We propose a novel approach for coping with alternating quantification as the main source of nonelementary complexity of deciding WS1S formulae. Our approach is applicable within the state-of-the-art automata-based WS1S decision procedure implemented e.g. in Mona. The way in which the standard decision procedure processes quantifiers involves determinization, with its worst case exponential complexity, for every quantifier alternation in the prefix of a formula. Our algorithm avoids building the deterministic automata—instead, it constructs only those of their states needed for (dis)proving validity of the formula. It uses a symbolic representation of the states, which have a deeply nested structure stemming from the repeated implicit subset construction, and prunes the search space by a nested subsumption relation, a generalization of the one used by the so-called antichain algorithms for handling nondeterministic automata. We have obtained encouraging experimental results, in some cases outperforming Mona, and some of the other recently proposed approaches, by several orders of magnitude.

中文翻译：

---

WS1S 的嵌套反链

我们提出了一种新方法来处理交替量化作为决定 WS1S 公式的非基本复杂性的主要来源。我们的方法适用于最先进的基于自动机的 WS1S 决策程序，例如在 Mona 中实现的。标准决策程序处理量词的方式涉及确定化，其最坏情况是指数复杂性，对于公式前缀中的每个量词交替。我们的算法避免构建确定性自动机——相反，它只构建那些用于（反）证明公式有效性所需的状态。它使用状态的符号表示，这些状态具有源自重复隐式子集构造的深度嵌套结构，并通过嵌套包含关系修剪搜索空间，所谓的反链算法用于处理非确定性自动机的算法的泛化。我们已经获得了令人鼓舞的实验结果，在某些情况下比 Mona 和其他一些最近提出的方法好几个数量级。