This article investigates the satisfiability problem for Separation Logic with k record fields, with unrestricted nesting of separating conjunctions and implications. It focuses on prenex formulæ with a quantifier prefix in the language ∃*∀* that contain uninterpreted (heap-independent) predicate symbols. In analogy with first-order logic, we call this fragment Bernays-Schönfinkel-Ramsey Separation Logic [BSR(SL k )]. In contrast with existing work on Separation Logic, in which the universe of possible locations is assumed to be infinite, we consider both finite and infinite universes in the present article. We show that, unlike in first-order logic, the (in)finite satisfiability problem is undecidable for BSR(SL k ). Then we define two non-trivial subsets thereof, for which the finite and infinite satisfiability problems are PSPACE-complete, respectively, assuming that the maximum arity of the uninterpreted predicate symbols does not depend on the input. These fragments are defined by controlling the polarity of the occurrences of separating implications, as well as the occurrences of universally quantified variables within their scope. These decidability results have natural applications in program verification, as they allow to automatically prove lemmas that occur in, e.g., entailment checking between inductively defined predicates and validity checking of Hoare triples expressing partial correctness conditions.

中文翻译：

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带有未解释谓词的 Bernays-Schönfinkel-Ramsey 类分离逻辑

本文研究了分离逻辑的可满足性问题ķ记录字段，可以不受限制地嵌套分离连词和含义。它侧重于在语言 ∃*∀* 中带有量词前缀的前缀公式，其中包含未解释的（与堆无关的）谓词符号。类比一阶逻辑，我们称这个片段Bernays-Schönfinkel-Ramsey 分离逻辑[BSR(SL ķ )]。与分离逻辑的现有工作（其中假设可能位置的宇宙是无限的）相比，我们在本文中考虑了有限和无限宇宙。我们表明，与一阶逻辑不同，(in) 有限可满足性问题对于 BSR(SL ķ ）。然后我们定义其两个非平凡子集，假设未解释谓词符号的最大元数不依赖于输入，则有限和无限可满足性问题分别是 PSPACE 完全的。这些片段是通过控制分离含义出现的极性以及在其范围内普遍量化的变量的出现来定义的。这些可判定性结果在程序验证中具有自然的应用，因为它们允许自动证明出现在例如归纳定义的谓词之间的蕴涵检查和表示部分正确性条件的 Hoare 三元组的有效性检查中出现的引理。