Linear Temporal Logic LTLK extended by Multi-Agent Logic Kn with Interacting Agents

Abstract

We study an extension LTLK of the linear temporal logic LTLK by implementing multi-agent knowledge logic KD45m (which is often referred as multi-modal logic S5m). The temporal language of our logic adapts the operations U (until) and N (next) and uses new temporal operations: Uw—weak until, and Us—strong until. We also employ the standard agents’ knowledge operations Ki from the multi-agent logic KD45m and extend them with an operation IntK responsible for knowledge obtained via interaction of agents. The semantic models for LTLK are Kripke/Hintikka-like structures NC based on the linear time. Structures NC use i ∈ N as indexes for time, and the base set of any NC consists of clusters C(i) (for all i ∈ N) containing all possible states at the time i. Agents’ knowledge is modelled in time clusters C(i) via agents’ knowledge accessibility relations Rj. The logic LTLK is the set of all formulas which are valid (true) in all such models NC w.r.t. all possible valuations. We prove that LTLK is decidable: we reduce the decidability problem to verification of validity for special normal reduced forms of rules in specific models (not LTLK models) of size single-exponential in size of the rules. Furthermore, we extend these results to a linear temporal logic LTLK(Z) based on the time flow indexed by all integer numbers (with additional operations Since and Previous). Also we show that LTLK has the finite model property (fmp) while LTLK (Z) has no standard fmp.