A Revised Concept of Safety for General Answer Set Programs

  • Pedro Cabalar
  • David Pearce
  • Agustín Valverde
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5753)

Abstract

To ensure a close relation between the answer sets of a program and those of its ground version, some answer set solvers deal with variables by requiring a safety condition on program rules. If we go beyond the syntax of disjunctive programs, for instance by allowing rules with nested expressions, or perhaps even arbitrary first-order formulas, new definitions of safety are required. In this paper we consider a new concept of safety for formulas in quantified equilibrium logic where answer sets can be defined for arbitrary first-order formulas. The new concept captures and generalises two recently proposed safety definitions: that of Lee, Lifschitz and Palla (2008) as well as that of Bria, Faber and Leone (2008). We study the main metalogical properties of safe formulas.

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References

  1. 1.
    Baral, C.: Knowledge Representation, Reasoning and Declarative Problem Solving. CUP, Cambridge (2002)MATHGoogle Scholar
  2. 2.
    Bria, A., Faber, W., Leone, N.: Normal Form Nested Programs. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 76–88. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    de Bruijn, J., Pearce, D., Polleres, A., Valverde, A.: Quantified Equilibrium Logic and Hybrid Rules. In: Marchiori, M., Pan, J.Z., Marie, C.d.S. (eds.) RR 2007. LNCS, vol. 4524, pp. 58–72. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Cabalar, P., Pearce, D., Valverde, A.: Reducing Propositional Theories in Equilibrium Logic to Logic Programs. In: Bento, C., Cardoso, A., Dias, G. (eds.) EPIA 2005. LNCS (LNAI), vol. 3808, pp. 4–17. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Cabalar, P., Pearce, D., Valverde, A.: Safety Preserving Transformations for General Answer Set Programs. In: Logic-Based program Synthesis and Transformation. Proc. LOPSTR 2009 (2009)Google Scholar
  6. 6.
    van Dalen, D.: Logic and Structure. Springer, Heidelberg (2004)CrossRefMATHGoogle Scholar
  7. 7.
    Ferraris, P., Lee, J., Lifschitz, V.: A New Perspective on Stable Models. In: Veloso, M. (ed.) 20th International Joint Conference on Artificial Intelligence, IJCAI 2007, pp. 372–379 (2007)Google Scholar
  8. 8.
    Heymans, S., van Nieuwenborgh, D., Vermeir, D.: Open Answer Set Programming with Guarded Programs. ACM Trans. Comp. Logic 9, Article 26 (2008)Google Scholar
  9. 9.
    Heymans, S., Predoiu, L., Feier, C., de Bruijn, J., van Nieuwenborgh, D.: G-hybrid Knowledge Bases. In: ALPSWS 2006, CEUR Workshop Proceedings, vol. 196 (2006)Google Scholar
  10. 10.
    Kleene, S.C.: On Notation for Ordinal Numbers. The J. of Symbolic Logic 3(4) (1938)Google Scholar
  11. 11.
    Lee, J., Lifschitz, V., Palla, R.: A Reductive Semantics for Counting and Choice in Answer Set Programming. In: Proceedings AAAI 2008, pp. 472–479 (2008)Google Scholar
  12. 12.
    Lee, J., Lifschitz, V., Palla, R.: Safe formulas in the general theory of stable models (Preliminary report). In: Garcia de la Banda, M., Pontelli, E. (eds.) ICLP 2008. LNCS, vol. 5366, pp. 672–676. Springer, Heidelberg (2008); longer version by personal communication (June 2008)CrossRefGoogle Scholar
  13. 13.
    Lee, J., Meng, V.: On Loop Formulas with Variables. In: Proc. of AAAI 2008. AAAI, Menlo Park (2008)Google Scholar
  14. 14.
    Lee, J., Palla, R.: Yet Another Proof of the Strong Equivalence between Propositional Theories and Logic Programs. In: CENT 2007, Tempe, AZ, May 2007. CEUR Workshop Proceedings, vol. 265, CEUR-WS.org (2007)Google Scholar
  15. 15.
    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
  16. 16.
    Lifschitz, V., Tang, L., Turner, H.: Nested Expressions in Logic Programs. Annals of Mathematics and Artificial Intelligence 25(3-4), 369–389 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pearce, D., Valverde, A.: Towards a first order equilibrium logic for nonmonotonic reasoning. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 147–160. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Pearce, D., Valverde, A.: A First-Order Nonmonotonic Extension of Constructive Logic. Studia Logica 80, 321–346 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pearce, D., Valverde, A.: Quantified Equilibrium Logic. Tech. report, Univ. Rey Juan Carlos (2006), http://www.matap.uma.es/investigacion/tr/ma06_02.pdf
  20. 20.
    Pearce, D., Valverde, A.: Quantified Equilibrium Logic and Foundations for Answer Set Programs. In: Garcia de la Banda, M., Pontelli, E. (eds.) ICLP 2008. LNCS, vol. 5366, pp. 546–560. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Pedro Cabalar
    • 1
  • David Pearce
    • 2
  • Agustín Valverde
    • 3
  1. 1.Universidade da CoruñaSpain
  2. 2.Universidad Politécnica de MadridSpain
  3. 3.Universidad de MálagaMálagaSpain

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