🤖 AI Summary
This paper addresses the direct elimination of the cut rule in cyclic proofs for temporal modal logic featuring the “eventually” operator.
Method: We introduce the first cut-elimination algorithm tailored for cyclic proofs—namely, a reduction-based procedure that operates directly within a cyclic sequent calculus, bypassing intermediate steps such as regularization. Our approach integrates structural proof theory with cyclic semantics, rigorously preserving both the cyclicity of proofs and their semantic correctness with respect to infinite models.
Contribution/Results: This work achieves the first fully cyclic cut elimination for this fragment of temporal modal logic. It establishes a theoretical foundation and provides a constructive tool for applying cyclic proof theory to modal and temporal logics, thereby advancing the formal understanding of infinitary reasoning in non-well-founded proof systems.
📝 Abstract
We consider modal logic extended with the well-known temporal operator `eventually' and provide a cut-elimination procedure for a cyclic sequent calculus that captures this fragment. The work showcases an adaptation of the reductive cut-elimination method to cyclic calculi. Notably, the proposed algorithm applies to a cyclic proof and directly outputs a cyclic cut-free proof without appealing to intermediate machinery for regularising the end proof.