We show that the shrinkage exponent, under random restrictions, of formulae over a nite complete basis B of Boolean functions is strictly greater than 1 if and only if all the functions in B are unate, i.e., monotone increasing or monotone decreasing in each one of their variables. As a consequence, we get non-linear lower bounds on the formula complexity of the parity function over any basis composed only of unate functions.

The P vs NP problem arose from the question of whether exhaustive search is necessary for problems with short verifiable solutions. We do not know if even a slight algorithmic improvement over exhaustive search is universally possible for all NP problems, and to date no major consequences have been derived from the assumption that an improvement exists. We show that there are natural NP and BPP problems for which minor algorithmic improvements over the trivial deterministic simulation already entail lower bounds such as NEXP ̸ ⊆ P/poly and LOGSPACE ̸ = NP. These results are especially interesting given that similar improvements have been found for many other hard problems. Optimistically, one might hope our results suggest a new path to lower bounds; pessimistically, they show that carrying out the seemingly modest program of finding slightly better algorithms for all search problems may be extremely difficult (if not impossible). We also prove unconditional superpolynomial time-space lower bounds for improving on exhaustive search: there is a problem verifiable with k(n) length witnesses in O(n a) time (for some a and some function k(n) ≤ n) that cannot be solved in k(n) c n a+o(1) time and k(n) c n o(1) space, for every c ≥ 1. While such problems can always be solved by exhaustive search in O(2 k(n) n a) time and O(k(n) + n a) space, we can prove a superpolynomial lower bound in the parameter k(n) when space usage is restricted.

We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan [11] and the theory of stable set polytope. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko [13]. Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give integrally matching upper and lower bounds of their formula size. We also show monotone formula size lower bounds of balanced recursive ternary majority functions improved from the quantum adversary bound of Laplante, Lee and Szegedy [15].

We show that the shrinkage exponent, under random restrictions, of formulae over a nite complete basis B of Boolean functions is strictly greater than 1 if and only if all the functions in B are unate, i.e., monotone increasing or monotone decreasing in each one of their variables. As a consequence, we get non-linear lower bounds on the formula complexity of the parity function over any basis composed only of unate functions.