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Refutable inference is a process of inductive inference with refutability of hypothesis spaces and is essential in Machine Discovery(Mukouchi and Arikawa). Though some refutable inferabilities under l...anguage model are known, most scientific laws are represented by functions. Thus we investigated refutable inferability of function classes. In order to develop a realistic theory of function learning, we investigated the refutable inferability of primitive recursive functions computed by a concrete programming system, loop programs (Meyer and R.itchie). Let FLoop(n) be the set of all primitive recursive functions computed by a loop program with at most n nesting of loops. We show $ FLoop(n)(n gep 1) $ and a natural subclass of FLoop(1) are not refutably inferable. Thus the existent natural hierarchies of loop programs are shown to be not suitable for machine discovery. Then we construct two types of rich series of refutably inferable classes such that each union of the series is FLoop(l) and show the inside structures of these series.show more
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