The Entropy of the K-Satisfiability Problem

TL;DR

This paper investigates the entropy and phase transition behavior of the K-Satisfiability problem using statistical mechanics, revealing that the transition is caused by the sudden emergence of contradictions rather than a gradual solution count decline.

Contribution

It provides a novel physical interpretation of the SAT transition, showing entropy remains finite at the critical point and differs across K values.

Findings

Entropy is finite at the transition point.

Transition caused by abrupt contradictions, not solution count decrease.

Different behaviors for K=1, K=2, and K≥3.

Abstract

The threshold behaviour of the K-Satisfiability problem is studied in the framework of the statistical mechanics of random diluted systems. We find that at the transition the entropy is finite and hence that the transition itself is due to the abrupt appearance of logical contradictions in all solutions and not to the progressive decreasing of the number of these solutions down to zero. A physical interpretation is given for the different cases $K=1$, $K=2$ and $K \geq 3$.