Bounds for Hodes-Specker theorem

In [2] Hodes and Specker proved a theorem which implies that certain Boolean functions have nonlinear formula size complexity. I shall prove that the asymptotic bound for the theorem is n. log log n. § O. Introduction Let f be a Boolean function, i.e. f:[0,1} n-~ {0, I} for some positive integer n. The variables of f will be denoted usually by Xl, . . . . . . . ,x n. Given 1 L il<i2< <irOn, then flxll ...... xl2 Xlr denotes the function from {0,I~ r into t0, i} obtained by substituting O's for all the variables of f different from Xil ... Xir. In § 5 we shall need substitutions containing also l's; in such a case we add a superscript to the bar. A base is an arbitrary finite set of Boolean functions /I ; the elements of~'~ are called connectives. The (formula size) complexity of f in base .cA is the number L n(f) equal to the minimum of the total number of occurrences of variables in an expression over~'~ equivalent to f , (or co if such an expression does not exist). The theorem of Hodes and Specker [~ can be expressed as follows: If fl is the base of all binary connectives, (or equivalently /~ = (O,I,A,~))), then there exists a function S(r,n) such that for e v e r y r lim S (r,n) ~o ,