Characterizing non-deterministic circuit size

Consider the following simple communication problem. Fix a universe U and a family Q of subsets of U. Players I and II receive, respectively, an element a E U and a subset A E Q. Their task is to find a subset B of U such that 1A fl B/ is even and a E B. With every Boolean function f we associate a collection fl~ of subsets of U = f'1 (0), and prove that the (one round) communication complexity of the problem it defines completely determines the szze of the smallest nondeterministic circuit for f. We propose a linear algebraic variant to the general approximation method of Razborov, whtch has exponentially smaller description. We use it to derive four different combinatorial problems (like the one above) that characterize non-uniform NP. These are tight, in the sense that they can be used to prove super-linear circuit size lower bounds. Comb~ned with Ra.zborov]s method, they present a purely combinatorial framework in which to study the P vs. NP vs. co – NP question.