Separation logic (SL) is a logical formalism for reasoning about programs that use pointers to mutate data structures. It is successful for program verification as an assertion language to state properties about memory heaps using Hoare triples. Most of the proof systems and verification tools for ${\textrm{SL}}$ focus on the decidable but rather restricted symbolic heaps fragment. Moreover, recent proof systems that go beyond symbolic heaps are purely syntactic or labelled systems dedicated to some fragments of ${\textrm{SL}}$ and they mainly allow either the full set of connectives, or the definition of arbitrary inductive predicates, but not both. In this work, we present a labelled proof system, called ${\textrm{G}_{\textrm{SL}}}$, that allows both the definition of cyclic proofs with arbitrary inductive predicates and the full set of SL connectives. We prove its soundness and show that we can derive in ${\textrm{G}_{\textrm{SL}}}$ the built-in rules for data structures of another non-cyclic labelled proof system and also that ${\textrm{G}_{\textrm{SL}}}$ is strictly more powerful than that system.

中文翻译：

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分离逻辑的标记循环证明

分离逻辑 (SL) 是一种逻辑形式，用于推理使用指针来改变数据结构的程序。程序验证作为一种断言语言使用 Hoare 三元组来声明有关内存堆的属性是成功的。${\textrm{SL}}$ 的大多数证明系统和验证工具都集中在可判定但相当受限的符号堆片段上。此外，最近超越符号堆的证明系统是纯粹的语法或标记系统，专用于 ${\textrm{SL}}$ 的某些片段，它们主要允许完整的连接词集或任意归纳谓词的定义，但是不是都。在这项工作中，我们提出了一个标记证明系统，称为 ${\textrm{G}_{\textrm{SL}}}$，这允许使用任意归纳谓词定义循环证明和完整的 SL 连接词集。我们证明了它的合理性并表明我们可以在 ${\textrm{G}_{\textrm{SL}}}$ 中推导出另一个非循环标记证明系统的数据结构的内置规则，并且 ${\ textrm{G}_{\textrm{SL}}}$ 比那个系统更强大。