On the number of relevant variables for discrete functions

TL;DR

This paper explores different ways to measure the complexity of discrete functions by examining the relationship between relevant variables and degrees in Boolean and multi-valued functions.

Contribution

It establishes new relations between the number of relevant variables and the degrees of discrete functions, enhancing understanding of their complexity measures.

Findings

Relations between relevant variables and degrees are formalized.

Connections between two-valued and three-valued functions are clarified.

New bounds or characterizations of function complexity are proposed.

Abstract

We consider various definitions of degrees of discrete functions and establish relations between the number of relevant (essential) variables and degrees of two- and three-valued functions. Keywords: relevant variable, sensitivity, degree of Boolean function.