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		(a) Exhibit such that:

		(i) is identifiable; but

		(ii) no weakly memory-limited scientist identifies .

		(b) Exhibit such that:

		(i) some weakly memory-limited scientist identifies ; but

		(ii) no memory-limited scientist identifies .

		3-15 Prove that is identifiable if and only if is identifiable on fat texts.

		3-16 A text is called ''lean" just in case it never repeats a number. Prove that is identifiable if and only if is identifiable on lean text.

		3-17 (Advanced) Prove Proposition 3.37.

		3-18 Text T is called "ascending" just in case for all n, , if and and , then . is identifiable on ascending text just in case some scientist F identifies every ascending text for every . Prove:

		(a) There is such that:

		(i) is not identifiable, but

		(ii) is identifiable on ascending text.

		(b) There is such that is not identifiable on ascending text.

		3-19 Text T is called "strictly ascending" just in case # does not occur in T, and for all n, , if n < m, then T(n) < T(m). Show that is identifiable on strictly ascending text.

		3-20 Let us see what happens if we tighten our success criterion by requiring that scientists never converge to an incorrect. index.

		3.57 Definition (Minicozzi [132]) Let scientist F and be given. F identifies reliably just in case:

		(a) F identifies ;

		(b) for all texts T, if F converges on T to , then content(T) = Wi.

		In this case, is reliably identifiable.