  Imposing an extensional uniformity condition on a non-uniform circuit
complexity class C means simply intersecting C with a uniform class L. By
contrast, the usual intensional uniformity conditions require that a
resource-bounded machine be able to exhibit the circuits in the circuit family
defining C. We say that (C,L) has the "Uniformity Duality Property" if the
extensionally uniform class C \cap L can be captured intensionally by means of
adding so-called "L-numerical predicates" to the first-order descriptive
complexity apparatus describing the connection language of the circuit family
defining C.
  This paper exhibits positive instances and negative instances of the
Uniformity Duality Property.
