Free resolutions of function classes via order complexes

TL;DR

This paper explores algebraic structures of function classes in theoretical computer science, using order complexes to construct free resolutions and analyze properties like Betti numbers and VC dimension bounds.

Contribution

It introduces a method to describe cellular free resolutions of ideals associated with intersection-closed function classes using order complexes of posets, extending to matroids and Cohen-Macaulay posets.

Findings

Betti numbers are pure and combinatorially determined by M"obius functions.

Provides bounds on VC dimension for certain function classes.

Connects algebraic properties with learning theory metrics.

Abstract

Function classes are collections of Boolean functions on a finite set, which are fundamental objects of study in theoretical computer science. We study algebraic properties of ideals associated to function classes previously defined by the third author. We consider the broad family of intersection-closed function classes, and describe cellular free resolutions of their ideals by order complexes of the associated posets. For function classes arising from matroids, polyhedral cell complexes, and more generally interval Cohen-Macaulay posets, we show that the multigraded Betti numbers are pure, and are given combinatorially by the M\"obius functions. We then apply our methods to derive bounds on the VC dimension of some important families of function classes in learning theory.