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A proof of Beigel's cardinality conjecture

Martin Kummer

Journal of Symbolic Logic 57 (2):677-681 (1992)

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Extremes in the degrees of inferability.Lance Fortnow, William Gasarch, Sanjay Jain, Efim Kinber, Martin Kummer, Stuart Kurtz, Mark Pleszkovich, Theodore Slaman, Robert Solovay & Frank Stephan - 1994 - Annals of Pure and Applied Logic 66 (3):231-276.details Most theories of learning consider inferring a function f from either observations about f or, questions about f. We consider a scenario whereby the learner observes f and asks queries to some set A. If I is a notion of learning then I[A] is the set of concept classes I-learnable by an inductive inference machine with oracle A. A and B are I-equivalent if I[A] = I[B]. The equivalence classes induced are the degrees of inferability. We prove several results about (...) Download Export citation Bookmark 4 citations
Computable categoricity and the Ershov hierarchy.Bakhadyr Khoussainov, Frank Stephan & Yue Yang - 2008 - Annals of Pure and Applied Logic 156 (1):86-95.details In this paper, the notions of Fα-categorical and Gα-categorical structures are introduced by choosing the isomorphism such that the function itself or its graph sits on the α-th level of the Ershov hierarchy, respectively. Separations obtained by natural graphs which are the disjoint unions of countably many finite graphs. Furthermore, for size-bounded graphs, an easy criterion is given to say when it is computable-categorical and when it is only G2-categorical; in the latter case it is not Fα-categorical for any recursive (...) ordinal α. (shrink) Download Export citation Bookmark 4 citations

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