Exponentially Handsome Proof Nets and Their Normalization

Abstract

Handsome proof nets were introduced by Retoré as a syntax for multiplicative linear logic. These proof nets are defined by means of cographs (graphs representing formulas) equipped with a vertices partition satisfying simple topological conditions. In this paper we extend this syntax to multiplicative linear logic with units and exponentials. For this purpose we develop a new sound and complete sequent system for the logic, enforcing a stronger notion of proof equivalence with respect to the one usually considered in the literature. We then define combinatorial proofs, a graphical proof system able to capture syntactically the proof equivalence, for the cut-free fragment of the calculus. We conclude the paper by defining the exponentially handsome proof nets as combinatorial proofs with cuts and defining an internal normalization procedure for this syntax.

Towards a Denotational Semantics for Proofs in Constructive Modal Logic

“…Moreover, an additional problem seems to arise for CK which is similar to the well-known "jump-problem" for multiplicative linear logic proof nets with units [19]: permutations of W may re-assign which is introduced by a specific K as in the following example. Winning strategies for linear logic We foresee no difficulties in defining WISs for elementary and light linear logic adapting the techniques used for defining CPs for multiplicative and exponential linear logic in [2].…”

“…In particular, to recover the results of Murawski-Ong for light linear logic [34], it suffices to consider the modalities ! and § as instances of , to define a frame condition simplifying the one of CK-frames (since there are no ), and restrain skew-fibration allowing deep weakening and deep contraction only to !-formulas using techniques similar to the ones in [2].…”

“…The syntax of combinatorial proofs has been introduced to address the problem of proof equivalence for classical logic [23,24]. In the last years this syntax has been extended to modal logics [5], multiplicative linear logic with exponentials [2], relevant logics [4,8], first order logic [25], and intuitionistic propositional logic [39]. Combinatorial proofs allow to represent "syntax-free" proofs, that is, to represent proofs independently from a specific proof system [3,38].…”