In this paper, we investigate the model checking (MC) problem for Halpern and Shoham's modal logic of time intervals (HS) and its fragments, where labeling of intervals is defined by regular expressions. The MC problem for HS has recently emerged as a viable alternative to the traditional (point-based) temporal logic MC. Most expressiveness and complexity results have been obtained by imposing suitable restrictions on interval labeling, namely, by either defining it in terms of interval endpoints, or by constraining a proposition letter to hold over an interval if and only if it holds over each component state (homogeneity assumption). In both cases, the expressiveness of HS gets noticeably limited, in particular when fragments of HS are considered.

A possible way to increase the expressiveness of interval temporal logic MC was proposed by Lomuscio and Michaliszyn, who suggested to use regular expressions to define interval labeling, i.e., the properties that hold true over intervals/computation stretches, based on their component points/system states. In this paper, we provide a systematic account of decidability and complexity issues for model checking HS and its fragments extended with regular expressions. We first prove that MC for (full) HS extended with regular expressions is decidable by an automaton-theoretic argument. Though the exact complexity of full HS MC remains an open issue, the complexity of all relevant proper fragments of HS is here determined. In particular, we provide an asymptotically optimal bound to the complexity of the two syntactically maximal fragments $\mathsf{A}\bar{\mathsf{A}}\mathsf{B}\bar{\mathsf{B}}\bar{\mathsf{E}}$ and $\mathsf{A}\bar{\mathsf{A}}\mathsf{E}\bar{\mathsf{B}}\bar{\mathsf{E}}$, by showing that their MC problem is $\mathbf{AEX}{\mathbf{P}}_{\mathbf{pol}}$-complete ($\mathbf{AEX}{\mathbf{P}}_{\mathbf{pol}}$ is the complexity class of problems decided by exponential-time bounded alternating Turing Machines making a polynomially bounded number of alternations). Moreover, we show that a better result holds for $\mathsf{A}\bar{\mathsf{A}}\mathsf{B}\bar{\mathsf{B}}$, $\mathsf{A}\bar{\mathsf{A}}\mathsf{E}\bar{\mathsf{E}}$ and all their sub-fragments, whose MC problem turns out to be PSPACE-complete.

在本文中，我们研究了Halpern和Shoham的时间间隔（HS）及其片段的模态逻辑的模型检查（MC）问题，其中间隔的标记由正则表达式定义。HS的MC问题最近成为传统（基于点）时间逻辑MC的可行替代方案。通过在间隔标签上施加适当的限制，即通过根据间隔端点来定义它，或者通过（如果且仅当）在每个组件状态上都保持命题字母以保持在一个间隔上，才能获得大多数表现力和复杂性结果。同质性假设）。在这两种情况下，HS的表达能力都受到明显限制，尤其是在考虑了HS片段的情况下。

Lomuscio和Michaliszyn提出了一种提高间隔时间逻辑MC表达能力的可能方法，他们建议使用正则表达式来定义间隔标签，即基于其组成点/系统在间隔/计算范围内均成立的属性状态。在本文中，我们为模型检查HS及其使用正则表达式扩展的片段提供了关于可判定性和复杂性问题的系统说明。我们首先证明，用正则表达式扩展的（完整）HS的MC是由自动机理论论点确定的。尽管完整HS MC的确切复杂度仍然是一个未解决的问题，但是这里确定了HS所有相关适当片段的复杂度。特别是，我们提供了两个语法上最大片段的复杂度的渐近最优约束$\mathsf{一种}\stackrel{〜}{\mathsf{一种}}\mathsf{乙}\stackrel{〜}{\mathsf{乙}}\stackrel{〜}{\mathsf{Ë}}$ 和 $\mathsf{一种}\stackrel{〜}{\mathsf{一种}}\mathsf{Ë}\stackrel{〜}{\mathsf{乙}}\stackrel{〜}{\mathsf{Ë}}$，表明他们的MC问题是 $\mathbf{AEX}{\mathbf{P}}_{\mathbf{波尔}}$-完成（$\mathbf{AEX}{\mathbf{P}}_{\mathbf{波尔}}$是由指数时间有界交替图灵机进行多项式有界数的交替决定的问题的复杂度类别）。而且，我们证明更好的结果对$\mathsf{一种}\stackrel{〜}{\mathsf{一种}}\mathsf{乙}\stackrel{〜}{\mathsf{乙}}$， $\mathsf{一种}\stackrel{〜}{\mathsf{一种}}\mathsf{Ë}\stackrel{〜}{\mathsf{Ë}}$及其所有子片段，其MC问题最终证明是PSPACE -complete。