We introduce the notion of adaptive synchronisation for pushdown automata, in which there is an external observer who has no knowledge about the current state of the pushdown automaton, but can observe the contents of the stack. The observer would then like to decide if it is possible to bring the automaton from any state into some predetermined state by giving inputs to it in an \emph{adaptive} manner, i.e., the next input letter to be given can depend on how the contents of the stack changed after the current input letter. We show that for non-deterministic pushdown automata, this problem is 2-EXPTIME-complete and for deterministic pushdown automata, we show EXPTIME-completeness. To prove the lower bounds, we first introduce (different variants of) subset-synchronisation and show that these problems are polynomial-time equivalent with the adaptive synchronisation problem. We then prove hardness results for the subset-synchronisation problems. For proving the upper bounds, we consider the problem of deciding if a given alternating pushdown system has an accepting run with at most $k$ leaves and we provide an $n^{O(k^2)}$ time algorithm for this problem.

中文翻译：

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下推自动机的自适应同步

我们介绍了下推自动机的自适应同步的概念，其中有一个外部观察者不了解下推自动机的当前状态，但是可以观察堆栈的内容。然后，观察者想决定是否可以通过以\ emph {adaptive}方式给自动机输入信息来使自动机从任何状态进入某个预定状态，即，下一个要输入的字母可以取决于当前输入字母之后，堆栈的内容已更改。我们表明对于非确定性下推自动机，此问题是2-EXPTIME完全；对于确定性下推自动机，我们显示EXPTIME完整性。为了证明下界，我们首先介绍子集同步（的不同变体），并证明这些问题与自适应同步问题是多项式时间等效的。然后，我们证明子集同步问题的硬度结果。为了证明上限，我们考虑以下问题：确定给定的交替下推系统是否具有最多$ k $个叶子的接受运行，并且我们为此问题提供了$ n ^ {O（k ^ 2）} $时间算法。