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- W3184611673 abstract "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 $k ge 2$, the number of $k$-SAT functions on $n$ variables is $(1+o(1))2^{binom{n}{k} + n}$, 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 Furedi 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." @default.
- W3184611673 created "2021-08-02" @default.
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- W3184611673 date "2021-07-19" @default.
- W3184611673 modified "2023-09-25" @default.
- W3184611673 title "Enumerating k-SAT functions" @default.
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- W3184611673 doi "https://doi.org/10.48550/arxiv.2107.09233" @default.
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