Reducing Inductive Definitions to Propositional Satisfiability

  • Nikolay Pelov
  • Eugenia Ternovska
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3668)


The FO(ID) logic is an extension of classical first-order logic with a uniform representation of various forms of inductive definitions. The definitions are represented as sets of rules and they are interpreted by two-valued well-founded models. For a large class of combinatorial and search problems, knowledge representation in FO(ID) offers a viable alternative to the paradigm of Answer Set Programming. The main reasons are that (i) the logic is an extension of classical logic and (ii) the semantics of the language is based on well-understood principles of mathematical induction.

In this paper, we define a reduction from the propositional fragment of FO(ID) to SAT. The reduction is based on a novel characterization of two-valued well-founded models using a set of inequality constraints on level mappings associated with the atoms. We also show how the reduction to SAT can be adapted for logic programs under the stable model semantics. Our experiments show that when using a state of the art SAT solver both reductions are competitive with other answer set programming systems — both direct implementations and SAT based.


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  1. 1.
    Baral, C.: Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, Cambridge (2003)zbMATHCrossRefGoogle Scholar
  2. 2.
    Ben-Eliyahu, R., Dechter, R.: Propositional semantics for disjunctive logic programs. Annals of Mathematics and Artificial Intelligence 12(1–2), 53–87 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brachman, R.J., Levesque, H.: Competence in knowledge representation. Proc. of the National Conference on Artificial Intelligence, 189–192 (1982)Google Scholar
  4. 4.
    Denecker, M.: Extending classical logic with inductive definitions. In: Palamidessi, C., Moniz Pereira, L., Lloyd, J.W., Dahl, V., Furbach, U., Kerber, M., Lau, K.-K., Sagiv, Y., Stuckey, P.J. (eds.) CL 2000. LNCS (LNAI), vol. 1861, pp. 703–717. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Denecker, M., Ternovska, E.: Inductive situation calculus. In: Principles of Knowledge Representation and Reasoning: Proc. of the 9th International Conference, pp. 545–553. AAAI Press, Menlo Park (2004)Google Scholar
  6. 6.
    Denecker, M., Ternovska, E.: A logic of non-monotone inductive definitions and its modularity properties. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 47–60. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Denecker, M., Theseider Dupré, D., Van Belleghem, K.: An inductive definition approach to ramifications. Linköping Electronic Articles in Computer and Information Science 3(7), 1–43 (1998), Google Scholar
  8. 8.
    Erdem, E., Lifschitz, V.: Tight logic programs. Theory and Practice of Logic Programming 3(4–5), 499–518 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fages, F.: Consistency of Clark’s completion and existence of stable models. Journal of Methods of Logic in Computer Science 1, 51–60 (1994)Google Scholar
  10. 10.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Logic Programming, 5th International Conference and Symposium, pp. 1070–1080 (1988)Google Scholar
  11. 11.
    Janhunen, T.: Representing normal programs with clauses. In: Proc. of the 16th European Conference on Artificial Intelligence, pp. 358–362 (2004)Google Scholar
  12. 12.
    Lierler, Y., Maratea, M.: Cmodels-2: SAT-based answer set solver enhanced to non-tight programs. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 346–350. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Lin, F., Zhao, J.: On tight logic programs and yet another translation from normal logic programs to propositional logic. In: International Joint Conference on Artificial Intelligence, pp. 853–858. Morgan Kaufmann, San Francisco (2003)Google Scholar
  14. 14.
    Lin, F., Zhao, Y.: ASSAT: Computing answer sets of a logic program by SAT solvers. Artificial Intelligence 157(1–2), 115–137 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Linke, T., Tompits, H., Woltran, S.: On acyclic and head-cycle free nested logic programs. In: Demoen, B., Lifschitz, V. (eds.) ICLP 2004. LNCS, vol. 3132, pp. 225–239. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Mariën, M., Gilis, D., Denecker, M.: On the relation between ID-Logic and Answer Set Programming. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 108–120. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Mitchell, D., Ternovska, E.: A framework for representing and solving NP-search problems. In: Proc. of the National Conference on Artificial Intelligence, pp. 430–435 (2005)Google Scholar
  18. 18.
    Niemelä, I.: Logic programs with stable model semantics as a constraint programming paradigm. Annals of Mathematics and Artificial Intelligence 25(3,4), 241–273 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ryan, L.: Efficient algorithms for clause-learning SAT solvers. Master’s thesis, Simon Fraser University, Burnaby, Canada (2004)Google Scholar
  20. 20.
    Simons, P., Niemelä, I., Soininen, T.: Extending and implementing the stable model semantics. Artificial Intelligence 138(1–2), 181–234 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ternovskaia, E.: Causality via inductive definitions. In: Working Notes of ”Prospects for a Commonsense Theory of Causation”. AAAI Spring Symposium Series, pp. 94–100 (1998)Google Scholar
  22. 22.
    Ternovskaia, E.: ID-logic and the ramification problem for the situation calculus. In: Proc. of the 14th European Conference on Artificial Intelligence, pp. 563–567 (2000)Google Scholar
  23. 23.
    Van Gelder, A., Ross, K.A., Schlipf, J.S.: The well-founded semantics for general logic programs. Journal of the ACM 38(3), 620–650 (1991)zbMATHGoogle Scholar
  24. 24.
    Van Nuffelen, B., Cortés-Calabuig, A., Denecker, M., Arieli, O., Bruynooghe, M.: Data integration ising ID-logic. In: Persson, A., Stirna, J. (eds.) CAiSE 2004. LNCS, vol. 3084, pp. 67–81. Springer, Heidelberg (2004)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Nikolay Pelov
    • 1
  • Eugenia Ternovska
    • 1
  1. 1.School of Computing ScienceSimon Fraser UniversityVancouverCanada

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