Israel Journal of Mathematics

, Volume 133, Issue 1, pp 45–60 | Cite as

The number of 2-sat functions

  • Béla Bollobás
  • Graham R. Brightwell
  • Imre Leader


Our aim in this paper is to address the following question: of the 22 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 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+0(1)) n 2 2.


Partial Order Boolean Function Colour Graph Satisfying Assignment Monotone Formula 
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Copyright information

© Hebrew University 2003

Authors and Affiliations

  • Béla Bollobás
    • 1
    • 2
  • Graham R. Brightwell
    • 3
  • Imre Leader
    • 4
  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  2. 2.Trinity CollegeCambridgeU.K.
  3. 3.Department of MathematicsLondon School of EconomicsLondonU.K.
  4. 4.Trinity CollegeCambridgeU.K.

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