Propositional Abduction in Modal Logic

Abstract

In this work, the problem of performing abduction in modal logics is addressed, along the lines of 3], where a proof theoretical abduction method for full rst order classical logic is de ned, based on tableaux and Gentzen-type systems. This work applies the same methodology to face modal abduction. The non-classical context enforces the value of analytical proof systems as tools to face the meta-logical and proof-theoretical questions involved in abductive reasoning. The similarities and di erences between quanti ers and modal operators are investigated and proof theoretical abduction methods for the modal systems K, D, T and S4 are de ned, that are sound and complete. The construction of the abductive explanations is in strict relation with the expansion rules for the modal logics, in a modular manner that makes local modi cations possible. The method given in this paper is general, in the sense that it can be adapted to any propositional modal logic for which analytic tableaux are provided. Moreover, the way towards an extension to rst order modal logic is straightforward.

Prime Implicates and Prime Implicants: From Propositional to Modal Logic

“…The results of our paper can be directly applied to the problem of abduction in K: our notion of prime implicants can be used as a definition of abductive explanations in K, and our prime implicate generation algorithm provides a means of producing all of the abductive explanations to a given abduction problem. Moreover, because the notion of term underlying our definition of abductive explanations is more expressive than that used by Cialdea Mayer and Pirri (1995), we are able to find explanations which are overlooked by their method. For instance, if we look for an explanation of the observation c given the background information 2(a ∨ b) → c, we obtain 2(a ∨ b), whereas their framework yields 2a and 2b.…”

“…Moreover, if we were to define clauses in K as those first-order clauses which are representable in K, we would obtain a set of clauses containing no 3 modalities, thereby losing much of the expressivity of K. Mayer and Pirri (1995) in a paper on abductive reasoning in modal logic. They define terms to be the formulae which can be constructed from the propositional literals using only ∧, 2, and 3.…”

Identifying Efficient Abductive Hypotheses Using Multicriteria Dominance Relation

“…Various procedures for generation of abducibles have been proposed so far; some are designed for Classical Propositional Calculus (CPC for short) [1], others for more sophisticated propositional logics [29,26] or for first-order logic [16,25,28]. Those procedures, which are defined in a strictly logical setting, use, for example, the proof methods of analytic tableaux [1,26,25], of sequent calculi [26,25], or the dynamic proof method of adaptive logics [29,28].…”

“…Those procedures, which are defined in a strictly logical setting, use, for example, the proof methods of analytic tableaux [1,26,25], of sequent calculi [26,25], or the dynamic proof method of adaptive logics [29,28]. In this paper we will consider an implementation of a procedure generating abductive hypotheses for CPC.…”

Abduction for Learning Smart City Rules

“…Abductive reasoning (also called abduction [11], abductive inference, [11] is a form of logical inference which starts with an observation then seeks to find the simplest and most likely explanation. Abduction was studied in computer science logic [17] both for propositional logic e.g., [16] and first order logic e.g., [15]. One of the most useful application of abduction in computer science is for logic programming [12].…”