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		10.43 Definition (read: M on text T with positive additional information x and negative additional information y converges) just in case there is an i such that for all but finitely many n, M(x, y, T[n]) = i, in which case M(x, y, T) is defined to be this i. If M(x, y, T) fails to converge, then M(x, y, T) is undefined.

		10.44 Definition Let ) and d1, .

		(a) We say that a scientist M a language L (written: ) just in case, for all texts T for L, all p1 with is Wp1 is d1-language conforming with L, and all p2 with Wp2 is d2-language conforming with (N - L), we have and .

		(b)

		In , the positive additional information is of type Ap and negative additional information is of type Acp. The reader can similarly provide definitions for the following paradigms that arise by taking all combinations of the Ap and Uap types of positive additional information and the Acp and Uacp types of negative additional information:

		(a)

		(b)

		(c)

		Having defined these paradigms, we now are in a position to present some representative results. The reader may easily check that all of the results in Section 10.3.1 about the paradigms ApdExa and UapdExa have corresponding counterparts for the TxtApdExa and TxtUapdExa paradigms.

		We first prove that extra errors in the final inferred grammar cannot be compensated for by any amount of additional information. More precisely, we prove that there are collections of languages that can be identified by allowing up to k + 1 errors in the final grammar, but which cannot be identified by allowing up to k errors in the final grammar, even if the scientist is given the best possible additional positive information (Uap-type density 1) and the best possible additional negative information (Uacp-type density 1).

		10.45 Proposition

		Proof: Fix some m > 1. Let n0 = 0. For each , let n3i+l = n3i + mi, n3i+2 = n3i + mi and n3i+3 = n3i+2+1. Let , ,