  The Lov\'{a}sz Local Lemma (LLL) is a keystone principle in probability
theory, guaranteeing the existence of configurations which avoid a collection
$\mathcal B$ of "bad" events which are mostly independent and have low
probability. In its simplest form, it asserts that whenever a bad-event has
probability $p$ and affects at most $d$ other bad-events, and $e p (d+1) < 1$,
then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of
Moser & Tardos (2010) gives randomized algorithms for most constructions based
on the LLL. However, deterministic algorithms have lagged behind. Notably,
prior deterministic LLL algorithms have required stringent conditions on
$\mathcal B$; for example, they have required that events in $\mathcal B$ have
low decision-tree complexity or depend on a small number of variables. For this
reason, they can only be applied to small fraction of the numerous LLL
applications in practice. We describe an alternate deterministic parallel (NC)
algorithm for the LLL, based on a general derandomization method of Sivakumar
(2002) using log-space statistical tests. The only requirement here is that
bad-events should be computable via a finite automaton with $\text{poly}(d)$
states. This covers most LLL applications to graph theory and combinatorics. No
auxiliary information about the bad-events, including any conditional
probability calculations, are required. Additionally, the proof is a
straightforward combination of general derandomization results and high-level
analysis of the Moser-Tardos algorithm. We illustrate with applications to
defective vertex coloring, domatic partition, and independent transversals.
