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As an immediate corollary to the above, we have the following, which summarizes the advantages of allowing extra mind changes. |
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12.5 Corollary Let  . Then,  . |
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§12.3 Number of Examples Required |
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Another freedom available to a scientist in the identification paradigm is that she can see as much data as she wishes before settling on a final conjecture. Practical systems for empirical inquiry seldom have such luxury. It is thus worth considering modifications on the identification paradigm in which a scientist is limited by the number of examples she can see before settling on a final conjecture. Here we discuss a proposal due to Wiehagen. |
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A simplistic approach to bounding the number of data elements would be to define a paradigm under which a scientist can see no more than a predetermined number of examples before converging. This approach is not satisfactory, however, since it does not allow the bound to vary among different functions in the collection to be identified (after all, the graphs of distinct functions might coincide on arbitrary long initial segments of N). In order to describe a paradigm in which the bound depends on the underlying reality being identified, we need a mechanism for naming these possible realities. One way to achieve this is to include in the paradigm's definition a parameter y , which is the programming system (numbering) in which a scientist's conjectures are interpreted. (N.B. In this section we allow the possibility that  is a strict subset of  .) Until now such a parameterization was implicit and a scientist's hypotheses were interpreted in some fixed acceptable programming system for the partial recursive functions. The parameterized version of the Ex paradigm is as follows. |
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12.6 Definition Let y be a programming system. |
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(a) M Ex-identifies S with respect to y (written:  ) just in case for each  and y M(f) = f. |
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(b)  . |
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It is easy to check that for any two acceptable programming systems, y and y ', Ex y = Ex y '. Thus, if attention is restricted to acceptable programming systems, then choosing one over the other does not yield any extra learning ability. Most of the paradigms discussed in this book can straightforwardly be redefined with the underlying pro- |
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