  We prove that if $G$ and $H$ are $\alpha$-back-and-forth equivalent groups
(in the sense of computable structure theory) for some ordinal $\alpha \geq
\omega$, then their group von Neumann algebras $L(G)$ and $L(H)$ are also
$\alpha$-back-and-forth equivalent. In particular, if $G$ and $H$ are
$\omega$-back-and-forth-equivalent groups, then $L(G)$ and $L(H)$ are
elementarily equivalent; this is known to fail under the weaker hypothesis that
$G$ and $H$ are merely elementarily equivalent. We extend this result to
crossed product von Neumann algebras associated to Bernoulli actions of
back-and-forth equivalent groups.
