On the Expressive Power of Higher-Order Pushdown Systems

TL;DR

This paper demonstrates that second-order collapsible pushdown automata have greater expressive power than any higher-order pushdown automaton without collapse, revealing new insights into automata capabilities and recursion schemes.

Contribution

It proves the existence of languages recognized by second-order collapsible automata that are not recognizable by any higher-order pushdown automaton without collapse, and introduces a pumping lemma for these automata.

Findings

Second-order collapsible pushdown automata recognize more languages than any higher-order pushdown automaton.

A new pumping lemma for deterministic higher-order pushdown automata is presented.

Some trees generated by second-order collapsible systems cannot be generated by any safe recursion scheme.

Abstract

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.