Page 81

		4.47 Proposition is performable on [·]f.

		Further discussion of parameterized scientists may be found in Jantke [98].

		§4.5 Exact Identification

		In this final section of the chapter, we advance a paradigm designed to refine the constraints on comparative grammar provided by considerations of learnability (see Section 3.1.2, above). Only the identification of languages is at issue.

		§4.5.1 Exact Learning and Comparative Grammar

		The converse of the dictum that natural languages are learnable by children (via casual exposure) is that nonnatural languages are not learnable. Put differently, the natural languages are generally taken to be the largest collection of child-learnable languages. We are thus led to consider paradigms in which learners are required to respond successfully to all languages in a given collection and to respond unsuccessfully to all other languages. One such paradigm relies on a modification of the identification concept that is defined as follows.

		4.48 Definition

		(a) Let scientist and be given. F identifies exactly just in case for all texts T, F identifies T if and only if T is for some . In this case is exactly identifiable.

		(b) The class of all such that some computable identifies exactly is denoted by TxtExExact.

		For example, the scientist defined in the proof of Proposition 3.18 identifies exactly. Similarly, let g be some recursive function such that Wg( s ) = content( s ) for each and consider a scientist F that on emits g( s ). Clearly, this F identifies exactly. Hence .

		Two points should be borne in mind about Definition 4.48. First, if F identifies , then F identifies every subset of and may identify some proper supersets of as well (see Exercise 3-3). In contrast, if F identifies exactly, then F exactly identifies no collection . Second, if F identifies exactly, then there is a strong sense in which F fails to identify ; namely, F fails to identify any text for such an L,