Characterizing Non-Deterministic Circuit Size (Four variations on one theme)

Consider the following simple communication problem. Fix a universe U and a family Ω of subsets of U . Players I and II receive, respectively, an element a ∈ U and a subset A ∈ Ω. Their task is 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 Ωf of subsets of U = f−1(0), and prove that its (one round) communication complexity 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 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. ∗Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139. Partially supported by NSF grant CCR-9212184. †Department of Computer Science, Hebrew University, Jerusalem, Israel 91904.