论文标题
线性逻辑的线性代数模型作为Sigma-Semirings上的模块类别
Linear-Algebraic Models of Linear Logic as Categories of Modules over Sigma-Semirings
论文作者
论文摘要
许多线性逻辑模型基于或与线性代数密切相关，因为在适当的系数集上，形态为“矩阵”。示例包括基于相干空间，有限空间和概率相干空间以及关系和加权关系模型的模型。本文介绍了基于模块理论的统一框架，使上述模型的线性代数方面更加明确。具体来说，我们考虑了Sigma-Semirings $ R $的模块，该模块是具有部分定义可计数和的类似环的结构，并表明上述模型中的形态实际上是标准代数的$ R $ linear Maps，对于适当的$ r $。我们代数处理的一个优点是，$ r $ - 模型的类别在本地可以呈现，从中可以轻松地遵循该类别成为直觉线性逻辑的模型，并以COFREE指数为单位。然后，我们讨论古典模型的结构，并表明上述模型是我们结构的示例。
A number of models of linear logic are based on or closely related to linear algebra, in the sense that morphisms are "matrices" over appropriate coefficient sets. Examples include models based on coherence spaces, finiteness spaces and probabilistic coherence spaces, as well as the relational and weighted relational models. This paper introduces a unified framework based on module theory, making the linear algebraic aspect of the above models more explicit. Specifically we consider modules over Sigma-semirings $R$, which are ring-like structures with partially-defined countable sums, and show that morphisms in the above models are actually $R$-linear maps in the standard algebraic sense for appropriate $R$. An advantage of our algebraic treatment is that the category of $R$-modules is locally presentable, from which it easily follows that this category becomes a model of intuitionistic linear logic with the cofree exponential. We then discuss constructions of classical models and show that the above-mentioned models are examples of our constructions.
