The reachability problem for timed automata asks if a given automaton has a run leading to an accepting state, and the liveness problem asks if the automaton has an infinite run that visits accepting states infinitely often. Both of these problems are known to be P space -complete. We show that if P ≠P space , the liveness problem is more difficult than the reachability problem; in other words, we exhibit a family of automata for which solving the reachability problem with the standard algorithm is in P but solving the liveness problem is P space -hard. This leads us to revisit the algorithmics for the liveness problem. We propose a notion of a witness for the fact that a timed automaton violates a liveness property. We give an algorithm for computing such a witness and compare it to existing solutions.

中文翻译：

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为什么定时自动机的活跃度很难，我们能做些什么

这可达性问题for timed automata 询问给定的自动机是否有运行导致接受状态，并且活性问题询问自动机是否有无限运行，无限频繁地访问接受状态。这两个问题都已知为 P空间-完全的。我们证明如果 P ≠P空间，活性问题比可达性问题更难；换句话说，我们展示了一系列自动机，用标准算法解决可达性问题在 P 中，但解决活性问题是 P空间-难的。这导致我们重新审视活性问题的算法。我们提出了一个见证人的概念，因为定时自动机违反了活性属性。我们给出了一种计算这种见证的算法，并将其与现有解决方案进行比较。