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		gramming system as a parameter; we will denote the resulting classes simply by including the programming system in the subscript. For example, Exn,  y  denotes the collections of functions that can be identified with no more than n mind changes with respect to the acceptable programming system  y .

		Before formalizing paradigms with a bounded number of examples, we introduce a couple of technical tools. For each M and , define

		.

		Roughly, conv (M,  s ) is the last mind change in the sequence Given a scientist M and , we also define

		.

		Thus, cony(M, f) denotes the earliest convergence point, if any, for M on f.

		We now introduce identification from a bounded number of examples in a given programming system. N.B. The paradigm described below is dependent on the order in which the function is fed to the scientist. For convenience of investigation, we assume that the function is presented to the scientist in canonical order. Such a requirement is a weakness of this model.

		12.7 Definition (Wiehagen [199]) Let h be a partial recursive function and  y  a programming system.

		(a) S is said to be h-Bounded-Ex-identifiable with respect to  y  (written: ) just in case there exists a scientist M that Ex-identifies S with respect to  y  and, for all i with , we have .

		(b) .

		(c) .

		(d) .

		Surprisingly, Bounded-Ex is no less inclusive than Ex. Indeed:

		12.8 Proposition (Wiehagen [199]) Bounded-Ex = Ex.

		Proof: Fix an arbitrary M. We construct a programming system  y  and a scientist M' such that , where . This will establish the