Composition Limits and Separating Examples for Some Boolean Function Complexity Measures

Abstract

Block sensitivity (bs(f)), certificate complexity (C(f)) and fractional certificate complexity (C*(f)) are three fundamental combinatorial measures of complexity of a boolean function f. It has long been known that bs(f) ≤ C*f ≤ C(f) =O(bs(f) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). We provide an infinite family of examples for which C(f) grows quadratic ally in C*(f) (and also bs(f)) giving optimal separations between these measures. Previously the biggest separation known was C(f)=C*(f) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">log</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4.5</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> . We also give a family of examples for which C*(f)=Ω(bs(f) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> ). These examples are obtained by composing boolean functions in various ways. Here the composition f ο g of f with g is obtained by substituting for each variable of f a copy of g on disjoint sets of variables. To construct and analyse these examples we systematically investigate the behaviour under function composition of these measures and also the sensitivity measure s(f). The measures s(f), C(f) and C*(f) behave nicely under composition: they are sub multiplicative (where measure m is sub multiplicative if m(f ο g) ≤ m(f)m(g)) with equality holding under some fairly general conditions. The measure bs(f) is qualitatively different: it is not sub multiplicative. This qualitative difference was not noticed in the previous literature and we correct some errors that appeared in previous papers. We define the composition limit of a measure m at function f, m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lim</sup> (f) to be the limit as k grows of m(f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k)</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/k</sup> , where f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(k)</sup> is the iterated composition of f with itself k-times. For any function f we show that bs <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lim</sup> (f) = (C*) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lim</sup> (f) and characterize s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lim</sup> (f), (C*) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lim</sup> (f), and C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lim</sup> (f) in terms of the largest eigenvalue of a certain set of 2 × 2 matrices associated with f.