论文标题
具有归纳定义的分离逻辑中可决定的范围:超越已建立的系统
Decidable Entailments in Separation Logic with Inductive Definitions: Beyond Established Systems
论文作者
论文摘要
我们定义了一类分离逻辑公式,其需要的问题:给定公式$ ϕ,ψ_1,\ ldots,ψ_n$,是否$ $ ϕ $的每种型号都是$ψ_i$的模型?是2时组完。该类别的公式是对涉及谓词原子的结合进行了定量的,这些结合由满足一组归纳规则的最小储物堆层结构的解释,这也是累积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的变量最终都必须绑定到分配的位置,即从堆的域中绑定。特别是,这可以确保每个结构的树宽度受集合中最大规则的大小界定。相比之下,我们在这里表明,虽然足以实现可定义性(与另外两个自然条件一起),但这是不必通过提供一种称为方程限制的条件来进行的,该条件适用于句法,适用于(删除)平等性。在这种情况下,需要问题更为笼统,因为方程式限制的规则定义了富裕的结构类别,即无界的树宽。在本文中,我们表明(1)可以将所有已建立的规则集转换为一个方程式限制的规则,以及(2)在后一种情况下,零件问题是2Exptime-complete,因此与已建立的规则集合的复杂性相匹配。
We define a class of Separation Logic formulae, whose entailment problem: given formulae $ϕ, ψ_1, \ldots, ψ_n$, is every model of $ϕ$ a model of some $ψ_i$? 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.