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 }.
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Blum, A. (2003). Learning a Function of r Relevant Variables. In: Schölkopf, B., Warmuth, M.K. (eds) Learning Theory and Kernel Machines. Lecture Notes in Computer Science(), vol 2777. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45167-9_54
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DOI: https://doi.org/10.1007/978-3-540-45167-9_54
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