# Learning a Function of r Relevant Variables

• Avrim Blum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2777)

## Abstract

This problem has been around for a while but is one of my favorites. I will state it here in three forms, discuss a number of known results (some easy and some more intricate), and finally end with small financial incentives for various kinds of partial progress. This problem appears in various guises in [2, 3, 10]. To begin we need the following standard definition: a boolean function f over {0,1} n has (at most) r relevant variables if there exist r indices i 1, ..., i r such that $$f(x) = g(x_{i_1}, \ldots, x_{i_r})$$ for some boolean function g over {0,1} r . In other words, the value of f is determined by only a subset of r of its n input variables. For instance, the function $$f(x) = x_1\bar{x}_2 \vee x_2\bar{x}_5 \vee x_5\bar{x}_1$$ has three relevant variables. The “class of boolean functions with r relevant variables” is the set of all such functions, over all possible g and sets {i 1, ..., i r }.

## Keywords

Boolean Function Relevant Variable Target Function Polynomial Time Algorithm Statistical Query
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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