Infinite computable version of Lovasz Local Lemma

Lovász Local Lemma (LLL) is a probabilistic tool that allows us to prove the existence of combinatorial objects in the cases when standard probabilistic argument does not work (there are many partly independent conditions). LLL can be also used to prove the consistency of an infinite set of conditions, using standard compactness argument (if an infinite set of conditions is inconsistent, then some finite part of it is inconsistent, too, which contradicts LLL). In this way we show that objects satisfying all the conditions do exist (though the probability of this event equals 0). However, if we are interested in finding a computable solution that satisfies all the constraints, compactness arguments do not work anymore. Moser and Tardos [1] recently gave a nice constructive proof of LLL. Lance Fortnow asked whether one can apply Moser–Tardos technique to prove the existence of a computable solution. We show that this is indeed possible (under almost the same conditions as used in the non-constructive version). 1 Computable LLL: the statement. Let P be a sequence of mutually independent random variables; each of them has a finite range. (In the simplest case Pi are independent random bits.) We consider some family A of forbidden events; each of them depends on a finite set of variables, denoted vbl(A) (for event A). Informally speaking, the classical LLL together with the compactness argument guarantee that if the events are of small probability and each of them is mostly independent with the others, there exists an evaluation for all variables that avoids all the forbidden events. To make the statement exact, we need to introduce some terminology and notation. Two events A and B are disjoint if they do not share variables, i.e., if vbl(A)∩vbl(B) =∅. For every A ∈A let Γ(A) be the open (punctured) neighborhood of A, i.e., the set of all events E ∈A that share variables (are not disjoint) with A, except A itself. Theorem 1 (Infinite version of LLL). Suppose that for every event A ∈ A a rational number x(A) ∈ (0,1) is fixed such that Pr[A]≤ x(A) ∏ E∈Γ(A) (1− x(E)), for all A ∈A. Then there exists an evaluation of variables that avoids all A ∈A. ∗Supported by RFBR 0901-00709a and NAFIT ANR-08-EMER-008 grants.