Property testing and expansion in cubical complexes

TL;DR

This paper introduces a general method for property testing in cubical complexes using incidence geometry, enabling efficient testing of functions based on cohomology group descriptions, with applications demonstrated in 2-cell function testing.

Contribution

It develops a novel approach connecting cohomology group descriptions to property testing in cubical complexes, applicable in any dimension.

Findings

Established a method for property testing in cubical complexes.

Demonstrated the method on functions induced from edges in 2-cells.

Provided a linear error ratio in testing procedures.

Abstract

We consider expansion and property testing in the language of incidence geometry, covering both simplicial and cubical complexes in any dimension. We develop a general method for passing from an explicit description of the cohomology group, which need not be trivial, to a testability proof with linear ratio between errors. The method is demonstrated by testing functions on 2-cells in cubical complexes to be induced from the edges.