Rules admissible in transitive temporal logic $T_{S4}$, sufficient condition

Abstract

The paper¹ develops a technique for computation inference rules admissible in temporal logic $T_{S4}$. The problem whether there exists an algorithm recognizing inference rules admissible in $T_{S4}$ is a long-standing open problem. The logic $T_{S4}$ has neither the extension property nor the co-cover property which previously were central instruments for construction decision algorithms for admissibility in modal logics (e.g. reflexive and transitive modal logic $S4$). Our paper uses a linear-compression property, a zigzag-ray property and a zigzag stretching property which hold for $T_{S4}$. The main result of the paper is a sufficient condition for admissibility inference rules in $T_{S4}$. It is shown that all rules which are valid in special finite models (with an effective upper bound on size) must be admissible in $T_{S4}$.