The birth of the contradictory component in random 2-SAT

TL;DR

This paper analyzes the structure of contradictory components in random 2-SAT formulas during the subcritical phase, revealing their 3-regular kernel structure and relating it to complex components in random graphs.

Contribution

It establishes the 3-regular kernel structure of contradictory components in subcritical 2-SAT and links it to the complex components of random graphs, advancing understanding of phase transition similarities.

Findings

Contradictory components have 3-regular kernels in the subcritical phase.

The structure of the spine and distribution of contradictory variables are characterized.

A technique for full asymptotic expansion of satisfiability in the subcritical phase is described.

Abstract

We prove that, with high probability, the contradictory components of a random 2-SAT formula in the subcritical phase of the phase transition have only 3-regular kernels. This follows from the relation between these kernels and the complex component of a random graph in the subcritical phase. This partly settles the question about the structural similarity between the phase transitions in 2-SAT and random graphs. As a byproduct, we describe the technique that allows to obtain a full asymptotic expansion of the satisfiability in the subcritical phase. We also obtain the distribution of the number of contradictory variables and the structure of the spine in the subcritical phase.