  We define a class of functions termed "Computable in the Limit", based on the
Machine Learning paradigm of "Identification in the Limit". A function is
Computable in the Limit if it defines a property P_p of a recursively
enumerable class A of recursively enumerable data sequences S in A, such that
each data sequence S is generated by a total recursive function s that
enumerates . Let the index s represent the data sequence S. The property
P_p(s)=x is computed by a partial recursive function f_p(s,t) such that there
exists a u where f_p(s,u)=x and for all t>=u, f_p(s,t)=x if it converges. Since
the index s is known, this is not an identification problem - instead it is
computing a common property of the sequences in A. We give a Normal Form
Theorem for properties that are Computable in the Limit, similar to Kleene's
Normal Form Theorem. We also give some examples of sets that are Computable in
the Limit, and derive some properties of Canonical and Complexity Bound
Enumerations of classes of total functions, and show that no full enumeration
of all indices of Turing machines TM_i that compute a given total function can
be Computable in the Limit.
