Centre for Discrete and Applicable Mathematics |
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CDAM Research Report, LSE-CDAM-2001-08October 2001 |
Béla Bollobás, Graham R. Brightwell, and Imre Leader
Abstract
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
,
the number of 2-SAT
formulae. There is also an obvious
lower bound of
,
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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