Autoepistemic equilibrium logic and epistemic specifications

Abstract

Epistemic specifications extend disjunctive answer-set programs by an epistemic modal operator that may occur in the body of rules. Their semantics is in terms of world views, which are sets of answer sets, and the idea is that the epistemic modal operator quantifies over these answer sets. Several such semantics were proposed in the literature. We here propose a new semantics that is based on the logic of here-and-there: we add epistemic modal operators to its language and define epistemic here-and-there models. We then successively define epistemic equilibrium models and autoepistemic equilibrium models. The former are obtained from epistemic here-and-there models in exactly the same way as Pearce's equilibrium models are obtained from here-and-there models, viz. by minimising truth; they provide an epistemic extension of equilibrium logic. The latter are obtained from the former by maximising the set of epistemic possibilities, and they provide a new semantics for Gelfond's epistemic specifications. For both semantics we establish a strong equivalence result: we characterise strong equivalence of two epistemic programs by means of logical equivalence in epistemic here-and-there logic. We finally compare our approach to the existing semantics of epistemic specifications and discuss which formalisms provide more intuitive results by pointing out some formal properties a semantics proposal should satisfy.

Refining the Semantics of Epistemic Specifications

“…Pearce [19] basically proposed EL in order to provide a purely logical foundation of ASP. Inspired by its success as ASP's general framework, Fariñas et al [23,6,28] introduced an epistemic extension of EL, named ES 15 here, in order to suggest an alternative semantics not only for ES, but also for nested ELPs. This section briefly recalls the approach of ES 15 .…”

“…There are two important differences of EHT 20 -models from functional EHT 15 -models defined above: First, EHT 20 -models are almost the same as relational EHT 15 -models (see [28], Sect. 8) when we consider them simply as nonempty collections of arbitrary HT-models, but disregard the relations between these HT-models.…”

“…, where the last conjunct r * 4 is EHT-equivalent to K¬¬a, i.e., K(¬a → ⊥) by Coroll. 1 in [28]. So, one can interpret r 4 in the way of applying the constraint ⊥←nota everywhere.…”

“…We know that ¬¬ K, K¬¬ and ¬K¬ are all equivalent in EHT (see Prop. 5; [28]) So, using Coroll. 1, we deduce that ∆ * is equivalent to (a ∨ b) ∧ (b ∨ ¬ Ka) in EHT, and again by Prop.…”

“…The majority [12,21,14,13,25,26] are reduct-based world-view semantics. The rest [29,23,28,4] are inspired by Pearce's equilibrium-model approach [19], characterising answer sets on a purely logical domain through minimal model reasoning. They are based on epistemic extensions of equilibrium logic.…”