Disjunctive Programs with Set Constraints

  • Victor W. Marek
  • Jeffrey B. Remmel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7265)


We study an extension of disjunctive logic programs called set constraint disjunctive (SCD) programs where the clauses of the program are allowed to have a disjunction of monotone set constraints in their head and arbitrary monotone and antimonotone set constraints in their body. We introduce new class of models called selector stable models which represent all models which can be computed by an analogue the Gelfond-Lifschitz transform. We show that the stable models of disjunctive logic programs can be defined in terms of selector stable models and then extend this result to SCD logic programs. Finally we show that there is a natural proof theory associated with selector stable models.


Logic Program Minimal Model Stable Model Horn Clause Proof Scheme 
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  1. 1.
    Baral, C.: Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press (2003)Google Scholar
  2. 2.
    Brik, A., Remmel, J.B.: Computing Stable Models of Logic Programs Using Metropolis Type Algorithms. In: Proceedings of Workshop on Answer Set Programming and Other Computing Paradigms (ASPOCP) 2011, paper no. 6, 15 pgs (2011)Google Scholar
  3. 3.
    Eiter, T., Gottlob, G.: On the Computational Cost of Disjunctive Logic Programming: Propositional Case. Ann. Math. Artif. Intell. 15, 289–323 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eiter, T., Faber, W., Leone, N., Pfeifer, G.: Declarative Problem-solving in DLV. In: Minker, J. (ed.) Logic-based Artificial Intelligence, pp. 79–103 (2000)Google Scholar
  5. 5.
    Ferraris, P., Lifschitz, V.: Weight constraints as nested expressions. Theor. Pract. Logic Prog. 5, 45–74 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gelfond, M., Lifschitz, V.: The stable semantics for logic programs. In: Proceedings 5th Int’l. Symp. Logic Programming, pp. 1070–1080. MIT Press (1988)Google Scholar
  7. 7.
    Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Gen. Comput. 9, 365–385 (1991)CrossRefzbMATHGoogle Scholar
  8. 8.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Trans. Comput. Log. 2, 526–541 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu, L., Truszczynski, M.: Properties and Applications of Programs with Monotone and Convex Constraints. J. Artif. Intell. Res. 27, 299–334 (2006)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Liu, L., Pontelli, E., Son, T.C., Truszczynski, M.: Logic Programs with Abstract Constraint Atoms – the Role of Computations. Artif. Intell. 174, 295–315 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lobo, J., Minker, J., Rajasekar, A.: Foundations of Disjunctive Logic Programming. MIT Press (1992)Google Scholar
  12. 12.
    Marek, V.W.: Introduction to Mathematics of Satisfiability. CRC Press (2009)Google Scholar
  13. 13.
    Marek, V.W., Remmel, J.B.: Set Constraints in Logic Programming. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 167–179. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Marek, V.W., Remmel, J.B.: Effective Set Constraints (in preparation)Google Scholar
  15. 15.
    Marek, W., Nerode, A., Remmel, J.B.: Nonmonotonic rule systems I. Ann. Math. Artif. Intell. 1, 241–273 (1990)CrossRefzbMATHGoogle Scholar
  16. 16.
    Marek, W., Nerode, A., Remmel, J.B.: Logic Programs, Well-orderings, and Forward Chaining. Ann. Pure App. Logic 96, 231–276 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Minker, J.: Overview of Disjunctive Logic Programming. Ann. Math. Artif. Intell. 12, 1–24 (1994)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Niemelä, I., Simons, P., Soininen, T.: Stable Model Semantics of Weight Constraint Rules. In: Gelfond, M., Leone, N., Pfeifer, G. (eds.) LPNMR 1999. LNCS (LNAI), vol. 1730, pp. 317–331. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  19. 19.
    Son, T.C., Pontelli, E., Tu, P.H.: Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms. J. Artif. Intell. Res. 29, 353–389 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    van Emden, M.H., Kowalski, R.A.: The semantics of predicate logic as a programming language. J. ACM 23, 733–742 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Victor W. Marek
    • 1
  • Jeffrey B. Remmel
    • 2
  1. 1.Department of Computer ScienceUniversity of KentuckyLexingtonUSA
  2. 2.Departments of Mathematics and Computer ScienceUniversity of California at San DiegoLa JollaUSA

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