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		Convention: For the remainder of the present chapter, d, d0, d1, d2 . . . range over real numbers in the unit interval.

		§10.3.1 Function Identification in the Presence of Approximations

		In the following an approximation to a function f will be taken as a partial recursive function  h  that agrees with f to some extent. The following definition introduces two different formalizations of "extent of agreement."

		10.29 Definition Let , , and .

		(a) We say that  h  is d-conforming with f just in case:

		(1) , i.e.,  h  does not contradict f; and

		(2) .

		(b) We say that  h  is uniformly d-conforming with f just in case:

		(1) , i.e.,  h  does not contradict f; and

		(2) .

		Let us recall from Chapter 7 the motivation for uniform d-conformity. It is this: even if  h  is 1-conforming with f,  h  may be a bad approximation locally for many large intervals; uniform conformity renders the approximation more homogeneous.

		In the paradigms defined below, a scientist learning f is fitted out with a program for an approximation  h  to f, along with access to the graph of f. The approximation  h  will be either d-conforming or uniformly d-conforming. Thus, our scientists again compute mappings from N × SEG to N, and M(p,  s ) denotes the conjecture of M on prior information p plus evidential state  s  A definition similar to Definition 10.1 can be stated to describe the convergence of a scientist on information p and graph f.

		The next definition makes things official. It extends the Ex paradigm to situations in which the scientist has access to a program for a partial function that is d- conforming with the function to be learned.

		10.30 Definition Let and .

		(a) A scientist M ApdExa-identifies f (written: ) just in case, for all p such that  j p is d-conforming with f, and  j M(p,f) = a f.

		(b) .