Statistical mechanics of the random K-SAT model

TL;DR

This paper analyzes the statistical mechanics of the random K-SAT problem using replica symmetric methods, deriving exact solutions for different K values and identifying phase transitions including a replica symmetry breaking transition for K>=3.

Contribution

It introduces an exact iterative scheme for the replica symmetric order parameter in the K-SAT model, covering various K cases and revealing phase transition behaviors.

Findings

Exact solutions for the number of solutions across K values

Identification of a first order transition at the satisfiability threshold

Proof that the annealed approximation is exact for large K

Abstract

The Random K-Satisfiability Problem, consisting in verifying the existence of an assignment of N Boolean variables that satisfy a set of M=alpha N random logical clauses containing K variables each, is studied using the replica symmetric framework of diluted disordered systems. We present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest K=2, K>= 3 and K>>1. The calculation of the number of solutions, which allowed us [Phys. Rev. Lett. 76, 3881 (1996)] to predict a first order jump at the threshold where the Boolean expressions become unsatisfiable with probability one, is thoroughly displayed. In the case K=2, the (rigorously known) critical value (alpha=1) of the number of clauses per Boolean variable is recovered while for K>=3 we show that the system exhibits a replica symmetry breaking transition. The annealed approximation is proven to be exact for large K.