A Note on Switching Conditions for the Generalized Logical Connectives in Multiplicative Linear Logic

TL;DR

This paper proposes an alternative n-ary switching condition for multiplicative linear logic proof-structures, demonstrating that it supports the sequentialization theorem but not the expansion property, unlike previous definitions.

Contribution

It introduces a new natural n-ary switching definition that ensures the sequentialization theorem holds, contrasting with prior switchings that did not.

Findings

Sequentialization theorem holds for the new n-ary switching

The expansion property does not hold for the new switching

No n-ary switching satisfies both properties simultaneously except for tensor- or par-based connectives

Abstract

Danos and Regnier (1989) introduced the par-switching condition for multiplicative proof-structures and simplified the sequentialization theorem of Girard (1987) by means of par-switching. Danos and Regnier (1989) also generalized the par-switching to a switching for n-ary connectives (hereafter called an n-ary switching) and showed that the "expansion" property holds, namely that any "excluded-middle" formula admits a correct proof-net in the sense of their n-ary switching. They added a remark that the sequentialization theorem does not hold with their switching. Their definition of switching for n-ary connectives is a natural generalization of the original switching for the binary connectives. However, there are many other possible definitions of switching for n-ary connectives. We give an alternative and "natural" definition of n-ary switching, and we show that the proof of sequentialization theorem by Olivier Laurent with the par-switching works for our n-ary switching; Consequently, the sequentialization theorem holds for our n-ary switching. On the other hand, we remark that the "expansion" property no longer holds under our switching anymore. We point out that no definition of n-ary switching satisfies both the sequentialization theorem and the "expansion" property at the same time except for the purely tensor-based (or purely par-based) connectives.