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Switch to: References
Citations of:
d-computable Categoricity for Algebraic Fields
Journal of Symbolic Logic 74 (4):1325 - 1351 (2009)
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For a computable structure $\mathcal {M}$, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of $\mathcal {M}$. If the spectrum has a least degree, this degree is called the degree of categoricity of $\mathcal {M}$. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity. |
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In this paper, we show that for any computable ordinal α, there exists a computable tree of rank with strong degree of categoricity if α is finite, and with strong degree of categoricity if α is infinite. In fact, these are the greatest possible degrees of categoricity for such trees. For a computable limit ordinal α, we show that there is a computable tree of rank α with strong degree of categoricity (which equals ). It follows from our proofs that, (...) |
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Results of R. Miller in 2009 proved several theorems about algebraic fields and computable categoricity. Also in 2009, A. Frolov, I. Kalimullin, and R. Miller proved some results about the degree spectrum of an algebraic field when viewed as a subfield of its algebraic closure. Here, we show that the same computable categoricity results also hold for finite-branching trees under the predecessor function and for connected, finite-valence, pointed graphs, and we show that the degree spectrum results do not hold for (...) |
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We investigate effective categoricity for polymodal algebras. We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra. |
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Let K be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from K such that every structure in K is isomorphic to exactly one structure on the list. Such a list is called a computable classification of K, up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields and that with a 0'-oracle, we can obtain similar (...) |
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Defining the degree of categoricity of a computable structure ${\mathcal{M}}$ to be the least degree d for which ${\mathcal{M}}$ is d-computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that for all n, degrees d.c.e. in and above 0 (n) can be so realized, as can the degree 0 (ω). |
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