From Interpolating Formulas to Separating Languages and Back Again

2025-08-18

AI Summary

This work addresses the generalization and decidability of Craig interpolation (CIP) in non-classical logics: for logics lacking CIP, it characterizes the existence of interpolants for entailments and analyzes their computational complexity. It introduces and systematically studies *L/L′-interpolation*—a weak-language variant subsuming definability—and establishes its equivalence to formal language separation (e.g., separation of regular languages by first-order definable ones). Methodologically, it unifies proof-theoretic interpolation extraction, automata theory, and logical characterization techniques to construct a *decidability transfer framework*: from language separation to logical interpolation. The key contribution is proving the decidability of interpolation existence in Linear Temporal Logic (LTL), thereby revealing a deep unification between interpolation theory and language separation at the level of decidability, and advancing cross-domain integration among logic, automata theory, and formal verification.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Traditionally, research on Craig interpolation is concerned with (a) establishing the Craig interpolation property (CIP) of a logic saying that every valid implication in the logic has a Craig interpolant and (b) designing algorithms that extract Craig interpolants from proofs. Logics that lack the CIP are regarded as `pathological' and excluded from consideration. In this chapter, we survey variations and generalisations of traditional Craig interpolation. First, we consider Craig interpolants for implications in logics without the CIP, focusing on the decidability and complexity of deciding their existence. We then generalise interpolation by looking for Craig interpolants in languages L' that can be weaker than the language L of the given implication. Thus, do not only we restrict the non-logical symbols of Craig interpolants but also the logical ones. The resulting L/L'-interpolation problem generalises L/L'-definability, the question whether an L-formula is equivalent to some L'-formula. After that, we move from logical languages to formal languages where interpolation disguises itself as separation: given two disjoint languages in a class C, does there exist a separating language in a smaller class C'? This question is particularly well-studied in the case when the input languages are regular and the separating language is first-order definable. Finally, we connect the different research strands by showing how the decidability of the separation problem for regular languages can be used to prove the decidability of Craig interpolant existence for linear temporal logic LTL.

Problem

• Study Craig interpolants in logics lacking CIP
• Generalize interpolation using weaker languages L'
• Link separation problems to interpolant decidability in LTL

Innovation

• Study Craig interpolants in logics without CIP
• Generalize interpolation with weaker language constraints
• Link separation problems to interpolation decidability