On interval dynamic logic: Introducing quasi-action lattices

Abstract

In this paper we discuss the incompatibility between the notions of validity and impreciseness in the context of Dynamic Logics. To achieve that we consider the Lukasiewicz action lattice and its interval counterpart, we show how some validities fail in the context of intervals. In order to capture the properties of action lattices that remain valid for intervals we propose a new structure called Quasi-action Lattices which generalizes action lattices and is able to model both: The Lukasiewicz action lattice, L, and its interval counterpart, L. The notion of graded satisfaction relation is extended to quasi-action lattices. We demonstrate that, in the case of intervals, the relation of graded satisfaction is correct (c.f. Theorem 3) with respect to the graded satisfaction relation on the Lukasiewicz action lattice. Although this theorem guarantees that satisfiability is preserved on intervals, we show that validity is not. We propose, then, to weaken the notion of validity on action lattices to designated validity on quasi-action lattices. In this context, Theorem 4 guarantees that the dynamic formulae which are valid with respect to L will be designated valid with respect to L.A classic result (e.g. [10,21]) establishes that Kleene algebras are closed under formation of matrices. Fact 1The structure M n ( L) = (M n ( L), max, , 0, 1, * ) defined above is a Kleene algebra.This justifies the adoption of M n ( L) as a well behaved computational space for LDL.Definition 3. LDL-models for a set of propositions Prop and programs Π, denoted by Mod LDL (Π, Prop), consist of tuples:where W is a finite set (of states), V : Prop × W → [0, 1] is a function, and A π ∈ M n ( L), with n standing for the cardinality of W .

Relating Kleene Algebras with Pseudo Uninorms

“…As examples of the latter efforts, we can point out our recent development on the study of Kleene algebras to deal with "intervals as programs" [22], in order to deal with situations where the precise values of the transitions weights are not provided (e.g. entailed by the machine representation of an irrational number).…”

Introduction to reversal fuzzy switch graph

“…The valuation of the membership function can occur on any lattice, however, depending on this choice, the resulting formal logic must be adjusted. For example, if we consider the lattice of intervals with the Kulish-Miranker order, considering correctness, then the modal logic associated with the graph will have a non-residual implication [23].…”

A Fuzzy Semantic for BDI Logic

“…Others fuzzy (or many-valued) modal logics have already been defined (see for example [21][22][23][24][25][26][27]). In the approach proposed in [23,28], the propositional connectives are interpreted as t-norms, t-conorms, fuzzy implications, fuzzy negations are considered in such a way that the classical tautologies are preserved as in [20,29] and the modal connectives have a semantic similar to the proposed by Caicedo and Rodríguez [30].…”