We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric. We then consider three particular automata models (finite automata, transducers and automata weighted by integers here called sum-automata) and instantiate the generic logic for each of them. We give tight complexity results for the three logics with respect to the model-checking problem, depending on whether the formula is fixed or not. We study the expressiveness of our logics by expressing classical structural patterns characterising for instance finite ambiguity and polynomial ambiguity in the case of finite automata, determinisability and finite-valuedness in the case of transducers and sum-automata. As a consequence of our complexity results, we directly obtain that these classical properties can be decided in polynomial time.

中文翻译：

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带输出的自动机模式逻辑

我们引入了一种逻辑来表达具有字符串输入的自动机的结构属性，并且可能在某些幺半群中输出。在这个逻辑中，讨论输出值的谓词集是参数化的。然后我们考虑三个特定的自动机模型（有限自动机、换能器和由整数加权的自动机，这里称为 sum-automata）并为它们中的每一个实例化通用逻辑。我们给出了关于模型检查问题的三个逻辑的严格复杂性结果，具体取决于公式是否固定。我们通过表达经典结构模式来研究我们的逻辑的表达能力，例如在有限自动机的情况下的有限模糊性和多项式模糊性，在换能器和和自动机的情况下的可确定性和有限值。