Computational effects are commonly modelled by monads, but often a monad can be presented by an algebraic theory of operations and equations. This talk is about monads and algebraic theories for languages for inference, and their connections to semirings and tensors. A basic class of examples of algebraic theories comes from considering the theory of modules for a semiring, e.g. the theory of unnormalized distributions, where the semiring is that of the non-negative real numbers. We propose that an interesting perspective is given by studying theories via semirings, and to this end explore several examples of subtheories of module theories, mostly relating to probability. Our main contribution concerns the commutative combination of effects, as studied by Hyland, Plotkin and Power: we observe that while the semiring tensor does not in general determine the tensor of subtheories of module theories, it still does in several fundamental probabilistic examples.

翻译：计算效应通常通过单子建模，但单子往往可通过运算与方程构成的代数理论来呈现。本报告聚焦于推理语言中的单子与代数理论，及其与半环和张量的联系。代数理论的基础范例源于半环上的模理论——例如非归一化分布理论，其中半环对应非负实数集。我们提出通过半环研究代数理论这一视角具有启发性，并为此探究了若干主要涉及概率论的模理论子理论实例。本文的主要贡献在于效应交换组合问题（遵循Hyland、Plotkin与Power的研究框架）：我们发现，虽然半环张量一般无法确定模理论子理论的张量结构，但在多个基础概率示例中该对应关系仍然成立。