Kleene's T Predicate - Normal Form Theorem

Normal Form Theorem

The T predicate can be used to obtain Kleene's normal form theorem for computable functions (Soare 1987, p. 15). This states there exists a primitive recursive function U such that a function f of one integer argument is computable if and only if there is a number e such that for all n one has

,

where μ is the μ operator and holds if both sides are undefined or if both are defined and they are equal. Here U is a universal operation (it is independent of the computable function f) whose purpose is to extract, from the number x (encoding a complete computation history) returned by the operator μ, just the value f(n) that was found at the end of the computation.