arXiv 2107.092332021-07math.CO

枚举k-SAT函数

Enumerating k-SAT functions

Dingding Dong, Nitya Mani, Yufei Zhao

备注：: 50 pages incl. 3 page appendix. Conference version appeared in SODA '22

摘要：

$n$布尔变量上有多少个$k$ -SAT函数？一个典型的这样的函数是什么样子的？Bollobás、Brightwell和Leader指出，对于每个固定的$k \ge 2$，$n$变量上的$k$-SAT函数的数量是$(1+o(1))2^{\binom{n}{k} + n}$，或者等价地：所有$k$ -SAT函数中的$1-o(1)$部分是unate，即，在否定一些变量后单调。他们证明了$k=2$猜想的一个较弱版本。Allen对$k=2$和Ilinca和Kahn对$k=3$的研究证实了这一猜想。证明了部分有向超图的$k$ -SAT函数的计数问题等价于图的密度问题.我们的证明使用超图容器方法。此外，通过求解相应的Turán密度问题，我们证明了关于$k=4$的Bollobás-Brightwell-Leader猜想.我们的解决方案适用于最近的结果Füredi和Maleki的最小三角形的边密度在一个图的给定的边缘密度。在附录（Nitya Mani和Edward Yu）中，我们通过强力计算机搜索进一步证实了猜想的$k=5$情况。

英文摘要：

How many $k$-SAT functions on $n$ boolean variables are there? What does a typical such function look like? Bollobás, 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 Turán density problem for partially directed hypergraphs. Our proof uses the hypergraph container method. Furthermore, we confirm the Bollobás--Brightwell--Leader conjecture for $k=4$ by solving the corresponding Turán density problem. Our solution applies a recent result of Füredi 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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