•  5
    Computability-theoretic categoricity and Scott families
    with Ekaterina Fokina and Daniel Turetsky
    Annals of Pure and Applied Logic 170 (6): 699-717. 2019.
  • Induction, algorithmic learning theory, and philosophy (edited book)
    with Michele Friend and Norma B. Goethe
    Springer. 2007.
  • Downey, R., Fiiredi, Z., Jockusch Jr., CG and Ruhel, LA
    with W. I. Gasarch, A. C. Y. Lee, M. Groszek, T. Hummel, H. Ishihara, B. Khoussainov, A. Nerode, I. Kalantari, and L. Welch
    Annals of Pure and Applied Logic 93 263. 1998.
  • Belegradek, O., Verbovskiy, V. and Wagner, FO, Coset
    with J. Y. Halpern, B. M. Kapron, U. Kohlenbach, P. Oliva, F. Lucas, B. Luttik, P. Matet, and M. Pourmahdian
    Annals of Pure and Applied Logic 121 287. 2003.
  •  6
    Σ 1 0 and Π 1 0 equivalence structures
    with Douglas Cenzer and Jeffrey B. Remmel
    Annals of Pure and Applied Logic 162 (7): 490-503. 2011.
  •  12
    Degree spectra of the successor relation of computable linear orderings
    with Jennifer Chubb and Andrey Frolov
    Archive for Mathematical Logic 48 (1): 7-13. 2009.
    We establish that for every computably enumerable (c.e.) Turing degree b the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees
  •  10
    Uncountable degree spectra
    Annals of Pure and Applied Logic 54 (3): 255-263. 1991.
    We consider a recursive model and an additional recursive relation R on its domain, such that there are uncountably many different images of R under isomorphisms from to some recursive model isomorphic to . We study properties of the set of Turing degrees of all these isomorphic images of R on the domain of
  •  27
    Bounding Homogeneous Models
    with Barbara F. Csima, Denis R. Hirschfeldt, and Robert I. Soare
    Journal of Symbolic Logic 72 (1). 2007.
    A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram De (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogene…Read more
  •  9
    Dependence relations in computably rigid computable vector spaces
    with Rumen D. Dimitrov and Andrei S. Morozov
    Annals of Pure and Applied Logic 132 (1): 97-108. 2005.
    We construct a computable vector space with the trivial computable automorphism group, but with the dependence relations as complicated as possible, measured by their Turing degrees. As a corollary, we answer a question asked by A.S. Morozov in [Rigid constructive modules, Algebra and Logic, 28 570–583 ; 379–387 ]
  •  21
    Π 1 1 relations and paths through
    with Sergey S. Goncharov, Julia F. Knight, and Richard A. Shore
    Journal of Symbolic Logic 69 (2): 585-611. 2004.
  •  7
    with Douglas Cenzer, David Marker, and Carol Wood
    Archive for Mathematical Logic 48 (1): 1-6. 2009.
  •  20
    Turing degrees of certain isomorphic images of computable relations
    Annals of Pure and Applied Logic 93 (1-3): 103-113. 1998.
    A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let be a computable model and let R be an extra relation on the domain of . That is, R is not named in the language of . We define to be the set of Turing degrees of the images f under all isomorphisms f from to computable models. We investigate conditions on and R which are sufficient and necessary for to contain every Turing degree. These conditions imply that if every Turing degre…Read more
  •  201
    Frequency computations and the cardinality theorem
    with Martin Kummer and Jim Owings
    Journal of Symbolic Logic 57 (2): 682-687. 1992.
  •  5
    Π₁¹ Relations and Paths through ᵊ
    with Sergey S. Goncharov, Julia F. Knight, and Richard A. Shore
    Journal of Symbolic Logic 69 (2). 2004.
  •  9
    Partial automorphism semigroups
    with Jennifer Chubb, Andrei S. Morozov, Sarah Pingrey, and Eric Ufferman
    Annals of Pure and Applied Logic 156 (2): 245-258. 2008.
    We study the relationship between algebraic structures and their inverse semigroups of partial automorphisms. We consider a variety of classes of natural structures including equivalence structures, orderings, Boolean algebras, and relatively complemented distributive lattices. For certain subsemigroups of these inverse semigroups, isomorphism of the subsemigroups yields isomorphism of the underlying structures. We also prove that for some classes of computable structures, we can reconstruct a c…Read more
  •  9
    Effective categoricity of Abelian p -groups
    with Wesley Calvert, Douglas Cenzer, and Andrei Morozov
    Annals of Pure and Applied Logic 159 (1-2): 187-197. 2009.
    We investigate effective categoricity of computable Abelian p-groups . We prove that all computably categorical Abelian p-groups are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. We investigate which computable Abelian p-groups are categorical and relatively categorical
  •  6
    Some effects of Ash–Nerode and other decidability conditions on degree spectra
    Annals of Pure and Applied Logic 55 (1): 51-65. 1991.
    With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they prove that R is intrinsically r.e. if and only if a natural recursive-syntacti…Read more
  •  36
    Chains and antichains in partial orderings
    with Carl G. Jockusch and Julia F. Knight
    Archive for Mathematical Logic 48 (1): 39-53. 2009.
    We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is ${\Sigma _{1}^{1}}$ or ${\Pi _{1}^{1}}$ , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two ${\Pi _{1}^{1}}$ sets. Our main result is that there is a computa…Read more
  •  26
    Isomorphism relations on computable structures
    with Ekaterina B. Fokina, Sy-David Friedman, Julia F. Knight, Charles Mccoy, and Antonio Montalbán
    Journal of Symbolic Logic 77 (1): 122-132. 2012.
    We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all ${\mathrm{\Sigma }}_{1}^{1}$ equivalence relations on hyperarithmetical subsets of ω
  •  18
    Spectra of Structures and Relations
    with Russel G. Miller
    Journal of Symbolic Logic 72 (1). 2007.
    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, an…Read more
  •  14
    San Antonio Convention Center San Antonio, Texas January 14–15, 2006
    with Douglas Cenzer, C. Ward Henson, Michael C. Laskowski, Alain Louveau, Russell Miller, Itay Neeman, and Sergei Starchenko
    Bulletin of Symbolic Logic 12 (4). 2006.
  •  6
    Turing degrees of hypersimple relations on computable structures
    Annals of Pure and Applied Logic 121 (2-3): 209-226. 2003.
    Let be an infinite computable structure, and let R be an additional computable relation on its domain A. The syntactic notion of formal hypersimplicity of R on , first introduced and studied by Hird, is analogous to the computability-theoretic notion of hypersimplicity of R on A, given the definability of certain effective sequences of relations on A. Assuming that R is formally hypersimple on , we give general sufficient conditions for the existence of a computable isomorphic copy of on whose d…Read more
  • Π11 Relations And Paths Through
    with Sergey Goncharov, Julia Knight, and Richard Shore
    Journal of Symbolic Logic 69 (2): 585-611. 2004.
  •  13
    Computability of fraïssé limits
    with Barbara F. Csima, Russell Miller, and Antonio Montalbán
    Journal of Symbolic Logic 76 (1). 2011.
    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 defi…Read more
  •  8
    $\Pi _{1}^{0}$ Classes and Strong Degree Spectra of Relations
    with John Chisholm, Jennifer Chubb, Denis R. Hirschfeldt, Carl G. Jockusch, Timothy McNicholl, and Sarah Pingrey
    Journal of Symbolic Logic 72 (3). 2007.
    We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof