🤖 AI Summary
This work addresses the bidirectional translation between non-well-founded and cyclic proofs in Linear Temporal Logic (LTL) by proposing a unified framework based on linear nested sequent calculi. The central challenge lies in identifying and unfolding cycles, for which the authors introduce a saturated recursive normal form to effectively recognize cyclic structures and devise a rule-forwarding mechanism to syntactically reconstruct cyclic proofs into non-well-founded ones. This approach establishes a precise correspondence between the two proof systems, resolving a long-standing problem of mutual translatability in LTL. Moreover, it enriches the proof-theoretic toolkit for multi-conclusion sequent calculi and lays the groundwork for more expressive reasoning systems in temporal logic.
📝 Abstract
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.