Relations Intrinsically Recursive in Linear Orders

Moses, Michael

doi:10.1002/malq.19860322514

Cited by 21 publications

(14 citation statements)

References 3 publications

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“…(ii) For any unary relation R on si, the following are equivalent: (a) The degree spectrum DgSp& (R) is upward closed under Turing reducibility; (b) The relation R is not intrinsically computable; (c) R cannot be defined by a quantifier-free formula with parameters from dom(si). (The equivalence of (b) and (c) had already been shown by Moses in [20].) As a corollary, Harizanov and Miller obtained from Miller's result in [19] that there is a relation R on si such that DgSp^(i?)…”

mentioning

confidence: 58%

Computability of Fraïssé limits

Csima¹,

Harizanov²,

Miller

et al. 2011

J. symb. log.

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Fraïssé studied countable structures through analysis of the age of , i.e., the set of all finitely generated substructures of . We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.

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mentioning

confidence: 58%

Computability of Fraïssé limits

Csima¹,

Harizanov²,

Miller

et al. 2011

J. symb. log.

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“…The implications 1 =⇒ 2 and 3 =⇒ 1 are immediate. In fact, Moses proved in [18] that 1 ⇐⇒ 2. To prove that 2 =⇒ 3, fix any degrees d ≤ T c, and suppose (using Lemma 1.6) that S ∈ d and (L, R) ∼ = (L, S).…”

Section: Dgsp L (R) Is Upward-closed Under Turing Reducibilitymentioning

confidence: 99%

Spectra of structures and relations

Harizanov

Miller²

2007

J. symb. log.

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We consider embeddings of structures which preserve spectra: if g : M → S with S computable, then M should have the same Turing degree spectrum (as a structure) that g(M) has (as a relation on S). We show that the computable dense linear order L is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph G. Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on G. Finally, we consider the extent to which all spectra of unary relations on the structure L may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c ≥ T 0 .

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“…In [30], Moses showed that, for any computable relation U on a linear ordering L, either U is definable by a quantifier free formula in the language of L expanded by finitely many constants (in which case it is obviously intrinsically computable) or there is a function f such that f(L) is a computable structure and f(U ) is not computable. It is clear from the proof of this result that, in the latter case, f can be chosen to be Δ 0 2 .…”

Section: Corollary Let U Be An Invariant Computable Relation On the mentioning

confidence: 99%

Degree Spectra of Relations on Computable Structures

Hirschfeldt¹,

Shore²

2000

Bull. symb. log.

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There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. (A valuable reference is the handbook [7]. In particular, the introduction and the articles by Ershov and Goncharov and by Harizanov give useful overviews, while the articles by Ash and by Goncharov cover material related to the topic of this communication.)Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable (c. e.) degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough.

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Relations Intrinsically Recursive in Linear Orders

Cited by 21 publications

References 3 publications

Computability of Fraïssé limits

Computability of Fraïssé limits

Spectra of structures and relations

Degree Spectra of Relations on Computable Structures

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