Multi-modal and Temporal Logics with Universal Formula--Reduction of Admissibility to Validity and Unification

Abstract

The article studies multi-modal (in particular temporal, tense, logics) possessing universal formulas. We prove (Theorems 11 and 12) that the admissibility problem for inference rules (inference rules with parameters) is decidable for all decidable multi-modal (in particular, temporal) logics possessing an universal formula and decidable w.r.t. unification (unification with parameters). These theorems use characterizations of admissible rules in terms of valid rules and unifiable formulas. Results are applied to wide range of multi-modal logics, including all linear transitive temporal logics, all temporal logics with bounded zigzag, all multi-modal logics with explicit universal modality. As consequence, we show that the admissibility problem for all such logics is decidable (e.g. for all logics of the series S4nU, n∈ N).