An axiomatization for the linear logic of knowledge and time $LTK_r$ with intransitive time relation

We study the problem of the axiomatization of the linear multimodal logic of knowledge and time $LTK_r$ with reflexive intransitive time relation. The logic is defined semantically as the set of formulas true on frames of a special kind. The $LTK_r$-frames are linear chains of clusters connected by a reflexive intransitive relation $R_T$ which simulates time. Elements inside a cluster are connected by several equivalence relations imitating the knowledge of different agents. The main result of the article is the proof of the fact that the finite set of formulas proposed by the authors is an axiomatization of the logic $LTK_r$ with reflexive intransitive time relation.