We prove the existence of two thresholds regarding the compilability of random 2-CNF formulas to OBDDs. The formulas are drawn from $\mathcal{F}_2(n,δn)$, the uniform distribution over all 2-CNFs with $δn$ clauses and $n$ variables, with $δ\geq 0$ a constant. We show that, with high probability, the random 2-CNF admits OBDDs of size polynomial in $n$ if $0 \leq δ< 1/2$ or if $δ> 1$. On the other hand, for $1/2 < δ< 1$, with high probability, the random $2$-CNF admits only OBDDs of size exponential in $n$. It is no coincidence that the two ``compilability thresholds'' are $δ= 1/2$ and $δ= 1$. Both are known thresholds for other CNF properties, namely, $δ= 1$ is the satisfiability threshold for 2-CNF while $δ= 1/2$ is the treewidth threshold, i.e., the point where the treewidth of the primal graph jumps from constant to linear in $n$ with high probability.

翻译：我们证明了随机2-CNF公式到OBDD可编译性的两个阈值存在性。公式取自均匀分布$\mathcal{F}_2(n,δn)$，该分布涵盖所有具有$δn$个子句和$n$个变量的2-CNF，其中$δ\geq 0$为常数。研究表明：当$0 \leq δ< 1/2$或$δ> 1$时，随机2-CNF以高概率可被大小为$n$多项式的OBDD表示；而当$1/2 < δ< 1$时，随机2-CNF以高概率仅能被大小为$n$指数的OBDD表示。两个“可编译性阈值”$δ= 1/2$与$δ= 1$并非偶然——它们均为已知的其他CNF性质阈值：$δ= 1$是2-CNF的可满足性阈值，而$δ= 1/2$是树宽阈值，即在此点原始图的树宽以高概率从常数跃升至$n$的线性阶。