Core Concepts

The authors mechanise the computation of propositional quantifiers and prove the correctness of uniform interpolation in the Coq proof assistant for three modal logics: the classical modal logic K, Gödel-Löb logic GL, and the intuitionistic modal logic iSL.

Abstract

The paper presents mechanised proofs of the uniform interpolation property for three modal logics:
Modal Logic K: The authors follow the strategy in [3] using the sequent calculus KS and provide a formalisation in Coq. They define a complete and terminating proof search strategy for KS, and then construct the uniform interpolants using the function AK
p.
Gödel-Löb Logic GL: The authors start from a sequent-style proof of uniform interpolation for GL from [5], but uncover an incompleteness in the original proof. They modify the definition of the function AGL
p to address this issue and provide a corrected version of the construction.
Intuitionistic Strong Löb Logic iSL: This is the first proof-theoretic construction of uniform interpolants for iSL. The authors extend the syntactic method of Pitts for intuitionistic logic IL, taking advantage of a recently developed sequent calculus G4iSLt for iSL.
The authors formalise all definitions and proofs in the constructive setting of the Coq proof assistant, ensuring that the definitions of the uniform interpolants are effective and can be used to extract an OCaml program for computing interpolants.

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Deeper Inquiries

The uniform interpolation results for modal logics like K, GL, and iSL can be extended to other modal logics or fragments by following a similar proof-theoretic approach. The key lies in defining the propositional quantifiers for the specific logic, ensuring correctness and termination of the proof search strategy, and proving the uniform interpolation property. By adapting the definitions and proof techniques to the specific axioms and rules of the target logic, one can establish uniform interpolation for a wide range of modal logics. This process may involve modifying the functions that compute the propositional quantifiers, adjusting the termination criteria, and proving the necessary implications and uniformity properties for the logic in question.

The verified uniform interpolation algorithms have several potential applications beyond the theoretical results presented in the paper. Some of these applications include:
Automated Reasoning: The algorithms can be used in automated theorem proving systems to enhance reasoning capabilities in modal logics. By computing uniform interpolants, these systems can provide more efficient and accurate proofs.
Model Checking: Uniform interpolation can be utilized in model checking algorithms to improve the verification process for systems with modal properties. It can help in reducing the complexity of model checking tasks and enhancing the scalability of the verification process.
Knowledge Representation: In artificial intelligence and knowledge representation systems, uniform interpolation algorithms can aid in representing and reasoning about complex modal knowledge bases. This can lead to more robust and efficient knowledge-based systems.
Natural Language Processing: Uniform interpolation techniques can be applied in natural language processing tasks that involve modal logic reasoning, such as semantic analysis and understanding of modal statements in text.

The complexity bounds of the computed uniform interpolants using syntactic approaches are generally lower than those obtained through semantic methods. Syntactic approaches, such as the proof-theoretic method used in the paper, provide more precise complexity bounds and guarantee termination of the algorithms. This can be advantageous in practical applications where efficiency and predictability are crucial. Semantic approaches, on the other hand, may involve more complex computations based on model theory or bisimulation relations, leading to potentially higher computational complexity.
The practical implications of these differences lie in the efficiency and scalability of the algorithms. Syntactic approaches are more suitable for automated reasoning systems and formal verification tools where fast computation and termination are essential. Semantic approaches may offer more insights into the semantics of the logic but could be computationally more demanding and harder to implement in automated systems. Ultimately, the choice between syntactic and semantic approaches depends on the specific requirements of the application and the trade-offs between complexity and precision.

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