Characterizing non-deterministic circuit size

Consider the following simple communication problem. Fix a universe V and a family f2 of subsets of U. Players I and II receive, respectively, an element a € U and a subset A € Ω. Their task ts to find a subset B of U such that \A ∩ B\ is even and a € B. With every Boolean function f we associate a collection Qf of subsets of U = /"!(0), and prove that the (one round) communication complexity of the problem it defines completely determines the size of the smallest nondeterministic circuit for f. We propose a linear algebraic variant to the general approximation method of Razborov, which 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. Combined with Razborov's method, they present a purely combinatorial framework in which to study the P vs. NP vs. co - NP question.