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way in which infinitely many stages can exist is by the success of step (2b) infinitely often. |
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Case 2: Some stage s is started but never left. In this case, let  and  . Note that L0 and L1 are infinitely different from each other. Let  , and  , where  and  are as defined in step 1 of stage s. Now, T0 is a text for L0 and T1 is a text for L1. However, M converges to the same grammar for both T0 and T1, thus M does not TxtEx*-identify  . |
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The above cases imply that  . |
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Having investigated noisy versus incomplete data, it is natural to consider next the relationship between incomplete and imperfect texts. How much more damage to learning results from adding noise to an environment already missing data? In this matter only approximate results arc known; the exact relationship is still open. We provide a list of some sample results. Proofs for some of them are developed in the exercises. |
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8.24 Proposition For all i > 0,  . |
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8.25 Proposition For all i > 0 and  ,  . |
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8.26 Proposition For all i,  ,  . |
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§8.2 Texts with Additional Structure |
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The paradigms discussed until now suppose that an input text can be ordered arbitrary, and thus require scientists to succeed regardless of the manner in which their data are presented. This is, however, a questionable representation of many natural environments. It seems unlikely, for example, that children face an entirely arbitrary order of linguistic input, since simpler sentences usually arrive earlier than complex ones. The present section briefly discusses some paradigms that begin to respond to this consideration. |
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It is often claimed that children encounter simple sentences first and difficult ones later. Complexity may be measured by the length of the sentence, degree of embedding, inflection etc. For our purposes we assume that a sentence whose code is smaller is simpler. |
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