Supportedly Stable Answer Sets for Logic Programs with Generalized Atoms

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9209)

Abstract

Answer Set Programming (ASP) is logic programming under the stable model or answer set semantics. During the last decade, this paradigm has seen several extensions by generalizing the notion of atom used in these programs. Among these, there are dl-atoms, aggregate atoms, HEX atoms, generalized quantifiers, and abstract constraints. In this paper we refer to these constructs collectively as generalized atoms. The idea common to all of these constructs is that their satisfaction depends on the truth values of a set of (non-generalized) atoms, rather than the truth value of a single (non-generalized) atom. Motivated by several examples, we argue that for some of the more intricate generalized atoms, the previously suggested semantics provide unintuitive results and provide an alternative semantics, which we call supportedly stable or SFLP answer sets. We show that it is equivalent to the major previously proposed semantics for programs with convex generalized atoms, and that it in general admits more intended models than other semantics in the presence of non-convex generalized atoms. We show that the complexity of supportedly stable answer sets is on the second level of the polynomial hierarchy, similar to previous proposals and to answer sets of disjunctive logic programs.

References

  1. 1.
    Alviano, M., Faber, W.: Semantics and compilation of answer set programming with generalized atoms. In: Konieczny, S., Tompits, H. (eds.) 15th International Workshop on Nonmonotonic Reasoning (NMR 2014), Number 1843–14-01 in INFSYS Research Reports, Institut für Informationssysteme, pp. 214–222, July 2014Google Scholar
  2. 2.
    Niemelä, I., Simons, P., Soininen, T.: Stable model semantics of weight constraint rules. In: Gelfond, M., Leone, N., Pfeifer, G. (eds.) LPNMR 1999. LNCS, vol. 1730, pp. 317–331. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Niemelä, I., Simons, P.: Extending the smodels system with cardinality and weight constraints. In: Minker, J. (ed.) Logic-Based Artificial Intelligence, pp. 491–521. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  4. 4.
    Dell’Armi, T., Faber, W., Ielpa, G., Leone, N., Pfeifer, G.: Aggregate functions in disjunctive logic programming: semantics, complexity, and implementation in DLV. In: Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI) 2003. Morgan Kaufmann Publishers, Acapulco, Mexico, pp. 847–852, August 2003Google Scholar
  5. 5.
    Faber, W., Pfeifer, G., Leone, N., Dell’Armi, T., Ielpa, G.: Design and implementation of aggregate functions in the DLV system. Theory Pract. Logic Program. 8(5–6), 545–580 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Calimeri, F., Cozza, S., Ianni, G.: External sources of knowledge and value invention in logic programming. Ann. Math. Artif. Intell. 50(3–4), 333–361 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Eiter, T., Lukasiewicz, T., Schindlauer, R., Tompits, H.: Combining answer set programming with description logics for the semantic web. In: Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR2004), Whistler, Canada, pp. 141–151 (2004). Extended Report RR-1843-03-13, Institut für Informationssysteme, TU Wien, 2003Google Scholar
  8. 8.
    Eiter, T., Ianni, G., Lukasiewicz, T., Schindlauer, R., Tompits, H.: Combining answer set programming with description logics for the semantic web. Artif. Intell. 172(12–13), 1495–1539 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eiter, T., Ianni, G., Schindlauer, R., Tompits, H.: A uniform integration of higher-order reasoning and external evaluations in answer set programming. In: International Joint Conference on Artificial Intelligence (IJCAI) 2005, pp. 90–96, Edinburgh, UK, August 2005Google Scholar
  10. 10.
    Pelov, N.: Semantics of logic programs with aggregates. Ph.D. thesis, Katholieke Universiteit Leuven, Leuven, Belgium, April 2004Google Scholar
  11. 11.
    Pelov, N., Denecker, M., Bruynooghe, M.: Well-founded and Stable semantics of logic programs with aggregates. Theory Pract. Logic Program. 7(3), 301–353 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Son, T.C., Pontelli, E.: A constructive semantic characterization of aggregates in ASP. Theory Pract. Logic Program. 7, 355–375 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Faber, W., Leone, N., Pfeifer, G.: Recursive aggregates in disjunctive logic programs: semantics and complexity. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 200–212. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  14. 14.
    Faber, W., Leone, N., Pfeifer, G.: Semantics and complexity of recursive aggregates in answer set programming. Artif. Intell. 175(1), 278–298 (2011). Special Issue: John McCarthy’s LegacyMathSciNetCrossRefGoogle Scholar
  15. 15.
    Ferraris, P.: Logic programs with propositional connectives and aggregates. ACM Trans. Comput. Log. 12(4), 25 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Alviano, M., Faber, W.: The complexity boundary of answer set programming with generalized atoms under the FLP semantics. In: Cabalar, P., Son, T.C. (eds.) LPNMR 2013. LNCS, vol. 8148, pp. 67–72. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  17. 17.
    Shen, Y., Wang, K., Eiter, T., Fink, M., Redl, C., Krennwallner, T., Deng, J.: FLP answer set semantics without circular justifications for general logic programs. Artif. Intell. 213, 1–41 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Clark, K.L.: Negation as failure. In: Gallaire, H., Minker, J. (eds.) Logic and Data Bases, pp. 293–322. Plenum Press, New York (1978)CrossRefGoogle Scholar
  19. 19.
    Liu, L., Truszczyński, M.: Properties and applications of programs with monotone and convex constraints. J. Art. Intell. Res. 27, 299–334 (2006)Google Scholar
  20. 20.
    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) CrossRefGoogle Scholar
  21. 21.
    Alviano, M., Faber, W., Leone, N., Perri, S., Pfeifer, G., Terracina, G.: The disjunctive datalog system DLV. In: de Moor, O., Gottlob, G., Furche, T., Sellers, A. (eds.) Datalog 2010. LNCS, vol. 6702, pp. 282–301. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Gebser, M., Kaufmann, B., Schaub, T.: Conflict-driven answer set solving: From theory to practice. Artif. Intell. 187, 52–89 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Alviano, M., Dodaro, C., Faber, W., Leone, N., Ricca, F.: WASP: a native ASP solver based on constraint learning. In: Cabalar, P., Son, T.C. (eds.) LPNMR 2013. LNCS, vol. 8148, pp. 54–66. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  25. 25.
    Alviano, M., Dodaro, C., Ricca, F.: Anytime computation of cautious consequences in answer set programming. TPLP 14(4–5), 755–770 (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of CalabriaRendeItaly
  2. 2.University of HuddersfieldHuddersfieldUK

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