Local Correction of Juntas

TL;DR

This paper studies the local correctability of Boolean functions called juntas, showing that most require only O(k log k) queries for correction, while some need exponentially many, highlighting typical versus worst-case complexities.

Contribution

It establishes that almost all k-juntas are efficiently locally correctable with O(k log k) queries, contrasting with certain worst-case examples requiring exponential queries.

Findings

Most k-juntas are O(k log k)-locally correctable.

Some k-juntas require exponential queries for correction.

Typical juntas are easier to correct than worst-case examples.

Abstract

A Boolean function f over n variables is said to be q-locally correctable if, given a black-box access to a function g which is "close" to an isomorphism f_sigma of f, we can compute f_sigma(x) for any x in Z_2^n with good probability using q queries to g. We observe that any k-junta, that is, any function which depends only on k of its input variables, is O(2^k)-locally correctable. Moreover, we show that there are examples where this is essentially best possible, and locally correcting some k-juntas requires a number of queries which is exponential in k. These examples, however, are far from being typical, and indeed we prove that for almost every k-junta, O(k log k) queries suffice.