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The characterization of [Ex]Popperian says that the collections of functions that Popperian scientists can identify are precisely the collections of functions that are subsets of r.e. indexable collections of functions. |
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5.65 Proposition (Barzdins
* and Freivalds [15], Case and Smith [35])  . |
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Proof: Suppose  . Fix a Popperian scientist M that identifies each function in  . Let  . It is easy to see that  and that  is r.e. indexable. |
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Suppose  is an r.e. indexable class of total functions. Choose a  such that  . Let M be a scientist that, for each  , |
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 . |
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It is easy to verify that M is Popperian and that  . |
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Reliable scientists were first studied by Minicozzi [132]. Reliability in the context of languages guarantees that if a scientist converges on a text for a language, then the scientist converges to an index for that language. The following formalizes reliability in the context of functions. |
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5.66 Definition (Minicozzi [132]) |
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(a) M is reliable on f just in case  . |
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(b) M is reliable on  just in case M is reliable on each  . |
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(c) M is recursive-reliable just in case M is reliable on  . |
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(d)  . |
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The next proposition shows that recursire-reliable scientists pay a price for this useful property. |
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5.67 Proposition  . |
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Proof: Clearly,  . Below we argue that  Our argument uses the following claim. |
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