In this paper, we revisit Moggi's celebrated calculus of computational effects from the perspective of logic of monoidal action (actegory). Our development takes the following steps. Firstly, we perform proof-theoretic reconstruction of Moggi's computational metalanguage and obtain a type theory with a modal type $\rhd$ as a refinement. Through the proposition-as-type paradigm, its logic can be seen as a decomposition of lax logic via Benton's adjoint calculus. This calculus models as a programming language a weaker version of effects, which we call \emph{semi-effects}. Secondly, we give its semantics using actegories and equivariant functors. Compared to previous studies of effects and actegories, our approach is more general in that models are directly given by equivariant functors, which include Freyd categories (hence strong monads) as a special case. Thirdly, we show that categorical gluing along equivariant functors is possible and derive logical predicates for $\rhd$-modality. We also show that this gluing, under a natural assumption, gives rise to logical predicates that coincide with those derived by Katsumata's categorical $\top\top$-lifting for Moggi's metalanguage.

中文翻译：

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等变函子的计算半效应和分类粘合的逻辑

在本文中，我们从幺半群行动（行为）逻辑的角度重新审视 Moggi 著名的计算效果演算。我们的开发采取以下步骤。首先，我们对 Moggi 的计算元语言进行了证明理论重建，并获得了一个以模态类型 $\rhd$ 作为细化的类型理论。通过命题作为类型范式，其逻辑可以看作是通过本顿伴随演算对松散逻辑的分解。这种微积分将一种较弱的效果建模为编程语言，我们称之为 \emph{semi-effects}。其次，我们使用行为和等变函子给出其语义。与之前对效应和行为的研究相比，我们的方法更通用，因为模型直接由等变函子给出，其中包括 Freyd 类别（因此是强单子）作为特例。第三，我们证明了沿着等变函子的分类粘合是可能的，并推导出 $\rhd$-modality 的逻辑谓词。我们还表明，在自然假设下，这种粘合产生的逻辑谓词与 Katsumata 对 Moggi 元语言的分类 $\top\top$-lifting 派生的逻辑谓词一致。