Non-Wellfounded and Cyclic Proofs for LTL: A Syntactic Correspondence with Linear Nested Sequents
For proof theorists, this work extends techniques for cycle recognition and unraveling from Gentzen sequents to more expressive linear nested sequents, addressing a gap in the literature.
The paper introduces non-wellfounded and cyclic linear nested sequent calculi for LTL, addressing cycle recognition and unraveling. It shows completeness of non-wellfounded proofs via saturation recurrence and provides a procedure to reconstruct non-wellfounded proofs from cyclic ones, establishing a syntactic correspondence.
We introduce and investigate non-wellfounded and cyclic linear nested sequent calculi, and, as a case study, develop such systems for linear temporal logic (LTL). The paper addresses two central problems, which we call 'cycle recognition' and 'unraveling.' Cycle recognition concerns identifying cycles in non-wellfounded proofs in order to extract corresponding cyclic proofs, while unraveling studies the converse transformation, from cyclic proofs to non-wellfounded ones. Although these processes are well understood for Gentzen sequents, they have received little attention for more expressive sequent formalisms and become more challenging in the linear nested sequent setting. To address cycle recognition, we show the completeness of non-wellfounded proofs relative to a particular normal form exhibiting a property we call 'saturation recurrence,' which enables the systematic extraction of cyclic proofs. To address unraveling, we introduce a specialized procedure that shifts rule applications forward along linear nested sequents, allowing non-wellfounded proofs to be reconstructed from cyclic ones. Overall, our work provides new proof-theoretic techniques for cycle recognition and unraveling in expressive multisequent formalisms.