Strictness of the Collapsible Pushdown Hierarchy

TL;DR

This paper introduces a pumping lemma for collapsible pushdown graph hierarchy levels, demonstrating that higher orders generate strictly more complex graphs and trees, confirming an open conjecture in the field.

Contribution

It provides the first separation examples for levels of collapsible pushdown hierarchies, advancing understanding of their expressive power.

Findings

Established a pumping lemma for each hierarchy level.

Provided the first examples separating hierarchy levels.

Confirmed that higher orders generate more complex structures.

Abstract

We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author's pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples that separate the levels of the collapsible pushdown graph hierarchy and of the collapsible pushdown tree hierarchy, i.e., the hierarchy of trees generated by higher-order recursion schemes. This confirms the open conjecture that higher orders allow one to generate more graphs and more trees.