A cardinality version of Beigel's nonspeedup theorem

This paper studies the Turing degrees of various properties defined for universal numberings, that is, for numberings which list all partial-recursive functions. In particular properties relating to the domain of the corresponding functions are investigated like the set DEQ of all pairs of indices of functions with the same domain, the set DMIN of all minimal indices of sets and DMIN * of all indices which are minimal with respect to equality of the domain modulo finitely many differences. A partial solution to a question of Schaefer is obtained by showing that for every universal numbering with the Kolmogorov property, the set DMIN * is Turing equivalent to the double jump of the halting problem. Furthermore, it is shown that the join of DEQ and the halting problem is Turing equivalent to the jump of the halting problem and that there are numberings for which DEQ itself has 1-generic Turing degree.

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“…The following proofs make use of Owings' Cardinality Theorem [17]. This says that whenever there is an m > 0 and a B-recursive {0, 1, 2, .…”

Section: Proposition 5 For Every Domain-universal Numberingmentioning

confidence: 99%

“…,m} − {K (a 1 ) + K (a 2 ) + · · · + K (a m )}. Owings' Cardinality Theorem [7,17] states that the existence of such a function h implies K T DMIN * ψ . It is well known that DMIN * Proof.…”

Section: Proposition 5 For Every Domain-universal Numberingmentioning

confidence: 99%

Index sets and universal numberings

Jain

Stephan

Teutsch

2011

Journal of Computer and System Sciences

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“…Since the tree has exponentially many paths, we will need to identify some of the paths as dead-ends. The following combinatorial lemmas [Bei91,Owi89] will help us prune the tree and maintain a polynomial bound on the running time of the tree traversal.…”

Section: -2mentioning

confidence: 99%

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Beals¹,

Chang²,

Gasarch³

et al. 1999

Chicago J. of Theoretical Comp. Sci.

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In computational complexity theory, a function f is called b(n)-enumerable if there exists a polynomial-time function that can restrict the output of f (x) to one of b(n) possible values. This paper investigates #GA, the function that computes the number of automorphisms of an undirected graph, and GI, the set of pairs of isomorphic graphs. The results in this paper show the following connections between the enumerability of #GA and the computational complexity of GI. 1. #GA is exp(O(√ n log n))-enumerable. 2. If #GA is polynomially enumerable, then GI ∈ R. 3. For < 1 2 , if #GA is n-enumerable, then GI ∈ P. Chicago Journal of Theoretical Computer Science 1 Not to be confused with recursive enumerability in recursive function theory or countable (denumerable) sets in set theory.

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“…In fact, this holds also for r.e. trees of bounded width [37]. We determine, for both the recursive and the r.e.…”

Section: Trees Of Bounded Widthmentioning

confidence: 99%

Learning Recursive Functions from Approximations

Case

Kaufmann

Kinber

et al. 1997

Journal of Computer and System Sciences

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This article investigates algorithmic learning, in the limit, of correct programs for recursive functions f from both inputÂoutput examples of f and several interesting varieties of approximate additional (algorithmic) information about f. Specifically considered, as such approximate additional information about f, are Rose's frequency computations for f and several natural generalizations from the literature, each generalization involving programs for restricted trees of recursive functions which have f as a branch. Considered as the types of trees are those with bounded variation, bounded width, and bounded rank. For the case of learning final correct programs for recursive functions, EX-learning, where the additional information involves frequency computations, an insightful and interestingly complex combinatorial characterization of learning power is presented as a function of the frequency parameters. For EX-learning (as well as for BC-learning, where a final sequence of correct programs is learned), for the cases of providing the types of additional information considered in this paper, the maximal probability is determined such that the entire class of recursive functions is learnable with that probability. ] 1997 Academic Press

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A cardinality version of Beigel's nonspeedup theorem

Cited by 30 publications

References 2 publications

Index sets and universal numberings

Index sets and universal numberings

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Learning Recursive Functions from Approximations

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