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.
KeywordsPartial Order Boolean Function Colour Graph Satisfying Assignment Monotone Formula
Unable to display preview. Download preview PDF.
- B. Bollobás,Modern Graph Theorys, Springer-Verlag, Berlin, 1998, xiii+394pp.Google Scholar
- B. Bollobás and G. R. Brightwell,The number of k-SAT functions, submitted for publication. (Available as CDAM Research Report LSE-CDAM-2001-11, London School of Economics.)Google Scholar
- B. Bollobás and A. Thomason,Hereditary and monotone properties of graphs, inThe Mathematics of Paul Erdős (R. L. Graham and J. Nešetřil, eds.), Springer-Verlag, Berlin, 1997, pp. 70–78.Google Scholar
- D. Du, J. Gu and P. M. Pardalos (eds.),Satisfiability Problem: Theory and Applications, Papers from the DIMACS Workshop held at Rutgers University, Piscataway, NJ, March 11–13, 1996, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 35, American Mathematical Society, Providence, RI, 1997, xvi+724pp.Google Scholar
- P. Erdős, D. J. Kleitman and B. L. Rothschild,Asymptotic enumeration of K n-free graphs, Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973) Tomo II, 19–27. Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Rome, 1976.Google Scholar
- S. Janson, T. Łuczak and A. Ruciński,Random Graphs, Wiley Interscience Series in Discrete Mathematics and Optimization, 2000, xi+333pp.Google Scholar
- U. Martin, personal communication.Google Scholar