Conditional Simple Temporal Networks (CSTNs) is a constraint based graph-formalism for conditional temporal planning. Three notions of consistency arise for CSTNs: weak, strong, and dynamic. Dynamic-Consistency (DC) is the most interesting notion, but it is also the most challenging. In order to address the DC-Checking problem, Comin and Rizzi [12] introduced ε-DC (a refined, more realistic, notion of DC), providing an algorithmic solution to it. Next, given that DC implies ε-DC for some sufficiently small $\epsilon >0$, and that for every $\epsilon >0$ it holds that ε-DC implies DC, it was offered a sharp lower bounding analysis on the critical value of the reaction-time $\hat{\epsilon }$ under which the two notions coincide. This delivered the first (pseudo) singly-exponential time algorithm for the DC-Checking of CSTNs. However, the ε-DC notion is interesting per se, and the ε-DC-Checking algorithm of Comin and Rizzi [12] rests on the assumption that the reaction-time satisfies $\epsilon >0$; leaving unsolved the question of what happens when $\epsilon =0$. In this work, we introduce and study π-DC, a sound notion of DC with an instantaneous reaction-time (i.e., one in which the planner can react to any observation at the same instant of time in which the observation is made). Firstly, we demonstrate by a counter-example that π-DC is not equivalent to 0-DC, and that 0-DC is actually inadequate for modeling DC with an instantaneous reaction-time. This shows that the main results obtained in our previous work do not apply directly, as they were formulated, to the case of $\epsilon =0$. Motivated by this observation, as a second contribution, our previous tools are extended in order to handle π-DC, and the notion of ps-tree is introduced, also pointing out a relationship between π-DC and consistency of Hyper Temporal Networks. Thirdly, a simple reduction from π-DC to (classical) DC is identified. This allows us to design and to analyze the first sound-and-complete π-DC-Checking algorithm. Remarkably, the time complexity of the proposed algorithm remains (pseudo) singly-exponential in the number of propositional letters. Finally, it is observed that the technique can be leveraged to actually reduce from π-DC to 1-DC, this allows us to further improve the exponents in the time complexity.

条件简单时态网络（CSTN）是用于条件时态规划的基于约束的图形式主义。CSTN出现三个一致性概念：弱，强和动态。动态一致性（DC）是最有趣的概念，但也是最具挑战性的。为了解决DC-Checking问题，Comin和Rizzi [12]引入了ε- DC（一种精致，更现实的DC概念），为它提供了一种算法解决方案。接下来，假设DC对于某些足够小的隐含ε- DC$\epsilon >0$，而且每个 $\epsilon >0$它认为ε- DC表示DC，因此对反应时间的临界值进行了尖锐的下界分析 $\hat{\epsilon }$这两个概念是一致的。这为CSTN的DC检查提供了第一个（伪）单指数时间算法。然而，ε- DC概念本身很有趣，Comin和Rizzi [12]的ε -DC-Checking算法基于反应时间满足的假设。$\epsilon >0$; 解决了什么时候会发生的问题$\epsilon =0$。在这项工作中，我们引入并研究π - DC，它是具有瞬时反应时间的DC的合理概念（即，计划者可以在进行观测的同一时刻对任何观测做出反应）。首先，我们通过一个反例证明π -DC不等于0-DC，并且0-DC实际上不足以对具有瞬时反应时间的DC进行建模。这表明，我们先前工作中获得的主要结果并不直接适用于以下情况：$\epsilon =0$。出于这种观察的动机，作为第二个贡献，我们扩展了先前的工具以处理π- DC，并引入了ps-tree的概念，还指出了π- DC与Hyper Temporal Networks一致性之间的关系。第三，确定了从π- DC到（经典）DC的简单还原。这使我们能够设计和分析第一个声音和完整的π -DC-Checking算法。值得注意的是，所提出算法的时间复杂度在命题字母的数量上保持（伪）单指数。最后，观察到可以利用该技术从π实际减少-DC到1-DC，这使我们可以进一步提高时间复杂度的指数。