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Stage n + 1: Let  be the least number such that  is found in  steps. (Such a number must exist, since M is accountable.) If (i)  , then let  . Else if (ii)  , then let  . |
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Let  and We = content(T). It is clear that  . It is also clear that M fails to identify T, since for each n, either (i)  (by not being single valued), or else (ii)  . |
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Suppose scientist M is defined on  . We say that M makes a "wild guess" if M does not identify WM( s ). Here we consider scientists that never respond to their data with a wild guess. |
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(a) M is said to be prudent just in case for all  , if M( s ) is defined, then M identifies WM( s ). |
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(b)  . |
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In other words, prudent learners only conjecture grammars for languages they are prepared to learn. It is easy to verify that the collection of finite languages, and the collection of co-singleton languages, can be identified by prudent scientists. |
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If "prestorage" models of linguistic development are correct, then children acquiring language may well be prudent learners. A prestorage model posits an internal list of candidate grammars that coincides exactly with the natural languages. Language acquisition amounts to the selection of a grammar from this list in response to linguistic input. Such a prestorage learner is prudent in as much as his or her linguistic hypotheses are limited to grammars from the list, that is, to grammars corresponding to natural (i.e., learnable) languages. The next proposition reveals that prudence has no impact on learnability. |
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5.20 Proposition(Fulk [71])  . |
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The proposition is an immediate corollary of the following lemmas. We defer the proofs of these lemmas to Section 5.4.1. To help state these lemmas, we introduce the following definition. |
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