Extended Bounded Response LTL with Past (LTLEBR+P) is a safety fragment of Linear Temporal Logic with Past (LTL+P) that has been recently introduced in the context of reactive synthesis. The strength of LTLEBR+P is a fully symbolic compilation of formulas into symbolic deterministic automata. Its syntax is organized in four levels. The first three levels feature (a particular combination of) future temporal modalities, the last one admits only past temporal operators. At the base of such a structuring there are algorithmic motivations: each level corresponds to a step of the algorithm for the automaton construction. The complex syntax of LTLEBR+P made it difficult to precisely characterize its expressive power, and to compare it with other LTL+P safety fragments. In this paper, we first prove that LTLEBR+P is expressively complete with respect to the safety fragment of LTL+P, that is, any safety language definable in LTL+P can be formalized in LTLEBR+P, and vice versa. From this, it follows that LTLEBR+P and Safety-LTL are expressively equivalent. Then, we show that past modalities play an essential role in LTLEBR+P: we prove that the future fragment of LTLEBR+P is strictly less expressive than full LTLEBR+P.

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扩展有界响应 LTL 的表现力

带过去的扩展有界响应 LTL (LTLEBR+P) 是带过去的线性时间逻辑 (LTL+P) 的安全片段，最近在反应合成的背景下引入。LTLEBR+P 的优势在于将公式完全符号化地编译为符号确定性自动机。它的语法分为四个级别。前三个级别以（特定组合）未来的时间模态为特征，最后一个级别只接受过去的时间操作符。在这种结构的基础上有算法动机：每个级别对应于自动机构造算法的一个步骤。LTLEBR+P 的复杂语法使其难以精确表征其表达能力，并将其与其他 LTL+P 安全片段进行比较。在本文中，我们首先证明 LTLEBR+P 就 LTL+P 的安全片段而言是表达完备的，也就是说，任何可在 LTL+P 中定义的安全语言都可以在 LTLEBR+P 中形式化，反之亦然。由此，LTLEBR+P 和 Safety-LTL 在表达上是等价的。然后，我们证明过去的模态在 LTLEBR+P 中起着至关重要的作用：我们证明 LTLEBR+P 的未来片段比完整的 LTLEBR+P 表现力更差。