Advertisement

Equivalence between Extended Datalog Programs — A Brief Survey

  • Stefan Woltran
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6702)

Abstract

This paper gives a brief overview about the research field on equivalences in Answer-Set Programming. More precisely, we are concerned here with disjunctive logic programs under the stable-model semantics. Such programs can be understood as extended datalog queries (i.e., datalog augmented by default negation and disjunction). In particular, we shall report on characterizations and complexity results for the notions of strong and respectively uniform equivalence. Most notably, uniform equivalence becomes undecidable in the presence of default negation, while strong equivalence remains decidable for full disjunctive datalog. We also consider a restricted setting where the arity of predicates is bounded by a fixed constant.

Keywords

Logic Program Ground Atom Disjunctive Program Stable Model Semantic Disjunctive Logic Program 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baral, C.: Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  2. 2.
    Dantsin, E., Eiter, T., Gottlob, G., Voronkov, A.: Complexity and Expressive Power of Logic Programming. ACM Computing Surveys 33(3), 374–425 (2001)CrossRefGoogle Scholar
  3. 3.
    de Jongh, D., Hendriks, L.: Characterizations of Strongly Equivalent Logic Programs in Intermediate Logics. Theory and Practice of Logic Programming 3(3), 259–270 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Denecker, M., Vennekens, J., Bond, S., Gebser, M., Truszczynski, M.: The Second Answer Set Programming Competition. In: Erdem, E., Lin, F., Schaub, T. (eds.) LPNMR 2009. LNCS, vol. 5753, pp. 637–654. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Eiter, T., Faber, W., Fink, M., Woltran, S.: Complexity Results for Answer Set Programming with Bounded Predicate Arities and Implications. Annals of Mathematics and Artificial Intelligence 51(2-4), 123–165 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Eiter, T., Fink, M.: Uniform Equivalence of Logic Programs under the Stable Model Semantics. In: Palamidessi, C. (ed.) ICLP 2003. LNCS, vol. 2916, pp. 224–238. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Eiter, T., Fink, M., Tompits, H., Traxler, P., Woltran, S.: Replacements in Non-Ground Answer-Set Programming. In: Proc. KR 2006, pp. 340–351. AAAI Press, Menlo Park (2006)Google Scholar
  8. 8.
    Eiter, T., Fink, M., Tompits, H., Woltran, S.: On Eliminating Disjunctions in Stable Logic Programming. In: Proc. KR 2004, pp. 447–458. AAAI Press, Menlo Park (2004)Google Scholar
  9. 9.
    Eiter, T., Fink, M., Tompits, H., Woltran, S.: Simplifying Logic Programs Under Uniform and Strong Equivalence. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 87–99. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Eiter, T., Fink, M., Tompits, H., Woltran, S.: Strong and Uniform Equivalence in Answer-Set Programming: Characterizations and Complexity Results for the Non-Ground Case. In: Proc. AAAI 2005, pp. 695–700. AAAI Press, Menlo Park (2005)Google Scholar
  11. 11.
    Eiter, T., Fink, M., Tompits, H., Woltran, S.: Complexity Results for Checking Equivalence of Stratified Logic Programs. In: Proc. IJCAI 2007, pp. 330–335. AAAI Press, Menlo Park (2007)Google Scholar
  12. 12.
    Eiter, T., Fink, M., Woltran, S.: Semantical Characterizations and Complexity of Equivalences in Answer Set Programming. ACM Transactions on Computational Logic 8(3), pages 53 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eiter, T., Gottlob, G., Gurevich, Y.: Normal Forms for Second-Order Logic over Finite Structures, and Classification of NP Optimization Problems. Annals of Pure and Applied Logic 78(1-3), 111–125 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Eiter, T., Gottlob, G., Mannila, H.: Disjunctive Datalog. ACM Transactions on Database Systems 22(3), 364–418 (1997)CrossRefGoogle Scholar
  15. 15.
    Eiter, T., Tompits, H., Woltran, S.: On Solution Correspondences in Answer Set Programming. In: Proc. IJCAI 2005, pp. 97–102. Professional Book Center (2005)Google Scholar
  16. 16.
    Faber, W., Pfeifer, G., Leone, N., Dell’Armi, T., Ielpa, G.: Design and Implementation of Aggregate Functions in the DLV System. Theory and Practice of Logic Programming 8(5-6), 545–580 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Feder, T., Saraiya, Y.: Decidability and Undecidability of Equivalence for Linear Datalog with Applications to Normal-Form Optimizations. In: Hull, R., Biskup, J. (eds.) ICDT 1992. LNCS, vol. 646, pp. 297–311. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  18. 18.
    Fink, M.: A General Framework for Equivalences in Answer-Set Programming by Countermodels in the Logic of Here-and-There. CoRR, abs/1006.3021 (2010) (to appear); Theory and Practice of Logic ProgrammingGoogle Scholar
  19. 19.
    Fink, M., Pichler, R., Tompits, H., Woltran, S.: Complexity of Rule Redundancy in Non-Ground Answer-Set Programming over Finite Domains. In: Baral, C., Brewka, G., Schlipf, J. (eds.) LPNMR 2007. LNCS (LNAI), vol. 4483, pp. 123–135. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  20. 20.
    Gaifman, H., Mairson, H., Sagiv, Y., Vardi, M.: Undecidable Optimization Problems for Database Logic Programs. Journal of the ACM 40(3), 683–713 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gebser, M., Liu, L., Namasivayam, G., Neumann, A., Schaub, T., Truszczynski, M.: The First Answer Set Programming System Competition. In: Baral, C., Brewka, G., Schlipf, J. (eds.) LPNMR 2007. LNCS (LNAI), vol. 4483, pp. 3–17. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  22. 22.
    Gelfond, M.: Representing Knowledge in A-Prolog. In: Kakas, A.C., Sadri, F. (eds.) Computational Logic: Logic Programming and Beyond. LNCS (LNAI), vol. 2408, pp. 413–451. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  23. 23.
    Gelfond, M., Leone, N.: Logic Programming and Knowledge Representation - The A-Prolog Perspective. Artificial Intelligence 138(1-2), 3–38 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gelfond, M., Lifschitz, V.: The Stable Model Semantics for Logic Programming. In: Proc. ICLP 1988, pp. 1070–1080. MIT Press, Cambridge (1988)Google Scholar
  25. 25.
    Gelfond, M., Lifschitz, V.: Classical Negation in Logic Programs and Disjunctive Databases. New Generation Computing 9, 365–385 (1991)CrossRefzbMATHGoogle Scholar
  26. 26.
    Halevy, A., Mumick, I., Sagiv, Y., Shmueli, O.: Static Analysis in Datalog Extensions. Journal of the ACM 48(5), 971–1012 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hillebrand, G., Kanellakis, P., Mairson, H., Vardi, M.: Tools for Datalog Boundedness. In: Proc. PODS 1991, pp. 1–12. ACM Press, New York (1991)Google Scholar
  28. 28.
    Inoue, K., Sakama, C.: Equivalence of Logic Programs Under Updates. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 174–186. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  29. 29.
    Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S., Scarcello, F.: The DLV System for Knowledge Representation and Reasoning. ACM Transactions on Computational Logic 7(3), 499–562 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Levy, A., Sagiv, Y.: Queries Independent of Updates. In: Proc. VLDB 1993, pp. 171–181. Morgan Kaufmann, San Francisco (1993)Google Scholar
  31. 31.
    Lifschitz, V.: Answer Set Programming and Plan Generation. Artificial Intelligence 138, 39–54 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly Equivalent Logic Programs. ACM Transactions on Computational Logic 2(4), 526–541 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lifschitz, V., Pearce, D., Valverde, A.: A Characterization of Strong Equivalence for Logic Programs with Variables. In: Baral, C., Brewka, G., Schlipf, J. (eds.) LPNMR 2007. LNCS (LNAI), vol. 4483, pp. 188–200. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  34. 34.
    Lin, F.: Reducing Strong Equivalence of Logic Programs to Entailment in Classical Propositional Logic. In: Proc. KR 2002, pp. 170–176. Morgan Kaufmann, San Francisco (2002)Google Scholar
  35. 35.
    Lin, F., Chen, Y.: Discovering Classes of Strongly Equivalent Logic Programs. Journal of Artificial Intelligence Research 28, 431–451 (2007)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Liu, L., Truszczynski, M.: Properties and Applications of Programs with Monotone and Convex Constraints. Journal of Artificial Intelligence Research 27, 299–334 (2006)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Maher, M.: Equivalences of Logic Programs. In: Minker, J. (ed.) Foundations of Deductive Databases and Logic Programming, pp. 627–658. Morgan Kaufmann, San Francisco (1988)CrossRefGoogle Scholar
  38. 38.
    Marek, V., Truszczyński, M.: Stable Models and an Alternative Logic Programming Paradigm. In: Apt, K., Marek, V.W., Truszczyński, M., Warren, D.S. (eds.) The Logic Programming Paradigm – A 25-Year Perspective, pp. 375–398. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  39. 39.
    Minker, J.: Overview of Disjunctive Logic Programming. Annals of Mathematics and Artificial Intelligence 12, 1–24 (1994)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Niemelä, I.: Logic Programming with Stable Model Semantics as Constraint Programming Paradigm. Annals of Mathematics and Artificial Intelligence 25(3-4), 241–273 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Nogueira, M., Balduccini, M., Gelfond, M., Watson, R., Barry, M.: An A-Prolog Decision Support System for the Space Shuttle. In: Gupta, G. (ed.) PADL 1999. LNCS, vol. 1551, pp. 169–183. Springer, Heidelberg (1999)Google Scholar
  42. 42.
    Oetsch, J., Tompits, H.: Program Correspondence under the Answer-Set Semantics: The Non-ground Case. In: Garcia de la Banda, M., Pontelli, E. (eds.) ICLP 2008. LNCS, vol. 5366, pp. 591–605. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  43. 43.
    Oikarinen, E., Janhunen, T.: Modular Equivalence for Normal Logic Programs. In: Proc. ECAI 2006, pp. 412–416. IOS Press, Amsterdam (2006)Google Scholar
  44. 44.
    Pearce, D., Tompits, H., Woltran, S.: Characterising Equilibrium Logic and Nested Logic Programs: Reductions and Complexity. Theory and Practice of Logic Programming 9(5), 565–616 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Pearce, D., Valverde, A.: Uniform Equivalence for Equilibrium Logic and Logic Programs. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 194–206. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  46. 46.
    Sagiv, Y.: Optimising DATALOG Programs. In: Minker, J. (ed.) Foundations of Deductive Databases and Logic Programming, pp. 659–698. Morgan Kaufmann, San Francisco (1988)CrossRefGoogle Scholar
  47. 47.
    Schaub, T.: Making Your Hands Dirty Inspires Your Brain! Or How to Switch ASP into Production Mode. In: Erdem, E., Lin, F., Schaub, T. (eds.) LPNMR 2009. LNCS (LNAI), vol. 5753, pp. 631–633. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  48. 48.
    Shmueli, O.: Decidability and Expressiveness Aspects of Logic Queries. In: Proc. PODS 1987, pp. 237–249. ACM Press, New York (1987)Google Scholar
  49. 49.
    Truszczynski, M., Woltran, S.: Relativized Hyperequivalence of Logic Programs for Modular Programming. Theory and Practice of Logic Programming 9(6), 781–819 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Turner, H.: Strong Equivalence Made Easy: Nested Expressions and Weight Constraints. Theory and Practice of Logic Programming 3(4-5), 602–622 (2003)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Vardi, M.: The Complexity of Relational Query Languages (Extended Abstract). In: Proc. STOC 1982, pp. 137–146. ACM, New York (1982)Google Scholar
  52. 52.
    Vardi, M.: On the Complexity of Bounded-Variable Queries. In: Proc. PODS 1995, pp. 266–276. ACM Press, New York (1995)Google Scholar
  53. 53.
    Woltran, S.: Characterizations for Relativized Notions of Equivalence in Answer Set Programming. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 161–173. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  54. 54.
    Woltran, S.: 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 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Wong, K.-S.: Sound and Complete Inference Rules for SE-Consequence. Journal of Artificial Intelligence Research 31, 205–216 (2008)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Stefan Woltran
    • 1
  1. 1.Institute of Information SystemsTechnische Universität WienViennaAustria

Personalised recommendations