A NON-UNIFORM VIEW OF CRAIG INTERPOLATION IN MODAL LOGICS WITH LINEAR FRAMES

Abstract Normal modal logics extending the logic $\mathsf {K4.3}$ of linear transitive frames are known to lack the Craig interpolation property (CIP), except some logics of bounded depth such as $\mathsf {S5}$ . We turn this ‘negative’ fact into a research question and pursue a non-uniform approach to Craig interpolation by investigating the following interpolant existence problem: decide whether there exists a Craig interpolant between two given formulas in any fixed logic above $\mathsf {K4.3}$ . Using a bisimulation-based characterisation of interpolant existence for descriptive frames, we show that this problem is decidable and coNP -complete for all finitely axiomatisable normal modal logics containing $\mathsf {K4.3}$ . It is thus not harder than entailment in these logics, which is in sharp contrast to other recent non-uniform interpolation results. We also extend our approach to Priorean temporal logics (with both past and future modalities) over the standard time flows—the integers, rationals, reals, and finite strict linear orders—none of which is blessed with the CIP.