 22, 319–351.CrossRefGoogle Scholar
Beame, P. and Pitassi, T. 1998. Propositional proof complexity: Past, present, and future. Bulletin of the EATCS 65, 66–89.Google Scholar
Ben-Eliyahu, R. and Dechter, R. 1994. Propositional semantics for disjunctive logic programs. Annals of Mathematics and Artificial Intelligence 12 (1–2), 53–87.CrossRefGoogle Scholar
Ben-Sasson, E., Impagliazzo, R. and Wigderson, A. 2004. Near optimal separation of tree-like and general resolution. Combinatorica 24 (4), 585–603.CrossRefGoogle Scholar
Brass, S. and Dix, J. 1995. Characterizations of the stable semantics by partial evaluation. In Proceedings of the 3rd International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 1995), Marek, V. W. and Nerode, A., Eds. Lecture Notes in Computer Science, vol. 928. Springer, Heidelberg, Germany, 85–98.CrossRefGoogle Scholar
Brooks, D. R., Erdem, E., Erdogan, S. T., Minett, J. W. and Ringe, D. 2007. Inferring phylogenetic trees using answer set programming. Journal of Automated Reasoning 39 (4), 471–511.CrossRefGoogle Scholar
Clark, K. L. 1978. Negation as failure. In Logic and Data Bases, Gallaire, H. and Minker, J., Eds. Plenum Press, New York, USA, 293–322.CrossRefGoogle Scholar
Cook, S. A. 1976. A short proof of the pigeon hole principle using extended resolution. SIGACT News 8 (4), 28–32.CrossRefGoogle Scholar
Cook, S. A. and Reckhow, R. A. 1979. The relative efficiency of propositional proof systems. Journal of Symbolic Logic 44 (1), 36–50.CrossRefGoogle Scholar
Davis, M., Logemann, G. and Loveland, D. 1962. A machine program for theorem proving. Communications of the ACM 5 (7), 394–397.CrossRefGoogle Scholar
Davis, M. and Putnam, H. 1960. A computing procedure for quantification theory. Journal of the ACM 7 (3), 201–215.CrossRefGoogle Scholar
Eiter, T., Fink, M., Tompits, H. and Woltran, S. 2004. 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. Lecture Notes in Computer Science, vol. 2923. Springer, Heidelberg, Germany, 87–99.Google Scholar
Erdem, E., Lifschitz, V. and Ringe, D. 2006. Temporal phylogenetic networks and logic programming. Theory and Practice of Logic Programming 6 (5), 539–558.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
Gebser, M., Kaufmann, B., Neumann, A. and Schaub, T. 2007. Conflict-driven answer set solving. In Proceedings of the 20th International Joint Conference on Articifial Intelligence (IJCAI 2007), Veloso, M. M., Ed., AAAI Press, California, USA, 286–392.Google Scholar
Gebser, M. and Schaub, T. 2006a. Characterizing ASP inferences by unit propagation. In ICLP Workshop on Search and Logic: Answer Set Programming and SAT, Giunchiglia, E., Marek, V., Mitchell, D. and Ternovska, E., Eds., AAAI Press, California, USA, 41–56.Google Scholar
Gebser, M. and Schaub, T. 2006b. Tableau calculi for answer set programming. In Proceedings of the 22nd International Conference on Logic Programming (ICLP 2006), Etalle, S. and Truszczynski, M., Eds. Lecture Notes in Computer Science, vol. 4079. Springer, Heidelberg, Germany, 11–25.Google Scholar
Gebser, M. and Schaub, T. 2007. Generic tableaux for answer set programming. In Proceedings of the 23rd International Conference on Logic Programming (ICLP 2007), Dahl, V. and Niemelä, I., Eds. Lecture Notes in Computer Science, vol. 4670. Springer, Heidelberg, Germany, 119–133.Google Scholar
Gelfond, M. and Leone, N. 2002. Logic programming and knowledge representation – The A-Prolog perspective. Artificial Intelligence 138 (1–2), 3–38.CrossRefGoogle Scholar
Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Proceedings of the 5th International Conference and Symposium on Logic Programming (ICLP/SLP 1988), Kowalski, R. A. and Bowen, K. A., Eds. MIT Press, Cambridge, Massachusetts, USA, 1070–1080.Google Scholar
Giunchiglia, E., Lierler, Y. and Maratea, M. 2006. Answer set programming based on propositional satisfiability. Journal of Automated Reasoning 36 (4), 345–377.CrossRefGoogle Scholar
Giunchiglia, E. and Maratea, M. 2005. On the relation between answer set and SAT procedures (or, between CMODELS and SMODELS). In Proceedings of the 21st International Conference on Logic Programming (ICLP 2005), Gabbrielli, M. and Gupta, G., Eds. Lecture Notes in Computer Science, vol. 3668. Springer, Heidelberg, Germany, 37–51.Google Scholar
Hai, L., Jigui, S. and Yimin, Z. 2003. Theorem proving based on the extension rule. Journal of Automated Reasononing 31 (1), 11–21.CrossRefGoogle Scholar
Haken, A. 1985. The intractability of resolution. Theoretical Computer Science 39 (2–3), 297–308.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
Järvisalo, M. and Junttila, T. in press. Limitations of restricted branching in clause learning. Constraints.Google Scholar
Järvisalo, M. and Oikarinen, E. 2007. Extended ASP tableaux and rule redundancy in normal logic programs. In Proceedings of the 23rd International Conference on Logic Programming (ICLP 2007), Dahl, V. and Niemelä, I., Eds. Lecture Notes in Computer Science, vol. 4670. Springer, Heidelberg, Germany, 134–148.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
Lifschitz, V. 2002. Answer set programming and plan generation. Artificial Intelligence 138 (1–2), 39–54.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
Lifschitz, V. and Turner, H. 1994. Splitting a logic program. In Proceedings of the 11th International Conference on Logic Programming, Hentenryck, P. V., Ed. MIT Press, Cambridge, Massachusetts, USA, 23–37.Google 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 18th International Joint Conference on Artificial Intelligence (IJCAI 2003), Gottlob, G. and Walsh, T., Eds. Morgan Kaufmann, San Francisco, California, USA, 853–858.Google Scholar
Lin, F. and Zhao, Y. 2004. ASSAT: Computing answer sets of a logic program by SAT solvers. Artificial Intelligence 157 (1–2), 115–137.CrossRefGoogle Scholar
Marek, V. W. and Truszczyński, M. 1999. Stable models and an alternative logic programming paradigm. In The Logic Programming Paradigm: A 25-Year Perspective, Apt, K. R., Marek, V. W., Truszczyński, M., and Warren, D. S., Eds. Springer, Heidelberg, Germany, 375–398.CrossRefGoogle Scholar
Moskewicz, M. W., Madigan, C. F., Zhao, Y., Zhang, L. and Malik, S. 2001. Chaff: Engineering an efficient SAT solver. In Proceedings of the 38th Design Automation Conference (DAC 2001). IEEE, USA, 530–535.Google Scholar
Niemelä, I. 1999. Logic programs with stable model semantics as a constraint programming paradigm. Annals of Mathematics and Artificial Intelligence 25 (3–4), 241–273.CrossRefGoogle Scholar
Nogueira, M., Balduccini, M., Gelfond, M., Watson, R. and Barry, M. 2001. An A-Prolog decision support system for the space shuttle. In Proceedings of the 3rd International Symposium on Practical Aspects of Declarative Languages (PADL 2001), Ramakrishnan, I. V., Ed. Lecture Notes in Computer Science, vol. 1990. Springer, Heidelberg, Germany, 169–183.CrossRefGoogle Scholar
Simons, P., Niemelä, I. and Soininen, T. 2002. Extending and implementing the stable model semantics. Artificial Intelligence 138 (1–2), 181–234.CrossRefGoogle Scholar
Soininen, T., Niemelä, I., Tiihonen, J. and Sulonen, R. 2001. Representing configuration knowledge with weight constraint rules. In Proceedings of the 1st International Workshop on Answer Set Programming: Towards Efficient and Scalable Knowledge (ASP 2001), Provetti, A. and Son, T. C., Eds.Google Scholar
Tseitin, G. S. 1969. On the complexity of derivation in propositional calculus. In Studies in Constructive Mathematics and Mathematical Logic, Part II, Slisenko, A., Ed. Seminars in Mathematics, V.A. Steklov Mathematical Institute, Leningrad, vol. 8. Consultants Bureau, 115–125. English translation appears in Automation of Reasoning 2: Classical Papers on Computational Logic 1967–1970 Siekmann, J. and Wrightson, G., Eds. Springer (1983), 466–483.Google Scholar