The Satisfiability Modulo Theories (SMT) issue concerns the satisfiability of formulae from multiple background theories, usually expressed in the language of first-order predicate logic with equality. SMT solvers are often based on variants of the Nelson-Oppen combination method, a solver for the quantifier-free fragment of the combination of theories with disjoint signatures, via cooperation among their decision procedures. When each of the theories to be combined by the Nelson-Oppen method is convex (that is, any conjunction of its literals can imply a disjunction of equalities only when it implies at least one of the equalities) and decidable in polynomial time, the running time of the combination procedure is guaranteed to be polynomial in the size of the input formula. In this paper, we prove the convexity of a fragment of Zermelo-Fraenkel set theory, called Multi-Level Syllogistic, most of whose polynomially decidable fragments we have recently characterized.

中文翻译：

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在 Nelson-Oppen 框架内应用的纯集合论片段的凸性

可满足性模理论 (SMT) 问题涉及来自多个背景理论的公式的可满足性，通常用具有相等性的一阶谓词逻辑语言表达。SMT 求解器通常基于 Nelson-Oppen 组合方法的变体，这是一种求解器，通过决策程序之间的合作，对具有不相交特征的理论组合的无量词片段进行求解。当 Nelson-Oppen 方法要组合的每个理论都是凸的（即，其文字的任何合取只有在它暗示至少一个等式时才能暗示等式的分离）并且在多项式时间内可判定，运行组合过程的时间保证是输入公式大小的多项式。在本文中，