We present a sublinear time algorithm that gives random local access to the uniform distribution over satisfying assignments to an arbitrary k-SAT formula Φ, at exponential clause density. Our algorithm provides memory-less query access to variable assignments, such that the output variable assignments consistently emulate a single global satisfying assignment whose law is close to the uniform distribution over satisfying assignments to Φ. Random local access and related models have been studied for a wide variety of natural Gibbs distributions and random graphical processes. Here, we establish feasibility of random local access models for one of the most canonical such sample spaces, the set of satisfying assignments to a k-SAT formula. Our algorithm proceeds by leveraging the local uniformity of the uniform distribution over satisfying assignments to Φ. We randomly partition the variables into two subsets, so that each clause has sufficiently many variables from each set to preserve local uniformity. We then sample some variables by simulating a systematic scan Glauber dynamics backward in time, greedily constructing the necessary intermediate steps. We sample the other variables by first conducting a search for a polylogarithmic-sized local component, which we iteratively grow to identify a small subformula from which we can efficiently sample using the appropriate marginal distribution. This two-pronged approach enables us to sample individual variable assignments without constructing a full solution.

We prove that 1−o(1) fraction of all k-SAT functions on n Boolean variables are unate (i.e., monotone after first negating some variables), for any fixed positive integer k and as n → ∞. This resolves a conjecture by Bollobás, Brightwell, and Leader from 2003.

How many k-SAT functions on n boolean variables are there? What does a typical such function look like? Bollobás, Brightwell, and Leader conjectured that, for each fixed k ≥ 2, the number of k-SAT functions on n variables is , or equivalently: a 1–o(1) fraction of all k-SAT functions are unate, i.e., monotone after negating some variables. They proved a weaker version of the conjecture for k = 2. The conjecture was confirmed for k = 2 by Allen and k = 3 by Ilinca and Kahn. We show that the problem of enumerating k-SAT functions is equivalent to a Turán density problem for partially directed hypergraphs. Our proof uses the hypergraph container method. Furthermore, we confirm the Bollobás–Brightwell–Leader conjecture for k = 4 by solving the corresponding Turán density problem.