Linear Temporal Logic (LTL) is one of the most commonly used formalisms for representing and reasoning about temporal properties of computations. Its application domains range from formal verification to artificial intelligence. Many real-time extensions of LTL have been proposed over the years, including Timed Propositional Temporal Logic (TPTL), that makes it possible to constrain the temporal ordering of pairs of events as well as the exact time elapsed between them.

The paper focuses on TPTL and Bounded TPTL with Past (

), a bounded variant of TPTL enriched with past operators, which has been recently introduced to formalise a meaningful class of timeline-based planning problems.

allows one to refer to the past while keeping the computational complexity under control: in contrast to the full TPTL with Past (TPTL+P), whose satisfiability problem is non-elementary, the satisfiability problem for

is

-complete.

The paper deals with the satisfiability problem for TPTL and

by providing an original tableau system for each of them that suitably generalises Reynolds' one-pass and tree-shaped tableau for LTL. First, we show how to handle past operators, by devising a one-pass and tree-shaped tableau system for LTL with Past (LTL+P). Then, we adapt it to TPTL and

, providing full proofs of the soundness and completeness of the resulting systems. In particular, completeness is proved by exploiting a novel model-theoretic argument that, compared to the one originally employed for the LTL system, provides a deeper understanding of the crucial role of the prune rule of the system.

中文翻译：

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用于TPTL和TPTL _b + Past的单通和树形表格系统

线性时间逻辑（LTL）是表示和推理计算的时间特性的最常用形式形式之一。它的应用领域从形式验证到人工智能。多年来，人们提出了LTL的许多实时扩展，包括定时命题时间逻辑（TPTL），这使得可以限制事件对的时间顺序以及事件对之间的确切时间。

本文侧重于TPTL和有限 TPTL 与过去（

），这是TPTL的有界变体，它丰富了过去的运算符，最近已引入该规范以正式化一类有意义的基于时间轴的计划问题。

允许人们在控制计算复杂度的同时参考过去：与具有过去的完整TPTL （TPTL + P）相比，后者的可满足性问题不是基本的，而对于

是

-完全的。

本文讨论了TPTL和

通过为它们中的每提供原始画面系统，适当地可以推广Reynolds的一个通和树形画面为LTL。首先，我们展示如何通过使用过去（LTL + P）为LTL设计一遍树形的表格系统来处理过去的运算符。然后，我们将其调整为适用于TPTL和

，提供所产生系统的健全性和完整性的完整证明。特别地，通过利用一种新颖的模型理论论证来证明完整性，与最初用于LTL系统的论证相比，该论证提供了对系统修剪规则的关键作用的更深刻的理解。