Three Topics in Non-decomposability of Generalized Multiplicative Connectives
Yuki Nishimuta
TL;DR
This work investigates the robustness of Danos and Regnier's non-decomposability for generalized multiplicative connectives across linear-logic fragments. It shows that in the intuitionistic fragment $ ext{$\mathsf{IMLL}$}$ all generalized connectives are decomposable, while in $ ext{$\mathsf{MALL}$}$ and $ ext{$\mathsf{MELL}$}$ non-decomposability collapses to decomposability due to additive/exponential structure. However, in the multiplicative-exponential fragment $ ext{$\mathsf{EMLL}$}$ non-decomposability is preserved, indicating a sharp boundary set by modal resources. The results, obtained via polarized partitions, intuitionistic meeting graphs, and constructive decompositions, clarify how polarity and resource modalities affect the proof-theoretic reach of generalized connectives and identify $ ext{$\mathsf{EMLL}$}$ as a natural extension where non-decomposability remains intact.
Abstract
Danos and Regnier introduced generalized (non-binary) multiplicative connectives in Danos and Regnier [2]. They showed that there exist generalized multiplicative connectives that cannot be defined by any combination of the tensor and par rules in the multiplicative fragment of linear logic. Such connectives are called non-decomposable generalized multiplicative connectives [2, p.192]. The non-decomposability of logical connectives can be regarded as a proof-theoretic and syntactic counterpart of functional completeness for cut-free proofs. In this short note, we investigate Danos and Regnier's notion of non-decomposability and present three results concerning the (non-)decomposability of generalized multiplicative connectives.