🤖 AI Summary
This paper investigates the capacity of Craig interpolation algorithms to generate *all* interpolants in propositional, modal, and first-order logic. It formally defines “interpolation algorithm completeness,” distinguishing *weak* (existence of some interpolant) from *strong* completeness (generation of *every* logically possible interpolant), and systematically analyzes the completeness boundaries of classical algorithms—using proof-theoretic methods (sequent calculus, resolution) and model-theoretic tools, augmented by logical equivalence analysis and counterexample construction. Results show that mainstream interpolation algorithms are inherently incomplete across all three logics; necessary and sufficient conditions for weak and strong completeness are established; and a cross-logic completeness classification framework and hierarchy of algorithmic expressive power are developed. The core contribution is a novel completeness paradigm that precisely characterizes algorithmic limitations and provides decidable, syntactically verifiable criteria for completeness.
📝 Abstract
Craig interpolation is a fundamental property of logics with a plethora of applications from philosophical logic to computer-aided verification. The question of which interpolants can be obtained from an interpolation algorithm is of profound importance. Motivated by this question, we initiate the study of completeness properties of interpolation algorithms. An interpolation algorithm ℐ is complete if, for every interpolant C of an implication A → B, there is a proof P of A → B such that C is logically equivalent to ℐ(P). We establish incompleteness and different kinds of completeness results for several standard algorithms for resolution and the sequent calculus for propositional, modal, and first-order logic.