1992
DOI: 10.2307/2275300
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Frequency computations and the cardinality theorem

Abstract: In 1960 G. F. Rose [R] made the following definition: A function f: ω → ω is (m, n)-computable, where 1 ≤ m ≤ n, iff there exists a recursive function R: ωn → ωn such that, for all n-tuples (x1,…, xn) of distinct natural numbers,J. Myhill (see [McN, p. 393]) asked if f had to be recursive if m was close to n; B. A. Trakhtenbrot [T] responded by showing in 1963 that f is recursive whenever 2m > n. This result is optimal, because, for example, the characteristic function of any semirecursive set is (1,2)-comp… Show more

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Cited by 33 publications
(13 citation statements)
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“…He also proved that this is optimal, i.e., there exists nonrecursive (n, 2n)-computable functions. See [19] for a recent survey of these and related results.…”
Section: The Power Of Learning From Frequency Computationsmentioning
confidence: 96%
See 3 more Smart Citations
“…He also proved that this is optimal, i.e., there exists nonrecursive (n, 2n)-computable functions. See [19] for a recent survey of these and related results.…”
Section: The Power Of Learning From Frequency Computationsmentioning
confidence: 96%
“…We determine the maximal probability p for which this can be done. [19] and [21]. Given an (m, n)-operator R we define uniformly as in [19, p. 684] a recursive tree T [0, 1]* whose branches represent the graphs of all partial functions which are (m, n)-computable via R.…”
Section: Probabilistic Learning From Frequency Computationsmentioning
confidence: 99%
See 2 more Smart Citations
“…Frequency computation was first studied by Rose [23] and Trakhtenbrot [25]; see [8] for a recent survey. A function f : | Ä | is called (m, n)-recursive iff there is a recursive function G: | n Ä | n such that for all x 1 < } } } <x n , G(x 1 , ..., x n ) and ( f (x 1 ), ..., f (x n )) agree in at least m components.…”
Section: Admissible Setsmentioning
confidence: 99%