The Star Height Hierarchy Vs. The Variable Hierarchy

TL;DR

This paper compares the star height and variable hierarchies in logic, proving that the variable hierarchy refines the star height hierarchy and that their non-collapse properties are linked.

Contribution

It establishes that the variable hierarchy is a proper refinement of the star height hierarchy, connecting their non-collapse behaviors through combinatorial characterizations.

Findings

Variable hierarchy refines star height hierarchy

Non-collapse of variable hierarchy implies non-collapse of star height hierarchy

Hierarchies are characterized combinatorially

Abstract

The star height hierarchy (resp. the variable hierarchy) results in classifying $\mu$-terms into classes according to the nested depth of fixed point operators (resp. to the number of bound variables). We prove, under some assumptions, that the variable hierarchy is a proper refinement of the star height hierarchy. We mean that the non collapse of the variable hierarchy implies the non collapse of the star height hierarchy. The proof relies on the combinatorial characterization of the two hierarchies.