# Almost Optimal Testers for Concise Representations

## TL;DR

This paper presents nearly optimal property testers for a variety of Boolean function classes with concise representations, improving efficiency in uniform and distribution-free models.

## Contribution

It introduces improved testers for multiple Boolean function classes, achieving near-optimal query complexities and extending to functions approximable by few relevant variables.

## Key findings

- Achieves near-optimal testing for k-junta and related classes.
- Extends methods to functions over any domain with few relevant variables.
- Provides efficient algorithms with improved query complexities.

## Abstract

We give improved and almost optimal testers for several classes of Boolean functions on $n$ inputs that have concise representation in the uniform and distribution-free model. Classes, such as $k$-junta, $k$-linear functions, $s$-term DNF, $s$-term monotone DNF, $r$-DNF, decision list, $r$-decision list, size-$s$ decision tree, size-$s$ Boolean formula, size-$s$ branching programs, $s$-sparse polynomials over the binary field and function with Fourier degree at most $d$. The method can be extended to several other classes of functions over any domain that can be approximated by functions that have a small number of relevant variables.