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Degree spectra of prime models
 J. Symbolic Logic
, 2004
"... 2.1 Notation from model theory................... 4 2.2 F ..."
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2.1 Notation from model theory................... 4 2.2 F
Computable categoricity of trees of finite height
 Journal of Symbolic Logic
"... We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ0 3conditi ..."
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We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ0 3condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆0 n+1categorical but not ∆0 ncategorical.
Questions in Computable Algebra and Combinatorics
, 1999
"... this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these ..."
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this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25
On the complexity of the successivity relation in computable linear orderings, in preparation
"... Abstract. In this paper, we solve a longstanding open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum ..."
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Abstract. In this paper, we solve a longstanding open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing ∆ 0 3isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.
Computability of Fraïssé limits
 IN PREPARATION
"... Fraïssé studied countable structures S through analysis of the age of S, i.e., the set of all finitely generated substructures of S. 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 ..."
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Fraïssé studied countable structures S through analysis of the age of S, i.e., the set of all finitely generated substructures of S. 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 quantifierfree 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.
CUTS OF LINEAR ORDERS
"... Abstract. We study the connection between the number of ascending and descending cuts of a linear order, its classical size, and its effective complexity (how much [how little] information can be encoded into it). 1. ..."
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Abstract. We study the connection between the number of ascending and descending cuts of a linear order, its classical size, and its effective complexity (how much [how little] information can be encoded into it). 1.
Spectra of Structures and Relations
, 2006
"... 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 ..."
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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 ′ ′. 1
Russian Academy of Sciences, Siberian Branch
, 2010
"... We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of nonlown degrees. We then repeat these questions specifically for linear orders and for relations on the ..."
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We survey known results on spectra of structures and on spectra of relations on computable structures, asking when the set of all highn degrees can be such a spectrum, and likewise for the set of nonlown degrees. We then repeat these questions specifically for linear orders and for relations on the computable dense linear order Q. New results include realizations of the set of nonlown Turing degrees as the spectrum of a relation on Q for all n ≥ 1, and a realization of the set of all nonlown Turing degrees as the spectrum of a linear order whenever n ≥ 2. The state of current knowledge is summarized in a table in the concluding section.