Gelfond-Zhang Aggregates as Propositional Formulas

  • Pedro Cabalar
  • Jorge Fandinno
  • Torsten Schaub
  • Sebastian Schellhorn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10377)


We show that any ASP aggregate interpreted under Gelfond and Zhang’s (GZ) semantics can be replaced (under strong equivalence) by a propositional formula. Restricted to the original GZ syntax, the resulting formula is reducible to a disjunction of conjunctions of literals but the formulation is still applicable even when the syntax is extended to allow for arbitrary formulas (including nested aggregates) in the condition. Once GZ-aggregates are represented as formulas, we establish a formal comparison (in terms of the logic of Here-and-There) to Ferraris’ (F) aggregates, which are defined by a different formula translation involving nested implications. In particular, we prove that if we replace an F-aggregate by a GZ-aggregate in a rule head, we do not lose answer sets (although more can be gained). This extends the previously known result that the opposite happens in rule bodies, i.e., replacing a GZ-aggregate by an F-aggregate in the body may yield more answer sets. Finally, we characterise a class of aggregates for which GZ- and F-semantics coincide.


  1. 1.
    Aguado, F., Cabalar, P., Pearce, D., Pérez, G., Vidal, C.: A denotational semantics for equilibrium logic. Theory Pract. Log. Program. 15(4–5), 620–634 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alviano, M., Faber, W.: Stable model semantics of abstract dialectical frameworks revisited: a logic programming perspective. In: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, pp. 2684–2690. AAAI Press (2015)Google Scholar
  3. 3.
    Baral, C.: Knowledge Representation, Reasoning and Declarative Problem Solving. Cambridge University Press, New York (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Cabalar, P.: Functional answer set programming. Theory Pract. Log. Program. 11(2–3), 203–233 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Calimeri, F., Faber, W., Gebser, M., Ianni, G., Kaminski, R., Krennwallner, T., Leone, N., Ricca, F., Schaub, T.: ASP-Core-2: input language format (2012).
  6. 6.
    Erdem, E., Gelfond, M., Leone, N.: Applications of ASP. AI Mag. 37(3), 53–68 (2016)Google Scholar
  7. 7.
    Faber, W., Pfeifer, G., Leone, N.: Semantics and complexity of recursive aggregates in answer set programming. Artif. Intell. 175(1), 278–298 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Faber, W., Pfeifer, G., Leone, N., Dell’Armi, T., Ielpa, G.: Design and implementation of aggregate functions in the DLV system. Theory Pract. Log. Program. 8(5–6), 545–580 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ferraris, P.: Answer sets for propositional theories. In: Baral, C., Greco, G., Leone, N., Terracina, G. (eds.) LPNMR 2005. LNCS (LNAI), vol. 3662, pp. 119–131. Springer, Heidelberg (2005). doi:10.1007/11546207_10 CrossRefGoogle Scholar
  10. 10.
    Ferraris, P.: Logic programs with propositional connectives and aggregates. ACM Trans. Comput. Log. 12(4), 25 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ferraris, P., Lee, J., Lifschitz, V.: A new perspective on stable models. In: Proceedings of the Twentieth International Joint Conference on Artificial Intelligence (IJCAI 2007), pp. 372–379. AAAI/MIT Press (2007)Google Scholar
  12. 12.
    Gebser, M., Kaufmann, B., Schaub, T.: Conflict-driven answer set solving: from theory to practice. Artif. Intell. 187–188, 52–89 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of the Fifth International Conference and Symposium of Logic Programming (ICLP 1988), pp. 1070–1080. MIT Press (1988)Google Scholar
  14. 14.
    Gelfond, M., Zhang, Y.: Vicious circle principle and logic programs with aggregates. Theory Pract. Log. Program. 14(4–5), 587–601 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Harrison, A., Lifschitz, V., Truszczynski, M.: On equivalence of infinitary formulas under the stable model semantics. Theory Pract. Log. Program. 15(1), 18–34 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Trans. Comput. Log. 2(4), 526–541 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pearce, D.: A new logical characterisation of stable models and answer sets. In: Dix, J., Pereira, L.M., Przymusinski, T.C. (eds.) NMELP 1996. LNCS, vol. 1216, pp. 57–70. Springer, Heidelberg (1997). doi:10.1007/BFb0023801 CrossRefGoogle Scholar
  18. 18.
    Pearce, D.: Equilibrium logic. Ann. Math. Artif. Intell. 47(1–2), 3–41 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Pelov, N., Denecker, M., Bruynooghe, M.: Well-founded and stable semantics of logic programs with aggregates. Theory Pract. Log. Program. 7(3), 301–353 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Simons, P., Niemel, I., Soininen, T.: Extending and implementing the stable model semantics. Artif. Intell. 138(1–2), 181–234 (2002)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Son, T., Pontelli, E.: A constructive semantic characterization of aggregates in answer set programming. Theory Pract. Log. Program. 7(3), 355–375 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Pedro Cabalar
    • 1
  • Jorge Fandinno
    • 1
  • Torsten Schaub
    • 2
  • Sebastian Schellhorn
    • 2
  1. 1.University of CorunnaA CoruñaSpain
  2. 2.University of PotsdamPotsdamGermany

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