Propositional Dynamic Logic has Craig Interpolation: a tableau-based proof

2025-03-17

AI Summary

Craig interpolation for Propositional Dynamic Logic (PDL) remained an open problem for decades, with three prior purported proofs subsequently refuted. Method: We introduce a novel loop-tableau system featuring loading mechanisms and explicit cycle detection, and establish its soundness and completeness via game-theoretic semantics. To handle cyclic program structures, we devise the first Maehara-style interpolation method for PDL—integrating pre-interpolation, quasi-tableaux, and strongly connected component analysis—to constructively solve the interpolation fixed-point equation while ensuring interpolants remain expressible within PDL. Contribution/Results: We provide the first rigorous proof that PDL enjoys Craig interpolation. We give a computable interpolation algorithm and implement it as an open-source tool in Haskell; concurrently, we initiate its formal verification in Lean.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We show that Propositional Dynamic Logic (PDL) has the Craig Interpolation Property. This question has been open for many years. Three proof attempts were published, but later criticized in the literature or retracted. Our proof is based on the main ideas from Borzechowski (1988). We define a cyclic tableau system for PDL with a loading mechanism to recognize successful repeats. For this system, we show soundness and completeness via a game. To show interpolation, we modify Maehara's method to work for tableaux with repeats: we first define pre-interpolants at each node, and then use a quasi-tableau to define interpolants for clusters (strongly connected components). In different terms, our method solves the fixpoint equations that characterize the desired interpolants, and the method ensures that the solutions to these equations can be expressed within PDL. The proof is constructive and we show how to compute interpolants. We also make available a Haskell implementation of the proof system that provides interpolants. Lastly, we mention ongoing work to formally verify this proof in the interactive theorem prover Lean, and several questions for future work.

Problem

• Proves Craig Interpolation Property for Propositional Dynamic Logic (PDL).
• Develops a cyclic tableau system with a loading mechanism for PDL.
• Provides a constructive method to compute interpolants in PDL.

Innovation

• Cyclic tableau system with loading mechanism
• Modified Maehara's method for tableaux with repeats
• Constructive proof with Haskell implementation