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		x, there is at most one y such that (x, y) occurs in the sequence. We use G as a typical variable for texts over N × N.

		(b) Similarly to before, the set of pairs appearing in a text G is denoted by content(G).

		(c) Let total function , and text G be given. G is for f just in case content(G) = f.

		Thus, the following sequence represents a text for the squaring function.

		3.39 (1, 1) (0, 0) (3, 9) (2, 4) (5, 25) (4, 16) . . .

		Notation for texts introduced for languages in 3.6 carries over to functions. Thus, let text G and be given. The nth pair in G is denoted by G(n). The initial finite sequence of G of length n is denoted G[n].

		Texts for functions have a property not shared by texts for languages. Let G be a text for a total function, and let T be a text for a language. For any N × N, examination of some initial segment of G suffices to verify the presence or absence of (n, m) in G once and for all. In particular, if (n, m') is found in G, where , then content(G). In contrast, no finite examination of T can definitively verify the absence of a number from T. This asymmetry renders function discovery easier than language discovery, as will be seen shortly.

		§3.9.3 Function Identification: Scientists

		In order to specify the concept of "scientist" in the current paradigm, we remind the reader that a subset X of N × N is a (graph of a) function just in case X contains no two elements of the form (x, y), (x, z) with . Also, we record the fact:

		3.40 Lemma Every finite function over N × N is a subset of some member of .

		3.41 Definition The set { a text for some total function and is denoted by SEG; are variables over SEG. (It will be clear from context whether such variables are supposed to range over SEQ or SEG.)

		Notation for members of SEQ carries over to SEG. Thus, let be given. The length of  s  is denoted . The set of pairs appearing in  s  is denoted by content( s ). For , the nth pair appearing in  s  is denoted by  s [n], and the initial sequence of length n in  s  is denoted by  s [n]. For example, if G is the text of 3.39, then G(3) = (2,4), G[2] = (1, 1) (0,0), and content(G[3]) = {(1, 1), (0,0), (3,9)}. Note that SEG is