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：摘要：刻画定理在模型理论中具有重要工具性作用，可用于评估和比较用于规范与验证形式化方法性质的时态语言的表达力。尽管线性时态情形下时态逻辑、谓词逻辑、代数模型与自动机之间已建立起完整联系，但分支时态情形仍存在显著的结构碎片化。本研究通过识别两类被证明等价于这些逻辑的犹豫树自动机变体，为在任意分支树上解释的重要分支时态逻辑（即CTL*与ECTL*）提供了自动机理论刻画。该刻画同样适用于单子路径逻辑以及树结构上双模拟不变的单子链逻辑片段。这些结果拓展了分支时态情形的刻画图景，并解决了一个存在四十年之久的开放问题。