Papers › Enumerating k-SAT functions
Enumerating k-SAT functions
Dingding Dong, Nitya Mani, Yufei Zhao
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How many k-SAT functions on n boolean variables are there? What does a typical such function look like? Bollob\'as, Brightwell, and Leader conjectured that, for each fixed k ≥2, the number of k-SAT functions on n variables is (1+o(1))2^(nk + n), or equivalently: a 1-o(1) fraction of all k-SAT functions are unate, i.e., monotone after negating some variables. They proved a weaker version of the conjecture for k=2. The conjecture was confirmed for k=2 by Allen and k=3 by Ilinca and Kahn. We show that the problem of enumerating k-SAT functions is equivalent to a Tur\'an density problem for partially directed hypergraphs. Our proof uses the hypergraph container method. Furthermore, we confirm the Bollob\'as--Brightwell--Leader conjecture for k=4 by solving the corresponding Tur\'an density problem. Our solution applies a recent result of F\"uredi and Maleki on the minimum triangular edge density in a graph of given edge density. In an appendix (by Nitya Mani and Edward Yu), we further confirm the k=5 case of the conjecture via a brute force computer search.
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