Proof nets for multiplicative cyclic linear logic and Lambek calculus

Abstract

This paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The main technical achievement of this work is to propose a correctness criterion that allows for sequentialization (recovering a proof from a proof net) for all McyLL proof nets, including those containing cut links. This is achieved by adapting the idea of contractibility (originally introduced by Danos to give a quadratic time procedure for proof nets correctness) to cyclic LL. This paper also gives a characterization of McyLL proof nets for Lambek Calculus and thus a geometrical (i.e., non-inductive) way to parse phrases or sentences by means of Lambek proof nets.

A System of Interaction and Structure III: The Complexity of BV and Pomset Logic

“…(In contrast, pomset logic keeps , commutative while adding the new connective ◁, and the self-duality of ◁ does not permute its arguments.) The standard system with those properties is cyclic 29 linear logic; see [Yet90] for its sequent calculus and proof nets, [DG04] for a deep inference system and [AM19] for pointers to more recent work on cyclic MLL.…”