🤖 AI Summary
This paper addresses the cut-elimination problem for non-wellfounded proof systems in frugal logic, specifically under an interpretation of the exponential modality “!” as a finite datastream constructor—where global consistency and convergence must be ensured. We propose a progressing-criterion-based non-wellfounded cut-elimination method, yielding the first infinitary cut-elimination procedure in frugal logic that simultaneously preserves progressiveness and higher-order regularity. Using finite approximation techniques, we rigorously establish the convergence of this procedure to well-defined non-wellfounded proofs. Additionally, we develop a relational model semantics that provides a sound denotational foundation for the system. Our main contribution is the first structural proof-theoretic framework for frugal logic that jointly satisfies structural conservation (i.e., admissibility of cut), limit convergence of reduction sequences, and semantic soundness with respect to the relational model.
📝 Abstract
We investigate non-wellfounded proof systems based on parsimonious logic, a weaker variant of linear logic where the exponential modality ! is interpreted as a constructor for streams over finite data. Logical consistency is maintained at a global level by adapting a standard progressing criterion. We present an infinitary version of cut-elimination based on finite approximations, and we prove that, in presence of the progressing criterion, it returns well-defined non-wellfounded proofs at its limit. Furthermore, we show that cut-elimination preserves the progressive criterion and various regularity conditions internalizing degrees of proof-theoretical uniformity. Finally, we provide a denotational semantics for our systems based on the relational model.
