Satisfiability Checking for PC(ID)

  • Maarten Mariën
  • Rudradeb Mitra
  • Marc Denecker
  • Maurice Bruynooghe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3835)


The logic FO(ID) extends classical first order logic with inductive definitions. This paper studies the satisifiability problem for PC(ID), its propositional fragment. We develop a framework for model generation in this logic, present an algorithm and prove its correctness. As FO(ID) is an integration of classical logic and logic programming, our algorithm integrates techniques from SAT and ASP. We report on a prototype system, called MidL, experimentally validating our approach.


Logic Programming Propositional Formula Strongly Connect Component Situation Calculus Positive Cycle 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berman, K.A., Schlipf, J.S., Franco, J.V.: Computing the well-founded semantics faster. In: Marek, V.W., Truszczyński, M., Nerode, A. (eds.) LPNMR 1995. LNCS, vol. 928, pp. 113–126. Springer, Heidelberg (1995)Google Scholar
  2. 2.
    Brachman, R.J., Levesque, H.J.: Competence in knowledge representation. In: Proc. of the National Conference on Artificial Intelligence, pp. 189–192 (1982)Google Scholar
  3. 3.
    Davis, M., Longemann, G., Loveland, D.: A machine program for theorem proving. Communications of the ACM 5, 394–397 (1962)MATHCrossRefGoogle Scholar
  4. 4.
    Dell’Armi, T., Faber, W., Ielpa, G., Koch, C., Leone, N., Perri, S., Pfeifer, G.: System description: DLV. In: Eiter, T., Faber, W., Truszczyński, M. (eds.) LPNMR 2001. LNCS (LNAI), vol. 2173, pp. 424–428. Springer, Heidelberg (2001)Google Scholar
  5. 5.
    Denecker, M.: Knowledge Representation and Reasoning in Incomplete Logic Programming. PhD thesis, Department of Computer Science, K.U.Leuven (1993)Google Scholar
  6. 6.
    Denecker, M.: The well-founded semantics is the principle of inductive definition. In: Dix, J., Fariñas del Cerro, L., Furbach, U. (eds.) JELIA 1998. LNCS (LNAI), vol. 1489, pp. 1–16. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Denecker, M., Bruynooghe, M., Marek, V.: Logic programming revisited: logic programs as inductive definitions. ACM Transactions on Computational Logic 2(4), 623–654 (2001)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Denecker, M., De Schreye, D.: Justification semantics: a unifying framework for the semantics of logic programs. In: LPNMR, pp. 365–379. MIT Press, Cambridge (1993)Google Scholar
  10. 10.
    Denecker, M., Ternovska, E.: Inductive situation calculus. In: Dubois, D., Welty, C.A., Williams, M. (eds.) KR, pp. 545–553. AAAI Press, Menlo Park (2004)Google Scholar
  11. 11.
    East, D., Truszczyński, M.: dcs: An implementation of datalog with constraints. CoRR, cs.AI/0003061 (2000)Google Scholar
  12. 12.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Generation Computing 9, 365–387 (1991)CrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Lin, F., Zhao, Y.: ASSAT: computing answer sets of a logic program by sat solvers. Artif. Intell. 157(1-2), 115–137 (2004)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lonc, Z., Truszczyński, M.: On the problem of computing the well-founded semantics. 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. 673–687. Springer, Heidelberg (2000)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, vol. 3229, pp. 108–120. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Mariën, M., Mitra, R., Denecker, M., Bruynooghe, M.: Satisfiability checking for PC(ID). Technical Report CW426, K.U. Leuven (2005)Google Scholar
  18. 18.
    Marques-Silva, J.P., Sakallah, K.A.: GRASP: A search algorithm for propositional satisfiability. IEEE Trans. Computers 48(5), 506–521 (1999)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Mitchell, D.G., Ternovska, E.: A framework for representing and solving NP search problems. In: Twentieth National Conf. on Artificial Intelligence (AAAI 2005). AAAI Press/MIT Press (2005)Google Scholar
  20. 20.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: DAC, pp. 530–535. ACM, New York (2001)Google Scholar
  21. 21.
    Niemelä, I., Simons, P., Syrjänen, T.: Smodels: a system for answer set programming. In: NMR (2000)Google Scholar
  22. 22.
    Nuutila, E., Soisalon-Soininen, E.: On finding the strongly connected components in a directed graph. Inf. Process. Lett. 49(1), 9–14 (1994)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pelov, N., Ternovska, E.: Reducing inductive definitions to propositional satisfiability. In: Gabbrielli, M., Gupta, G. (eds.) ICLP 2005. LNCS, vol. 3668, pp. 221–234. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  24. 24.
    Ryan, L.: Efficient algorithms for clause-learning SAT solvers. Master’s thesis, Simon Fraser University (2004)Google Scholar
  25. 25.
    Sagonas, K.F., Swift, T., Warren, D.S.: XSB as an efficient deductive database engine. In: Snodgrass, R.T., Winslett, M. (eds.) SIGMOD Conference, pp. 442–453. ACM Press, New York (1994)Google Scholar
  26. 26.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Van Gelder, A.: The alternating fixpoint of logic programs with negation. Journal of Computer and System Sciences 47(1), 185–221 (1993)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    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)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Zhang, L., Madigan, C.F., Moskewicz, M.W., Malik, S.: Efficient conflict driven learning in a boolean satisfiability solver. In: ICCAD, pp. 279–285. ACM, New York (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Maarten Mariën
    • 1
  • Rudradeb Mitra
    • 1
  • Marc Denecker
    • 1
  • Maurice Bruynooghe
    • 1
  1. 1.Department of Computer ScienceKatholieke Universiteit LeuvenBelgium

Personalised recommendations