We present a new approach to the composition
of learning algorithms (in various models) for
classes of constant VC-dimension into learning algorithms for
more complicated classes.
We prove that if a class \CC is learnable
in time t from a hypothesis class \HH of constant VC-dimension
then the class \C^\star of all
functions F of the form F=f(g_1,\ldots,g_m),
where f is any function
and g_1,\ldots,g_m\in \C, is
learnable in time polynomial in t and m .
We also use a simple
argument to prove that the composition theorem cannot be extended
to classes with a nonconstant VC-dimension.

A composition theorem for the exact learning model
(with equivalence queries only)
is proven in [BBK97] {\it only} for classes \C
of constant VC-dimension
that have {\it constant space} learning algorithms.
Constant space algorithms are hard to find
and have large complexity.
Our algorithm is simple and has a
complexity lower than the algorithm in [BBK97].

We then show how to change a PAC-learning
algorithm of \CC from \HH to an
SQ-learning algorithm and to a PAC-learning
algorithm for \CC^\star with malicious noise
that achieves the optimal error rate \eta/(1-\eta)+\beta
for any \beta.
This, in particular, shows that if a class of
constant VC-dimension is PAC-learnable from a
class of constant VC-dimension then
it is SQ-learnable and PAC-learnable with malicious noise.
We apply this result for SQ-learning and PAC-learning with malicious
noise a general class of geometric objects.
This class includes the set of all geometric objects
in the constant dimensional space
that are bounded by m algebraic surfaces of constant
degree (for example, hyperplanes, spheres, etc.).
This result generalizes all the results known
from the literature about learning geometric objects
in the SQ-learning and PAC-learning models with malicious noise.