Declarative and computational properties of logic programs with aggregates

  • Authors:

  • Francesco Calimeri;Wolfgang Fabery;Nicola Leone;Simona Perri

  • Affiliations:

  • Department of Mathematics, University of Calabria, Rende, CS, Italy;Department of Mathematics, University of Calabria, Rende, CS, Italy;Department of Mathematics, University of Calabria, Rende, CS, Italy;Department of Mathematics, University of Calabria, Rende, CS, Italy

  • Venue:
  • IJCAI'05 Proceedings of the 19th international joint conference on Artificial intelligence
  • Year:
  • 2005

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Abstract

We investigate the properties of logic programs with aggregates. We mainly focus on programs with monotone and antimonotone aggregates (LPm,aA programs). We define a new notion of unfounded set for (LPm,aA programs, and prove that it is a sound generalization of the standard notion of unfounded set for aggregate-free programs. We show that the answer sets of an LPm,aA program are precisely its unfounded-free models. We define a well-founded operator WP for LPm,aA programs; we prove that its total fixpoints are precisely the answer sets of P, and its least fixpoint WPw(0) is contained in the intersection of all answer sets (if P admits an answer set). WPW(0) is efficiently computable, and for aggregate-free programs it coincides with the well-founded model. We carry out an in-depth complexity analysis in the general framework, including also nonmonotone aggregates. We prove that monotone and anti-monotone aggregates do not increase the complexity of cautious reasoning, which remains in co-NP. Nonmonotone aggregates, instead, do increase the complexity by one level in the polynomial hierarchy. Our results allow also to generalize and speed-up ASP systems with aggregates.