The number of 2-sat functions

Our aim in this paper is to address the following question: of the 2^{2 n} Boolean functions onn variables, how many are expressible as 2-SAT formulae? In other words, we wish to count the number of different instances of 2-SAT, counting two instances as equivalent if they have the same set of satisfying assignments. Viewed geometrically, we are asking for the number of subsets of then-dimensional discrete cube that are unions of (n-2)-dimensional subcubes.There is a trivial upper bound of 2^4(n/2), the number of 2-SAT formulae. There is also an obvious lower bound of 2^(n/2), corresponding to the monotone 2-SAT formulae. Our main result is that, rather surprisingly, this lower bound gives the correct speed: the number of 2-SAT functions is 2^(1+0(1)) n ^{2 2}.