On the Significance of the Collapse Operation

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