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Now suppose that scientist Ffinite of 3.9 is working on text T for  . By Proposition 3.15 Ffinite identifies T. However, no finite initial segment of T provides sufficient information to determine the moment at which Ffinite begins to identify T. This is because no T[n] in T excludes the possibility that the next element of T (namely, T(n)) lies outside of content(T[n]), thus falsifying Ffinite's present conjecture. |
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More generally, identification is a limiting process in the sense that it concerns the behavior of a learning function on an infinite subset of its domain.1 Because of the limiting nature of identification, the behavior of a scientist F on a text T cannot in general be predicted from F's behavior on any finite portion of T. The unpredictability is not connected to our external viewpoint, requiring us to observe scientists from the outside. To make this clear, let us briefly consider scientists that announce their own convergence, and may thus be considered to observe their own operation. |
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3.19 Definition (after Freivalds and Wiehagen [67]) A scientist F is self-monitoring just in case for all texts T, if F identifies T, then |
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(a) there is a unique  with F(T[n]) = 0, and |
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(b) for this n, F(T[i]) = F(T[n + 1]) for all i > n. |
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Intuitively, a scientist is self-monitoring just in case she signals her own convergence, where 0 serves as the signal. The following proposition is suggested by our earlier remarks. |
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3.20 Proposition No self-monitoring scientist identifies  . |
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Propositions 3.20 and 3.15 show that identifiability does not entail identifiability by a self-monitoring scientist. Informally, a scientist can identify a text without it being possible for her to ever know that she has done so. (Compare Section 1.4.4, above.) |
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The limiting nature of identification implies that the behavior of a scientist on short members of SEQ is irrelevant to the class of languages it identifies. To say this more clearly, call a scientist n-mute just in case it is undefined on all  with  . Then, it is easy to verify that for all  ,  is identifiable if and only if some n-mute scientist identifies  . This fact invites the thought that some scientists make more efficient use of data than others, and that a more realistic model of empirical discovery would integrate concern for the resources consumed by inquiry. Such concerns will in |
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1 For this reason Gold [80] refers to identification as "identification in the limit." |
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