Linear Intransitive Temporal Logic of Knowledge LTK_r, Decision Algorithms, Inference Rules

TL;DR

This paper explores the decidability of a linear intransitive temporal logic of knowledge, LTK_r, providing algorithms for determining admissible inference rules within this logical framework.

Contribution

It introduces a semantic model for LTK_r, proves its decidability, and develops an algorithm for checking the admissibility of inference rules.

Findings

LTK_r logic is decidable.

An algorithm for admissible inference rules is provided.

Special Kripke models facilitate the analysis of admissibility.

Abstract

Our paper investigates the linear logic of knowledge and time LTK_r with reflexive intransitive time relation. The logic is defined semantically, -- as the set of formulas which are true at special frames with intransitive and reflexive time binary relation. The LTK_r -frames are linear chains of clusters connected by a reflexive intransitive relation $R_T$. Elements inside a cluster are connected by several equivalence relations imitating the knowledge of different agents. We study the decidability problem for formulas and inference rules. Decidability for formulas follows from decidability w.r.t. admissible inference rules.To study admissibility, we introduce some special constructive Kripke models useful for description of admissibility of inference rules. With a special technique of definable valuations we find an algorithm determining admissible inference rules in LTK_r. That is, we show that the logic LTK_r is decidable and decidable with respect to admissibility of inference rules.