Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2001-08

October 2001

The number of 2-SAT functions

Béla Bollobás, Graham R. Brightwell, and Imre Leader


Our aim in this paper is to address the following question: of the 22n 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 $2^{4{n\choose 2}}$, the number of 2-SAT formulae. There is also an obvious lower bound of $2^{n\choose 2}$, 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+o(1))n2/2.

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