In a recent paper, the $\mathcal L^{\mathcal S}$-calculus has been defined. It is a proof-language for a significant fragment of intuitionistic linear logic. Its main feature is that the linearity properties can be expressed in its syntax, since it has interstitial logical rules whose proof-terms are a sum and a multiplication by scalar. The calculus is parametrized on the structure $\mathcal S$. This structure was originally identified with the field of complex numbers, since the calculus is designed as a quantum lambda calculus. However, in this paper we show that a semiring is enough, and we provide a categorical semantics for this calculus in the category of cancellative semimodules over the given semiring. We prove the semantics to be sound and adequate.
翻译：在最近的一篇论文中, $\ mathcal L ⁇ mathcal S} $- calculus 定义了 $\ mathcal L ⁇ mathcal S} $- calculus 。 它是一个证明性语言, 说明大量直观线性逻辑。 它的主要特征是直观性能可以用其语法表达, 因为其具有相互间的逻辑规则, 其验证性术语是总和和和乘以计算。 计算性在结构 $\ mathcal S$ 上是相容的 。 这个结构最初被确定为复杂数字的领域, 因为微积分是设计成量子的 ambambadad callulus 。 然而, 在本文中, 我们显示一个精度精度的精度已经足够了, 我们为在给定的半模量的半模量体中, 我们为该半模量的微积提供了绝对的精度的精度的精度。 我们证明这些精度是合理和充分的。
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