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10.41 Definition Let  and  |
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(a) A scientist M TxtAcpdExa-identifies L (written:  ) just in case, for all texts T for L and all p such that W<p is language d-conforming with L, we have  and W<SB >M(p,T) =a L. |
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(b) |
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Replacing d-conformity in the above definition by uniform d-conformity yields the paradigm TxtUapdExa. |
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Before presenting results about TxtApdExa and TxtUapdExa, let us take another step in the elaboration of paradigms. In both of the foregoing models, the advance information about the incoming language L is "positive" in the sense that it describes a subset of L rather than its complement. It can also be imagined that the child receives advance, negative information in the form of an index for a subset of N - L. (This possibility does not bear on the debate about the existence of negative information in texts; textual information is provided by caretakers whereas advance information is lodged in the child's mind through the genome.) We rely on the following definition to formulate the possibility of advance, negative information. |
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10.42 Definition Let  and  . |
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(a) A scientist M TxtAcpdExa-identifies L (written:  ) just in case, for all texts T for L and all p such that Wp is language d-conforming with (N - L), we have  and WM(p,T) =a L. |
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(b)  |
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Acp is an abbreviation for "Approximate complement partial additional information." Replacing d-conformity in the above definition by uniform d-conformity yields the paradigm TxtUacpdExa. |
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Finally, we introduce paradigms in which the learner receives advance information about both L and N - L, where L is the language being presented. Within these paradigms, scientists compute mappings from N × N × SEQ to N. The first argument provides advance information about a subset of L and the second about a subset of N - L. Thus, M(x, y, s ) denotes the conjecture of scientist M on a grammar x for a language L, plus a grammar y for an (r.e.) subset of N - L, plus an initial segment s of a text for L. Setting things down in an official way yields the following definitions. |
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