A Hybrid of Tense Logic $S4_T$ and Multi-Agent Logic with Interacting Agents

In this paper we introduce a temporal multi-agent logic $S4_T^\mathcal{IA}$, which implements interacting agents. Logic $S4_T^\mathcal{IA}$ is defined semantically as the set of all formulas of the appropriate propositional language that are valid in special Kripke models. The models are based on $S4$-like time frames, i.e., with reflexive and transitive time-accessibility relations. Agents knowledge-accessibility relations $R_i$, defined independently for each individual agent, are $S5$-relations on $R$-time clusters, and interaction of the agents consists of passing knowledge along arbitrary paths of such relations. The key result of the paper is an algorithm for checking satisfiability and recognizing theorems of $S4_T^\mathcal{IA}$. We also prove the effective finite model property for the logic $S4_T^\mathcal{IA}$.