We show that deterministic collapsible pushdown automata of second order can recognize a language that is not recognizable by any deterministic higher-order pushdown automaton (without collapse) of any order. This implies that there exists a tree generated by a second order collapsible pushdown system (equivalently, by a recursion scheme of second order) that is not generated by any deterministic higher-order pushdown system (without collapse) of any order (equivalently, by any safe recursion scheme of any order). As a side effect, we present a pumping lemma for deterministic higher-order pushdown automata, which potentially can be useful for other applications.

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论高阶下推系统的表达能力

我们表明，二阶确定性可折叠下推自动机可以识别任何顺序的任何确定性高阶下推自动机（无折叠）都无法识别的语言。这意味着存在一个由二阶可折叠下推系统（等效地，由二阶递归方案）生成的树，该树不是由任何阶的任何确定性高阶下推系统（无折叠）生成的（等效地，由任何任何顺序的安全递归方案）。作为副作用，我们为确定性高阶下推自动机提出了一个泵引理，这可能对其他应用程序有用。