🤖 AI Summary
Craig interpolation fails for most normal modal logics extending K4.3 (excluding S5) and for Priorean temporal logics over ℤ, ℚ, ℝ, and finite linear orders—posing a fundamental challenge to structural proof theory and model-theoretic analysis.
Method: We systematically investigate the decidability of *uniform* interpolation (i.e., existence of interpolants independent of proof details) and establish its first bisimulation-based semantic characterization. Using descriptive frames, canonical models, and model-theoretic techniques, we analyze interpolation across diverse temporal flows.
Contribution/Results: We prove that uniform interpolation existence is coNP-complete for every finitely axiomatizable modal logic containing K4.3—matching the complexity of logical consequence. This result uniformly covers standard temporal logics over ℤ, ℚ, ℝ, and finite linear orders. Our work establishes both decidability and precise computational complexity, surpassing prior work in both accuracy and generality for uniform interpolation in modal and temporal logics.
📝 Abstract
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.
