Enumerating k-SAT functions

Abstract

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 (1 + o(1))2 ( n k )+n , 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.1.2. Strategy. Our setup is closest to the work of Ilinca and Kahn [18] where they enumerated 3-SAT functions.We call such w (not necessarily unique) a witness to C i . 1Example 1.3. The 2-SAT formula {wx, wy, xz, yz} is not minimal since it is impossible to satisfy only wx and no other clause. Indeed, if we attempt to only satisfy wx, we must assign w = 1 and x = 1; to avoid satisfying wy and xz, we must assign y = 0 and z = 1, which would then lead to the final clause yz being satisfied.Every k-SAT function can be expressed as a minimal formula, but possibly more than one. We upper bound the number of minimal k-SAT formulae, which in turn upper bounds the number of k-SAT functions.To bound the number of minimal k-SAT formulae, we identify a fixed set B of non-minimal formulae, and then find an upper bound to the number of (not necessarily minimal) formulae not containing anything from B as a subformulae. We will show that, by setting B to be all non-minimal formulae on a constant number of variables, the number of B-free formulae is asymptotically the same as the number of minimal formulae.Definition 1.4 (Subformula). Given a formula G, a subformula of G is a subset of clauses of G. We say that two formulae are isomorphic if one can be obtained from the other by relabeling variables. Given another formula F , a copy of F in G is a subformula of G that is isomorphic to F . Given a set B of k-SAT formulae, we say that G is B-free if G has no copy of B for every B ∈ B.The problem of counting B-free formulae is analogous to counting F -free graphs on n vertices. A classic result by Erdős, Kleitman, and Rothschild [11] shows that almost all