The number of 2-SAT functions


Our aim in this paper is to address the following question: of the 22(n) Boolean functions on n 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 the n-dimensional discrete cube that are unions of (n - 2)-dimensional subcubes. There is a trivial upper bound of 24(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 + ο(1))n(2)/2.

Publication Title

Israel Journal of Mathematics