Concurrent separation logic is distinguished by transfer of state ownership upon parallel composition and framing. The algebraic structure that underpins ownership transfer is that of partial commutative monoids (PCMs). Extant research considers ownership transfer primarily from the logical perspective while comparatively less attention is drawn to the algebraic considerations. This paper provides an algebraic formalization of ownership transfer in concurrent separation logic by means of structure-preserving partial functions (i.e., morphisms) between PCMs, and an associated notion of separating relations. Morphisms of structures are a standard concept in algebra and category theory, but haven't seen ubiquitous use in separation logic before. Separating relations are binary relations that generalize disjointness and characterize the inputs on which morphisms preserve structure. The two abstractions facilitate verification by enabling concise ways of writing specs, by providing abstract views of threads' states that are preserved under ownership transfer, and by enabling user-level construction of new PCMs out of existing ones.

中文翻译：

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关于并发分离逻辑的代数抽象

并发分离逻辑的特点是在并行组合和框架上转移状态所有权。支持所有权转移的代数结构是部分可交换幺半群 (PCM)。现有研究主要从逻辑角度考虑所有权转移，而对代数考虑的关注相对较少。本文通过 PCM 之间保持结构的部分函数（即态射）和相关的分离关系概念，提供了并发分离逻辑中所有权转移的代数形式化。结构的态射是代数和范畴论中的标准概念，但之前在分离逻辑中并没有普遍使用。分离关系是二元关系，可以概括不相交性并表征态射保留结构的输入。这两个抽象通过支持编写规范的简洁方式、提供在所有权转移下保留的线程状态的抽象视图以及从现有的用户级构建新的 PCM 来促进验证。