The problem if a given configuration of a pushdown automaton (PDA) is bisimilar with some (unspecified) finite-state process is shown to be decidable. The decidability is proven in the framework of first-order grammars, which are given by finite sets of labelled rules that rewrite roots of first-order terms. The framework is equivalent to PDA where also deterministic (i.e. alternative-free) epsilon-steps are allowed, hence to the model for which Sénizergues showed an involved procedure deciding bisimilarity (1998, 2005). Such a procedure is here used as a black-box part of the algorithm.

The result extends the decidability of the regularity problem for deterministic PDA that was shown by Stearns (1967), and later improved by Valiant (1975) regarding the complexity. The decidability question for nondeterministic PDA, answered positively here, had been open (as indicated, e.g., by Broadbent and Göller, 2012).

如果下推自动机（PDA）的给定配置与某些（未指定）有限状态过程是双相似的，则表明该问题是可以确定的。可判定性在一阶语法的框架中得到了证明，一阶语法由重写一阶项的根的有限规则的标记规则集提供。该框架等效于PDA，其中还允许确定性（即无替代的）ε步长，因此等效于Sénizergues显示了决定双相似性的过程的模型（1998年，2005年）。这样的过程在这里用作算法的黑匣子部分。

结果扩展了Stearns（1967）所显示的确定性PDA的正则性问题的可判定性，后来在复杂性方面被Valiant（1975）进行了改进。对于不确定性PDA的可判定性问题，在这里得到了肯定的回答，这一问题已经公开（例如，Broadbent和Göller，2012年指出）。