  Profinite algebras are exactly those that are isomorphic to inverse limits of
finite algebras. Such algebras are naturally equipped with Boolean topologies.
A variety $\mathcal V$ is standard if every Boolean topological algebra with
the algebraic reduct in $\mathcal V$ is profinite.
  We show that there is no algorithm which takes as input a finite algebra
$\mathbf A$ of a finite type and decide whether the variety ${\mathcal
V}({\mathbf A})$ generated by ${\mathbf A}$ is standard. We also show the
undecidability of some related properties. In particular, we solve a problem
posed by Clark, Davey, Freese and Jackson.
  We accomplish this by combining two results. The first one is Moore's result
saying that there is no algorithm which takes as input a finite algebra
$\mathbf A$ of a finite type and decides whether ${\mathcal V}({\mathbf A})$
has definable principal subcongruences. The second is our result saying that
possessing definable principal subcongruences yields possessing finitely
determined syntactic congruences for varieties. The latter property is known to
yield standardness.
