In this short paper we present a survey of some results concerning the random SAT problems. To elaborate, the Boolean Satisfiability (SAT) Problem refers to the problem of determining whether a given set of $m$ Boolean constraints over $n$ variables can be simultaneously satisfied, i.e. all evaluate to $1$ under some interpretation of the variables in $\{ 0,1\}$. If we choose the $m$ constraints i.i.d. uniformly at random among the set of disjunctive clauses of length $k$, then the problem is known as the random $k$-SAT problem. It is conjectured that this problem undergoes a structural phase transition; taking $m=\alpha n$ for $\alpha>0$, it is believed that the probability of there existing a satisfying assignment tends in the large $n$ limit to $1$ if $\alpha<\alpha_\mathrm{sat}(k)$, and to $0$ if $\alpha>\alpha_\mathrm{sat}(k)$, for some critical value $\alpha_\mathrm{sat}(k)$ depending on $k$. We review some of the progress made towards proving this and consider similar conjectures and results for the more general case where the clauses are chosen with varying lengths, i.e. for the so-called random mixed SAT problems.

翻译：在这篇短文中，我们综述了与随机SAT问题相关的若干结果。具体而言，布尔可满足性（SAT）问题是指判断一组给定的$n$个变量上的$m$个布尔约束能否同时满足，即所有约束在变量取自$\{0,1\}$的某种解释下均取值为$1$。若这$m$个约束是从长度为$k$的析取子句集合中独立同分布均匀随机选取，则该问题被称为随机$k$-SAT问题。据猜测，该问题存在结构相变：取$m=\alpha n$（$\alpha>0$），当$n$趋于无穷大时，若$\alpha<\alpha_\mathrm{sat}(k)$，则存在满足赋值的概率趋于$1$；若$\alpha>\alpha_\mathrm{sat}(k)$，则此概率趋于$0$，其中临界值$\alpha_\mathrm{sat}(k)$依赖于$k$。我们回顾了针对此猜想的部分证明进展，并探讨了更一般情形（即子句长度可变，即所谓的随机混合SAT问题）中的类似猜想与结果。