arXiv subjects

Yuki Nishimuta

Publications and source records attributed to Yuki Nishimuta.

3 recordsLinked to original sources

arXiv2026

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

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.

math.LO↗

arXiv2026

Deducibility of Identicals, Reflection Principle and Synthetic Connectives

Sambin et al. (2000) introduced Basic Logic as a uniform framework for various logics. At the same time, they also introduced the principle of reflection as a criterion for being a connective in Basic Logic. In this paper, we make explicit the relationship between Hacking's deducibility of identicals condition (Hacking, 1979) and the principle of reflection by proving their equivalence. Moreover, despite Sambin et al.'s conjecture that only six connectives satisfy the principle of reflection, we show that a logical connective satisfies the principle of reflection if and only if it is Girard's synthetic connective.

math.LO↗

arXiv2026

Three Topics in Non-decomposability of Generalized Multiplicative Connectives

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.

math.LO↗