We define a class of Separation Logic formulae, whose entailment problem: given formulae $\phi, \psi_1, \ldots, \psi_n$, is every model of $\phi$ a model of some $\psi_i$? is 2EXPTIME-complete. The formulae in this class are existentially quantified separating conjunctions involving predicate atoms, interpreted by the least sets of store-heap structures that satisfy a set of inductive rules, which is also part of the input to the entailment problem. Previous work consider established sets of rules, meaning that every existentially quantified variable in a rule must eventually be bound to an allocated location, i.e. from the domain of the heap. In particular, this guarantees that each structure has treewidth bounded by the size of the largest rule in the set. In contrast, here we show that establishment, although sufficient for decidability (alongside two other natural conditions), is not necessary, by providing a condition, called equational restrictedness, which applies syntactically to (dis-)equalities. The entailment problem is more general in this case, because equationally restricted rules define richer classes of structures, of unbounded treewidth. In this paper we show that (1) every established set of rules can be converted into an equationally restricted one and (2) the entailment problem is 2EXPTIME-complete in the latter case, thus matching the complexity of entailments for established sets of rules.

中文翻译：

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具有归纳定义的分离逻辑中的可判定内涵：超越既定系统

我们定义了一类分离逻辑公式，其蕴涵问题是：给定公式$\phi, \psi_1, \ldots, \psi_n$，$\phi$的每个模型是否都是某个$\psi_i$的模型？已完成 2EXPTIME。此类中的公式是涉及谓词原子的存在量化分离连词，由满足一组归纳规则的最少存储堆结构集解释，这也是蕴涵问题的输入的一部分。以前的工作考虑了已建立的规则集，这意味着规则中的每个存在量化变量最终都必须绑定到分配的位置，即来自堆的域。特别是，这保证了每个结构的树宽受集合中最大规则的大小限制。相比之下，我们在这里展示了建立，尽管对于可判定性（以及其他两个自然条件）来说是足够的，但通过提供一个称为等式限制性的条件，它在句法上适用于（不）等式，这不是必要的。在这种情况下，蕴涵问题更普遍，因为等式限制规则定义了更丰富的结构类，无界树宽。在本文中，我们表明（1）每个已建立的规则集都可以转换为等式限制的规则集，（2）在后一种情况下，蕴涵问题是 2EXPTIME 完全的，从而匹配已建立规则集的蕴涵的复杂性。因为等式限制规则定义了更丰富的结构类别，无限的树宽。在本文中，我们表明（1）每个已建立的规则集都可以转换为等式限制的规则集，（2）在后一种情况下，蕴涵问题是 2EXPTIME 完全的，从而匹配已建立规则集的蕴涵的复杂性。因为等式限制规则定义了更丰富的结构类别，无限的树宽。在本文中，我们表明（1）每个已建立的规则集都可以转换为等式限制的规则集，（2）在后一种情况下，蕴涵问题是 2EXPTIME 完全的，从而匹配已建立规则集的蕴涵的复杂性。