An exactly solvable random satisfiability problem

TL;DR

This paper introduces a new exactly solvable model for random satisfiability problems, extending hyper-SAT to q-state variables and providing precise analysis of phase transitions without replicas.

Contribution

It presents an extension of the hyper-SAT model to q-state variables, with exact solutions for critical behavior and a novel duality, advancing understanding of satisfiability phase transitions.

Findings

Exact calculation of SAT/UNSAT transition exponents

Introduction of a duality in the model

Analogy with the Random Energy Model

Abstract

We introduce a new model for the generation of random satisfiability problems. It is an extension of the hyper-SAT model of Ricci-Tersenghi, Weigt and Zecchina, which is a variant of the famous K-SAT model: it is extended to q-state variables and relates to a different choice of the statistical ensemble. The model has an exactly solvable statistic: the critical exponents and scaling functions of the SAT/UNSAT transition are calculable at zero temperature, with no need of replicas, also with exact finite-size corrections. We also introduce an exact duality of the model, and show an analogy of thermodynamic properties with the Random Energy Model of disordered spin systems theory. Relations with Error-Correcting Codes are also discussed.