Composition limits and separating examples for some boolean function complexity measures

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 ) 2 ). We provide an infinite family of examples for which C ( f ) grows quadratically in C *( f ) (and also bs ( f )) giving optimal separations between these measures. Previously the biggest separation known was C(f) = C*(f)^log _4,5 5 . We also give a family of examples for which C *( f )= Ω ( bs ( f ) 3/2 ). These examples are obtained by composing boolean functions in various ways. Here the composition fog 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 submultiplicative (where measure m is submultiplicative if m ( fog ) ≤ m ( f ) m ( g )) with equality holding under some fairly general conditions. The measure bs ( f ) is qualitatively different: it is not submultiplicative. 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 lim ( f ) to be the limit as k grows of m ( f (k) ) 1/ k , where f ( k ) is the iterated composition of f with itself k -times. For any function f we show that bs lim ( f ) = ( C *) lim ( f ) and characterize s lim ( f ); ( C *) lim ( f ), and C lim ( f ) in terms of the largest eigenvalue of a certain set of 2×2 matrices associated with f .