Ground Nonmonotonic Modal Logics

Abstract

In this paper we address ground logics, a family of nonmonotonic modal logics, and their usage in knowledge representation. In such a setting non-modal sentences are used to represent the knowledge of an agent about the world, while an epistemic operator provides the agent with autoepistemic or introspective knowledge. Ground logics are based on the idea of characterizing the knowledge of the agent by allowing it to make nonmonotonic assumptions only with respect to the knowledge about the world, i.e. expressed by nonmodal formulae. They are characterized by a fix-point equation which determines the set of formulae derivable from the agent's initial knowledge and which can be applied to different normal modal systems to obtain a variety of nonmonotonic modal logics. In the paper we address the semantical, computational and epistemological properties of ground logics. We provide a semantic characterization of ground logics by defining a preference relation on possible-world models based on the minimization of the knowledge expressed by nonmodal formulae. We analyze the computational complexity of reasoning in ground logics, providing both a lower bound through a reduction from quantified boolean formulae and an upper bound through an algorithm for computing logical entailment. We discuss the representational features of ground logics, in particular defaults, and provide a thorough comparison with McDermott and Doyle's logics.

Multi-modal nonmonotonic logics of minimal knowledge

“…Hence, any such conclusion may be retracted when new facts are added to the agent's knowledge. For this reason, many nonmonotonic modal formalisms have been proposed in order to characterize the reasoning abilities of an introspective agent: among them, we recall the nonmonotonic modal logics originally proposed by McDermott and Doyle [32,31,30], Moore's autoepistemic logic [35], Lifschitz's logic of minimal knowledge and negation as failure MKNF/MBNF [26,27], Levesque's logic of only knowing [24], and ground nonmonotonic modal logics [18,9,46]. …”

“…In particular, we point out that the semantic definition of S A n is based on a preference order on possible-world structures, following the studies on a model theory for nonomotonic modal logics [44,41,43,9]. Also, it can be seen as a generalization of the possible-world semantics of MKNF and MBNF [26,27].…”

“…The main difference with respect to such previous constructions is that here, due to the presence of multiple modal operators, we cannot impose the condition that the preferred models of a theory always correspond to structures in which each accessibility relation is total (which has a syntactic counterpart in the so-called stable sets of modal formulas [45]). Consequently, minimality of knowledge in the preferred models is imposed via a different, although simple, condition (formally stated by Definition 2), which can be seen as a generalization of analogous minimality criteria in previous, simpler nonmonotonic modal formalisms like MKNF [26] or ground nonmonotonic modal logics [9].…”

“…The family of ground nonmonotonic modal logics studied in [46,47,40,9] is also deeply related to the logics S A n . More precisely, it can be shown that the ground nonmonotonic modal logic based on the modal system S5 and known as S5 G [9] corresponds to the logic S5 A n , in the sense that, given a unimodal theory Σ, the S5 G -models of Σ coincide with the S5 A nmodels of the theory Σ obtained from Σ by replacing each occurrence of the modal operator with the modality K 1 .…”

A non-preferential semantics of non-monotonic modal logic

“…1 In particular, modal logics K, T, and S4 are characterized by classes of Kripke interpretations possessing the former property, and logics S4F, Sw5, and a very important KD45 (that underlies autoepistemic logic) are characterized by classes of Kripke interpretations possessing the latter.…”