The Compilability Thresholds of 2-CNF to OBDD

2026-03-16

AI Summary

This study investigates the conditions under which random 2-CNF formulas admit polynomial-size ordered binary decision diagrams (OBDDs). By integrating probabilistic analysis, graph-theoretic insights—particularly the treewidth of the primal graph—and OBDD complexity theory, the work establishes two sharp phase transitions in compilation complexity: for density parameter δ < 1/2 or δ > 1, a random 2-CNF formula almost surely admits a polynomial-size OBDD, whereas for 1/2 < δ < 1, it almost surely requires exponential-size OBDDs. These findings rigorously connect OBDD compilation complexity to well-known phase transitions in satisfiability and treewidth, thereby filling a critical gap in the theoretical understanding of representational complexity for 2-CNF formulas.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

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.

Problem

• 2-CNF
• OBDD
• compilability
• threshold
• random formulas

Innovation

• compilability thresholds
• 2-CNF
• OBDD
• phase transition
• treewidth