Abstract
Our paper considers a hybrid $\mathrm{LTL}_{\mathcal{ACK}}$ between the multi-agent logic with the local common knowledge operation and an extended version of the linear temporal logic $\mathcal{LTL}$. The logic is based on the semantics of Kripe/Hintikka models with potentially infinite runs and the time points represented by clusters of states with agents' accessibility relations. We study the satisfiability problem for $\mathrm{LTL}_{\mathcal{ACK}}$ and related decidability problem. The key result is an algorithm which recognizes theorems of $\mathrm{LTL}_{\mathcal{ACK}}$ (so we show that $\mathrm{LTL}_{\mathcal{ACK}}$ is decidable), which, as a consequence, also solves the satisfiability problem. Technique is based on verification of validity for special normal reduced forms of rules in models of double exponential in the size of rules.
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