It cannot be decided whether a pushdown automaton accepts using a pushdown height, which does not depend on the input length, i.e., when it accepts using constant height. Furthermore, when a pushdown automaton accepts in constant height, the height can be arbitrarily large with respect to the size of the description of the machine, namely it does not exist any recursive function in the size of the description of the machine bounding the height of the pushdown. In contrast, in the restricted case of pushdown automata over a one-letter input alphabet, i.e., unary pushdown automata, the situation is different. First, acceptance in constant height is decidable. Moreover, in the case of acceptance in constant height, the height is at most exponential with respect to the size of the description of the pushdown automaton. We also prove a matching lower bound. Finally, if a unary pushdown automaton uses nonconstant height to accept, then the height should grow at least as the logarithm of the input length. This bound is optimal.

无法确定下推自动机是否使用下推高度接受，这不取决于输入长度，即何时接受使用恒定高度。此外，当下推自动机接受恒定高度时，高度可以相对于机器描述的大小任意大，即在机器描述的大小中不存在任何递归函数限制高度下推。相反，在单字母输入字母表上的下推自动机的受限情况下，即一元下推自动机，情况就不同了。首先，恒定高度的接受度是可确定的。此外，在恒定高度接受的情况下，高度相对于下推自动机的描述大小至多是指数的。我们还证明了一个匹配的下界。最后，如果一元下推自动机使用非常量高度来接受，那么高度应该至少以输入长度的对数增长。这个界限是最优的。