Deciding formulas mixing arithmetic and uninterpreted predicates is of practical interest, notably for applications in verification. Some decision procedures consist in building by structural induction an automaton that recognizes the set of models of the formula under analysis, and then testing whether this automaton accepts a non-empty language. A drawback is that universal quantification is usually handled by a reduction to existential quantification and complementation. For logical formalisms in which models are encoded as infinite words, this hinders the practical use of this method due to the difficulty of complementing infinite-word automata. The contribution of this paper is to introduce an algorithm for directly computing the effect of universal first-order quantifiers on automata recognizing sets of models, for formulas involving natural numbers encoded in unary notation. This makes it possible to apply the automata-based approach to obtain implementable decision procedures for various arithmetic theories.

翻译：将算术与未解释谓词结合的公式判定具有实际意义，尤其在验证应用中。某些判定过程通过结构归纳法构建能识别公式模型集的自动机，再检验该自动机是否接受非空语言。其局限性在于全称量词通常通过归约为存在量词与补运算处理。对于模型编码为无限词的逻辑形式系统，由于无限词自动机补运算的困难性，该方法在实际应用中受到制约。本文贡献在于：针对采用一元编码的自然数公式，提出直接计算全称一阶量词对识别模型集的自动机作用的算法。这使得基于自动机的方法可应用于多种算术理论，实现可执行的判定过程。