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		§6.1.4 Vacillatory Function Identification

		The Bc criterion might appear overly liberal in allowing scientists to announce ever larger indexes for the target function. A more modest relaxation of Ex countenances endless shifts in hypotheses, but only within a fixed, finite set of alternatives. Thus, for scientist M to identify function f in this sense, there must be a finite set Df of indexes with two properties: first, all but finitely many of M's conjectures about f must be drawn from Df; second, every index in Df must be for f. It is easy to see that this new identification criterion is implied by Ex-identification (taking Df to be a singleton set) and implies Bc-identification. In order to facilitate comparison with earlier paradigms, we also incorporate anomalies into this success criterion. The formal definition is as follows.

		6.16 Definition (Barzdins * and Podnieks [16], Case and Smith [35]) Let .

		(a) M Fexa-identifies f (written: ) just in case there exists a nonempty finite set Df of a-error indexes for f such that for all but finitely many n, .

		(b) .

		So, for an M to Fexa-identify f, there must be some finite set Df of indexes to which M's conjectures must ultimately be limited — and Df must consist only of indexes for a-variants of f. Clearly, for all . Surprisingly, the inclusion is nowhere proper.

		6.17 Proposition (Barzdins* and Podnieks [16], Case and Smith [35]) For each , Exa = Fexa.

		Proof: Let . We argue Exn = Fexn. The * case is discussed later.

		As previously noted, , hence it suffices to show that . We do this by showing how to effectively transform any scientist M into a corresponding scientist M' such that .

		We employ an internal simulation argument that uses a cancellation technique. M' behaves as follows.

		On input  s , M' simulates M on  s  to find all the distinct indexes that M outputs on the initial sequences of  s . Let these indexes be p0, pl, . . ., pk. M' then cancels any index pi, , from this list that are found to be convergently different from  s  at more than n arguments in less than or equal to steps. More precisely, M' cancels any