Characterizations of the Disjunctive Well-Founded Semantics: Confluent Calculi and Iterated GCWA

  • Authors:

  • Stefan Brass;Jürgen Dix

  • Affiliations:

  • Universität Hannover, Institut für Informatik, Lange Laube 22, D-30159 Hannover, Germany, e-mail:sb@informatik.uni-hannover.de;Universität Koblenz, Institut für Informatik, Rheinau 1, D-56075 Koblenz, Germany, e-mail:dix@informatik.uni-koblenz.de

  • Venue:
  • Journal of Automated Reasoning
  • Year:
  • 1998

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Abstract

Recently Brass and Dix introduced the semantics D-WFS for generaldisjunctive logic programs. The interesting feature of this approach is thatit is both semantically and proof-theoretically founded. Semantically, D-WFSis invariant under some natural declarative principles. Proof-theoretically,any program Φ is associated a normalform Φ, called the residualprogram, by a nontrivial bottom-up construction using least fixpoints of twomonotonic operators.We show in this paper that the original calculus, consisting of somesimple transformations, has a very strong and appealing property: it isconfluent and terminating. This means that all the transformations can beapplied in any order: whenever we arrive at an irreducible program (no moretransformation is applicable), this program is already uniquely determinedand coincides with the normalform res(Φ) Moreover, for fair sequences itis also strongly terminating: every fair sequence of transformations leadsto normalform res(Φ). Another feature of our approach is that D-WFS canbe read off from res(Φ) immediately in a very simple way. No propersubset of the calculus has these properties – only when we restrict tocertain subclasses of programs.We also give an equivalent characterization of D-WFS in terms of iteratedminimal model reasoning with respect to positive programs. This constructionis a generalization of a description of the well-founded semantics: weintroduce a very simple and neat construction of a sequence D_ithat eventually stops and represents the set of derivable disjunctions.Both characterizations open the way for efficient implementations. Thefirst does so because the ordering of the transformations does not matter:we are free to choose always the “best” transformation, whichmaximally reduces the program. The second does so because special methodsfrom circumscription, in particular a sophisticated minimal model reasonerfor positive programs, might be useful.