**Undecidable Problems for Finite
Algebras **

R. McKenzie

These properties of a finite algebra **A**, referring
to the variety *V*(**A**) generated by **A**, are
undecidable: *V*(**A**) is residually finite, is residually
small, is finitely axiomatizable, has a model companion,
has type-set excluding the Boolean type (defined in
tame congruence theory).
All these results were proved in the past three years by
the same method, which successfully models the
computations of a Turing machine on an infinite tape by
using elements of a power algebra **A**^\omega
to represent instantaneous configurations in the computation,
and using algebraic operations to model the transition between
configurations produced by the Turing machine.

It is likely that this method can be
used to obtain further interesting undecidability results.
In the talk, I shall discuss some possible results of this kind,
and examine in some detail the application which is closest
to the origin of the method, namely the undecidability of
the residual smallness of *V*(**A**).