A Self-Tester for Linear Functions over the Integers with an Elementary Proof of Correctness

TL;DR

This paper introduces a straightforward self-testing algorithm for linear functions over integers, providing elementary correctness proofs and extending the approach to multidimensional vector spaces with query complexity independent of domain size.

Contribution

It offers simple, self-contained proofs for linearity testing algorithms and generalizes self-testing to homomorphisms on vector spaces, with query efficiency.

Findings

Self-testing algorithm for univariate linear functions

Elementary proof of correctness for the algorithms

Generalization to vector space homomorphisms

Abstract

We present simple, self-contained proofs of correctness for algorithms for linearity testing and program checking of linear functions on finite subsets of integers represented as n-bit numbers. In addition we explore a generalization of self-testing to homomorphisms on a multidimensional vector space. We show that our self-testing algorithm for the univariate case can be directly generalized to vector space domains. The number of queries made by our algorithms is independent of domain size.