Core Concepts

This work presents a syntactic cut-elimination procedure for a cyclic sequent calculus GKe that captures the modal logic MLe extended with the 'eventually' temporal operator.

Abstract

The content discusses cut elimination for cyclic proofs in the context of modal logic extended with the 'eventually' temporal operator. The key points are:
The authors consider modal logic MLe, which extends basic modal logic with the 'eventually' (F) operator. They provide a complete cyclic sequent calculus GKe for this logic.
The main challenge in cut elimination for cyclic proofs is that cuts can interfere with the global validity condition that ensures soundness of cyclic reasoning. The authors distinguish between "unimportant" cuts, which can be eliminated by standard techniques, and "important" cuts, which require a more involved treatment.
For important cuts, the authors develop a reductive cut-elimination procedure that works directly on cyclic proofs. This involves transforming the proof into a "traversed proof" representation, where cuts are isolated and can be eliminated recursively.
The authors show that their cut-elimination procedure preserves the cyclic structure of the proof, directly producing a cut-free cyclic proof without the need for intermediate steps to regularize the proof.
The work is presented as a case study, with the authors noting that the techniques developed here could potentially be applied to cut elimination for cyclic proofs in other modal fixpoint logics.

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Key Insights Distilled From

by Bahareh Afsh... at **arxiv.org** 05-06-2024

Deeper Inquiries

The cut-elimination technique developed in this work for cyclic proofs in the context of the logic MLe has several potential applications beyond this specific scenario.
Automated Reasoning: The cut-elimination procedure can be applied to automated reasoning systems to improve the efficiency and reliability of theorem proving in modal logics extended with fixpoint operators. This can enhance the automation of reasoning tasks in various domains such as verification, model checking, and artificial intelligence.
Program Verification: The technique can be utilized in program verification to ensure the correctness of software systems that involve temporal properties. By eliminating cuts in cyclic proofs, it can help in verifying complex programs with temporal logic specifications.
Semantic Analysis: The method can be used in semantic analysis of systems with temporal properties to ensure consistency and correctness in the interpretation of these systems. This can be valuable in areas like natural language processing, where temporal logic is used to model temporal relationships.
Model Checking: The cut-elimination technique can enhance model checking algorithms for systems with temporal specifications. By streamlining the proof process, it can improve the scalability and performance of model checking tools.

To extend the authors' approach to handle more complex modal fixpoint logics like the full modal μ-calculus while preserving direct cut-elimination on cyclic proofs, several considerations need to be taken into account:
Adaptation of Rules: The rules of the cyclic proof system GKe would need to be extended or modified to accommodate the additional constructs and operators present in the full modal μ-calculus. This may involve introducing new rules specific to the μ-calculus operators.
Handling Fixpoints: The technique would need to address the handling of fixpoint operators in the proof system. This could involve developing specialized rules for dealing with fixpoint formulas and ensuring their correct treatment during the cut-elimination process.
Complexity Analysis: The extension to more complex modal logics would require a thorough analysis of the computational complexity of the cut-elimination procedure. Ensuring that the extended approach remains efficient and effective for handling the increased expressiveness of the μ-calculus is crucial.
Proof-Theoretic Properties: Exploring the proof-theoretic properties specific to the full modal μ-calculus and how they interact with cyclic proofs would be essential. This includes investigating structural properties, normalization properties, and the relationship between fixpoints and cyclic derivations.

Beyond cut elimination, the techniques introduced in this work can be leveraged to investigate various other proof-theoretic properties of cyclic proof systems like GKe. Some of these properties include:
Normalization: Studying the normalization properties of cyclic proofs to ensure that every proof can be transformed into a normal form. This can help in simplifying proofs and understanding the structure of derivations in cyclic systems.
Structural Analysis: Analyzing the structural properties of cyclic proofs, such as the presence of cycles, the depth of cuts, and the distribution of formulas in the proof tree. Understanding these structural aspects can provide insights into the complexity and expressiveness of the proof system.
Subformula Properties: Investigating the behavior of subformulas in cyclic proofs, including their propagation through the proof tree, their impact on the proof structure, and their role in the cut-elimination process. This can shed light on the interaction between different parts of a proof and the overall coherence of the derivation.
Proof Complexity: Exploring the complexity of proofs in cyclic systems, including the size of proofs, the number of steps required for cut elimination, and the relationship between proof complexity and the properties of the underlying logic. Understanding proof complexity can help in optimizing proof search and verification processes.

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