Abstract
AbstractA common feature in Answer Set Programming is the use of a second negation, stronger than default negation and sometimes called explicit, strong or classical negation. This explicit negation is normally used in front of atoms, rather than allowing its use as a regular operator. In this paper we consider the arbitrary combination of explicit negation with nested expressions, as those defined by Lifschitz, Tang and Turner. We extend the concept of reduct for this new syntax and then prove that it can be captured by an extension of Equilibrium Logic with this second negation. We study some properties of this variant and compare to the already known combination of Equilibrium Logic with Nelson’s strong negation.
Highlights
The introduction of stable models (Gelfond and Lifschitz 1988) in logic programming was motivated by the search of a suitable semantics for default negation, their early application to knowledge representation revealed the need of a second negation to
To understand the difference for knowledge representation between default negation and explicit negation, a typical example is to distinguish the rule ¬train → cross, that captures the criterion “you can cross if you have no information on a train coming,” from the encoding ∼train → cross that means “you can cross if you have evidence that no train is coming.”
We have introduced a variant of constructive negation in Equilibrium Logic we called explicit negation
Summary
The introduction of stable models (Gelfond and Lifschitz 1988) in logic programming was motivated by the search of a suitable semantics for default negation, their early application to knowledge representation revealed the need of a second negation to. Once in NNF, the obtained equilibrium models coincide with answer sets for the syntactic fragments of nested expressions (Lifschitz et al 1999) or for regular programs (Gelfond and Lifschitz 1993) For this reason, most papers on Equilibrium Logic for ASP assumed a reduction to NNF from the very beginning, and little attention was paid to the behaviour of formulas in the scope of strong negation under a logic programming perspective. We prove that equilibrium models (with explicit negation) capture the answer sets for these extended nested expressions and, preserve the strong equivalences from (Lifschitz et al 1999) even for arbitrary formulas (including implication). Proofs can be found in the supplementary material corresponding to this paper at the TPLP archives
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