Andrea Montanari

Linear equations with Boolean variables

Solving a system of linear equations over a finite field F is arguably one of the most fundamental operations in mathematics. Several algorithms have been devised to accomplish such a task in polynomial time. The best known is Gauss elimination, that has O(N 3) complexity (here N is number of variables in the linear system, and we assume the number of equations to be M = Θ(N)). As a matter of fact, one can improve over Gaussian elimination, and the best existing algorithm for general systems has complexity O(N 2.376...). Faster methods do also exist for special classes of instances. The set of solutions of a linear system is an affine subspace of F N. Despite this apparent simplicity, the geometry of affine or linear subspaces of F N can be surprisingly rich. This observation is systematically exploited in coding theory. Linear codes are just linear spaces over finite fields. Nevertheless, they are known to achieve Shannon capacity on memoryless symmetric channels, and their structure is far from trivial, as we already saw in Ch. 11. From a different point of view, linear systems are a particular example of constraint satisfaction problems. We can associate with a linear system a decision problem (establishing whether it has a solution), a counting problem (counting the number of solutions), an optimization problem (minimize the number of violated equations). While the first two are polynomial, the latter is known to be NP-hard. In this chapter we consider a specific ensemble of random linear systems over Z 2 (the field of integers modulo 2), and discuss the structure of its set of solutions. The ensemble definition is mainly motivated by its analogy with other random constraint satisfaction problems, which also explains the name XOR-satisfiability (XORSAT). In the next section we provide the precise definition of the XORSAT ensemble and recall a few elementary properties of linear algebra. We also introduce one of the main objects of study of this chapter: the SAT-UNSAT threshold. Section 18.2 takes a detour into the properties of belief propagation for XORSAT. These are shown to be related to the correlation structure of the uniform measure over solutions and, in Sec. 18.3, to the appearance of a 2-core in the associated factor graph. Sections 18.4 and 18.5 build on these results to compute the SAT-UNSAT threshold and characterize the structure of the solution space. While many results can be derived rigorously, …