Characterisations theorems serve as important tools in model theory and can be used to assess and compare the expressive power of temporal languages used for the specification and verification of properties in formal methods. While complete connections have been established for the linear-time case between temporal logics, predicate logics, algebraic models, and automata, the situation in the branching-time case remains considerably more fragmented. In this work, we provide an automata-theoretic characterisation of some important branching-time temporal logics, namely CTL* and ECTL* interpreted on arbitrary-branching trees, by identifying two variants of Hesitant Tree Automata that are proved equivalent to those logics. The characterisations also apply to Monadic Path Logic and the bisimulation-invariant fragment of Monadic Chain Logic, again interpreted over trees. These results widen the characterisation landscape of the branching-time case and solve a forty-year-old open question.

翻译：特征刻画定理作为模型论中的重要工具，可用于评估和比较形式化方法中用于性质规约与验证的时态语言的表达能力。虽然线性时间情况下时态逻辑、谓词逻辑、代数模型与自动机之间已建立完整关联，但分支时间情况仍较为碎片化。本研究通过识别两种与这些逻辑等价的犹豫树自动机变体，提供了在任意分支树结构上解释的重要分支时间时态逻辑——即CTL*和ECTL*——的自动机理论刻画。该刻画同样适用于同样在树结构上解释的Monadic Path逻辑及双模拟不变片段Monadic Chain逻辑。这些结果拓展了分支时间情况的刻画图景，并解决了一个存在四十年之久的开放性问题。