A non-uniform view of Craig interpolation in modal logics with linear frames
For logicians and computer scientists working on modal logics, this provides a non-uniform interpolation result that is computationally tractable, contrasting with other recent results.
The paper addresses the lack of Craig interpolation in modal logics with linear frames by studying the interpolant existence problem. It shows that for all finitely axiomatisable normal modal logics containing K4.3, this problem is decidable and coNP-complete, no harder than entailment.
Normal modal logics extending the logic K4.3 of linear transitive frames are known to lack the Craig interpolation property, except some logics of bounded depth such as 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 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 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 Craig interpolation property.