# 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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