Overview
This study investigates a non-uniform approach to Craig interpolation within modal logics, specifically focusing on those containing the logic K4.3 (logics of linear transitive frames). While these logics are generally known to lack the Craig interpolation property, with exceptions like S5 (logics of bounded depth), the research re-frames this absence as a question concerning the decidability of interpolant existence for any fixed logic above K4.3. The study employs a bisimulation-based characterisation to address this problem.
Research Context
Normal modal logics that extend K4.3, which characterizes linear transitive frames, typically do not possess the Craig interpolation property. A notable exception acknowledged in the research is S5, a logic of bounded depth. This general lack of interpolation presents a challenge for certain applications of logic. Rather than seeking a universal interpolation property, this research investigates the "interpolant existence problem," which asks whether a Craig interpolant exists between two given formulas in a specific fixed logic from this class. This problem formulation represents a non-uniform approach to Craig interpolation.
Approach
The core approach for investigating the interpolant existence problem involved utilizing a bisimulation-based characterisation. This characterisation is specifically applicable to descriptive frames. The method was applied to finitely axiomatisable normal modal logics that contain K4.3. The study then extended this approach to address Priorean temporal logics. These temporal logics incorporate both past and future modalities and were examined over standard time flows, including the integers, rationals, reals, and finite strict linear orders. The listed standard time flows, much like the modal logics investigated, are also noted for not possessing the Craig interpolation property.
Findings
- For all finitely axiomatisable normal modal logics containing K4.3, the interpolant existence problem (determining if a Craig interpolant exists between two given formulas) is decidable.
- This problem is coNP-complete for the specified class of logics.
- The complexity of this problem is not greater than that of entailment in these logics.
- The findings contrast with other recent non-uniform interpolation results in terms of comparative hardness with entailment.
- The non-uniform approach was extended to Priorean temporal logics with both past and future modalities.
- This extended approach applies to Priorean temporal logics over standard time flows such as the integers, rationals, reals, and finite strict linear orders.
- None of the mentioned standard time flows possess the Craig interpolation property.