3 research outputs found
The satisfiability threshold for random linear equations
Let be a random matrix over the finite field with
precisely non-zero entries per row and let be a random vector
chosen independently of . We identify the threshold up to which the
linear system has a solution with high probability and analyse the
geometry of the set of solutions. In the special case , known as the
random -XORSAT problem, the threshold was determined by [Dubois and Mandler
2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof
technique was subsequently extended to the cases [Falke and Goerdt
2012]. But the argument depends on technically demanding second moment
calculations that do not generalise to . Here we approach the problem from
the viewpoint of a decoding task, which leads to a transparent combinatorial
proof
Belief Propagation on the random -SAT model
Corroborating a prediction from statistical physics, we prove that the Belief
Propagation message passing algorithm approximates the partition function of
the random -SAT model well for all clause/variable densities and all inverse
temperatures for which a modest absence of long-range correlations condition is
satisfied. This condition is known as "replica symmetry" in physics language.
From this result we deduce that a replica symmetry breaking phase transition
occurs in the random -SAT model at low temperature for clause/variable
densities below but close to the satisfiability threshold
The number of satisfying assignments of random 2-SAT formulas
We show that throughout the satisfiable phase the normalized number of satisfying assignments of a random 2-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to “true” under a uniformly random satisfying assignment