Propositional dynamic logic for searching games with errors

Abstract

We investigate some finitely-valued generalizations of propositional dynamic logic with tests. We start by introducing the n + 1valued Kripke models and a corresponding language based on a modal extension of Łukasiewicz many-valued logic. We illustrate the definitions by providing a framework for an analysis of the Rényi -Ulam searching game with errors.Our main result is the axiomatization of the theory of the n + 1valued Kripke models. This result is obtained through filtration of the canonical model of the smallest n+1-valued propositional dynamic logic.2010 Mathematics Subject Classification. 03B45, 03B70, 03B50. 2012 ACM Classification. Theory of computation → Logic and verification.

Dynamic Łukasiewicz logic and its application to immune system

“…Notice that the axiom Ax0 is well known as axiom of basic modal logic K , the axioms Ax1-Ax6 have been proposed by Segerberg in 1977 (see also Kozen and Parikh 1981) for dynamic logic, but in a terms, and Ax7-Ax8 are specific axioms for modal Łukasiewicz logic (Di Nola and Grigolia 2004;Hansoul and Teheux 2006). Notice also the works (Di Nola et al 2020;Teheux 2014) on dynamic Łukasiewicz logic.…”

Modal Definability Based on Łukasiewicz Validity Relations

“…(VI) Informally, the validity relation |= defined in section 2 permit talking about the set of possible truth values in worlds of L n -frames. This gain of expressive power could turn out to be interesting for application-oriented modal extensions of many-valued logics such as the many-valued generalization of Propositional Dynamic Logic developed in [19]. case of a unary modality was given in [14,Proposition 5.6], but the general case has not been published previously.…”

Modal extensions of Łukasiewicz logic for modelling coalitional power

“…Other systems have been considered, such as ATL [4,3] and its epistemic extensions [24]. A natural task could be to design the many-valued versions of those calculi in order to capture wider classes of games or protocols in which errors are allowed; see [22], for instance. (5) We did not consider the complexity issue of the satisfiability problem for the many-valued modal languages and models introduced in this paper.…”

“…On the logical side it corresponds to the axioms (1)-(2) of the logic P n in Definition 5.9. It is worth mentioning that homogeneity appears as an axiom in other modal languages as well; see [7,19,22]. The finiteness of our scale of truth degrees is essential for the completeness results, which usually fail badly for modal extensions of infinitely-valued Łukasiewicz logic [13].…”