The Space of Solutions of Coupled XORSAT Formulae

TL;DR

This paper investigates the solution space of coupled random XOR-satisfiability problems, revealing threshold saturation phenomena and characterizing the geometrical structure of solutions in these models.

Contribution

It introduces a coupled XORSAT ensemble and demonstrates the occurrence of threshold saturation, providing new insights into the solution space geometry of these models.

Findings

Threshold saturation occurs in coupled XORSAT models.

The solution space exhibits distinct geometrical transitions.

Characterization of solution space properties in coupled ensembles.

Abstract

The XOR-satisfiability (XORSAT) problem deals with a system of $n$ Boolean variables and $m$ clauses. Each clause is a linear Boolean equation (XOR) of a subset of the variables. A $K$-clause is a clause involving $K$ distinct variables. In the random $K$-XORSAT problem a formula is created by choosing $m$ $K$-clauses uniformly at random from the set of all possible clauses on $n$ variables. The set of solutions of a random formula exhibits various geometrical transitions as the ratio $\frac{m}{n}$ varies. We consider a {\em coupled} $K$-XORSAT ensemble, consisting of a chain of random XORSAT models that are spatially coupled across a finite window along the chain direction. We observe that the threshold saturation phenomenon takes place for this ensemble and we characterize various properties of the space of solutions of such coupled formulae.