Properties and applications of boolean function composition

Abstract

For Boolean functions f:{0,1}n -> {0,1} and g:{0,1}m -> {0,1}, the function composition of f and g denoted by f O g : {0,1}nm -> {0,1} is the value of f on n inputs, each of them is the calculation of g on a distinct set of m Boolean variables. Motivated by previous works that achieved some of the best separations between complexity measures such as sensitivity, block-sensitivity, degree, certificate complexity and decision tree complexity we show that most of these complexity measures behave multiplicatively under composition. We use this multiplicative behavior to establish several applications. First, we give a negative answer for Adam Kalai's question from [MOS04]: "Is it true that every Boolean function f:{0,1}n -> {0,1} with degree as a polynomial over the reals (denoted by deg(f)) at most n/3, has a restriction fixing 2n/3 of its variables under which f becomes a parity function?" This question was motivated by the problem of learning juntas as it suggests a simple algorithm, faster than that of Mossel et al. We give a counterexample for the question using composition of functions strongly related to the Walsh-Hadamard code. In fact, we show that for every constants ε,δ>0 there are (infinitely many) Boolean functions f: {0,1}n -> {0,1} such that deg(f) ≤ ε ⋅ n and under any restriction fixing less than (1-δ) ⋅ n variables, f does not become a parity function.