Given these definitions, we are now in a position to define the temporal classification.
Let 
be a set of streams with the same type. Let 
be a set of
labels of some kind, that describes the set of possible classes.
Define a function
which takes an element of and returns an element of
.
The goal is given a subset of the function 
(say 
)
produce a function 
which is a similar to as
possible
.
Again, it is hard to represent the notion of ``learning'' here; and
there are several intuitive aspects which the above definition does
not cover. Again, the channel type means more than just that the
channels have the same range; and is more than just a
random collection of streams. in some ways represents a
domain (in the machine learning sense) that we are interested in.
It also remains for similarity to be defined. For example, if we
are to continue the above example, one might be the set
of all possible signs. 
is not just a random
function, but a function that tells us what the class, or type, of a
given sign is.
Intuitively, our goal is: given a limited example of streams and their
classes, in other words, some subset of the function , say
, can we make an estimate of , say , that is close
to ? This is similar to the definition for pure inductive
inference.