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Signed Dual Tableaux for Kleene Answer Set Programs

  • Patrick Doherty
  • Andrzej Szałas
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 17)

Abstract

Dual tableaux were introduced by Rasiowa and Sikorski (1960) as a cut free deduction system for classical first-order logic. In the current paper, a sound and complete proof procedure based on dual tableaux is proposed for \({R}_3\), which is the standard Kleene logic augmented with a weak negation connective and an implication connective proposed, in another context, by Shepherdson (1989). \({R}_3\) is used as a basis for defining Kleene Answer Set Programs (\(\textsc {ASP}^{K}\)programs). The semantics for \(\textsc {ASP}^{K}\)programs is based on strongly supported models. Both entailment procedures and model generation procedures for normal and non-normal \(\textsc {ASP}^{K}\)programs are proposed based on the use of dual tableaux and a model filtering technique. The dual tableau proof procedure extended with a model filtering technique is shown to be sound and complete for \(\textsc {ASP}^{K}\)programs, both normal and non-normal. Since there is a direct relationship between answer sets for classical ASP programs and \({R}_3\) models for \(\textsc {ASP}^{K}\)programs, it can be shown that the sound and complete dual tableaux proof procedure with filtering for \(\textsc {ASP}^{K}\)programs is also sound and complete for classical normal ASP programs. For classical non-normal ASP programs, the proof procedure is only sound, since an alternative semantics for disjunction is used in \(\textsc {ASP}^{K}\).

Keywords

Signed tableaux Signed dual tableaux Answer set programming Kleene three-valued logic Strongly supported model 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Computer and Information ScienceLinköping UniversityLinköpingSweden
  2. 2.Institute of Informatics, University of WarsawWarsawPoland

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