•  12
    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 (1): 287. 2003.
  •  9
    On Cohesive Powers of Linear Orders
    with Rumen Dimitrov, Andrey Morozov, Paul Shafer, Alexandra A. Soskova, and Stefan V. Vatev
    Journal of Symbolic Logic 88 (3): 947-1004. 2023.
    Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$\omega $,$\zeta $, and$\eta $denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$\omega $. If$\mathcal {L}$is a computable copy of$\omega $that is computably isomorphic to the usual pre…Read more
  •  7
    Interpreting a Field in its Heisenberg Group
    with Rachael Alvir, Wesley Calvert, Grant Goodman, Julia Knight, Russell Miller, Andrey Morozov, Alexandra Soskova, and Rose Weisshaar
    Journal of Symbolic Logic 87 (3): 1215-1230. 2022.
    We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Mo…Read more
  •  4
    On the isomorphism problem for some classes of computable algebraic structures
    with Steffen Lempp, Charles F. D. McCoy, Andrei S. Morozov, and Reed Solomon
    Archive for Mathematical Logic 61 (5): 813-825. 2022.
    We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is \-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.
  •  2
    Logic and Algebraic Structures in Quantum Computing (edited book)
    with Jennifer Chubb and Ali Eskandarian
    Cambridge University Press. 2014.
    Experts in the field explore the connections across physics, quantum logic, and quantum computing.
  •  5
  •  15
    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.
  •  27
    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.
    We study computability theoretic properties of and equivalence structures and how they differ from computable equivalence structures or equivalence structures that belong to the Ershov difference hierarchy. Our investigation includes the complexity of isomorphisms between equivalence structures and between equivalence structures
  •  58
    Intrinsic bounds on complexity and definability at limit levels
    with John Chisholm, Ekaterina B. Fokina, Sergey S. Goncharov, Julia F. Knight, and Sara Quinn
    Journal of Symbolic Logic 74 (3): 1047-1060. 2009.
    We show that for every computable limit ordinal α, there is a computable structure A that is $\Delta _\alpha ^0 $ categorical, but not relatively $\Delta _\alpha ^0 $ categorical (equivalently. it does not have a formally $\Sigma _\alpha ^0 $ Scott family). We also show that for every computable limit ordinal a, there is a computable structure A with an additional relation R that is intrinsically $\Sigma _\alpha ^0 $ on A. but not relatively intrinsically $\Sigma _\alpha ^0 $ on A (equivalently,…Read more
  •  12
    The possible turing degree of the nonzero member in a two element degree spectrum
    Annals of Pure and Applied Logic 60 (1): 1-30. 1993.
    We construct a recursive model , a recursive subset R of its domain, and a Turing degree x 0 satisfying the following condition. The nonrecursive images of R under all isomorphisms from to other recursive models are of Turing degree x and cannot be recursively enumerable
  •  84
    Simple and immune relations on countable structures
    with Sergei S. Goncharov, Julia F. Knight, and Charles F. D. McCoy
    Archive for Mathematical Logic 42 (3): 279-291. 2003.
    Let ???? be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of ???? in which the image of R (¬R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on ???? and R, a necessary and sufficient condition for the existence of a computable copy ℬ of ???? in which the image of R (¬R, resp.) is simple (immune, resp.).
  •  29
    Spaces of orders and their Turing degree spectra
    with Malgorzata A. Dabkowska, Mieczyslaw K. Dabkowski, and Amir A. Togha
    Annals of Pure and Applied Logic 161 (9): 1134-1143. 2010.
    We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s …Read more
  •  29
    Effective categoricity of equivalence structures
    with Wesley Calvert, Douglas Cenzer, and Andrei Morozov
    Annals of Pure and Applied Logic 141 (1): 61-78. 2006.
    We investigate effective categoricity of computable equivalence structures . We show that is computably categorical if and only if has only finitely many finite equivalence classes, or has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Sin…Read more
  •  33
    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.
  • Sequences of $n$-diagrams
    with Julia Knight and Andrei Morozov
    Journal of Symbolic Logic 67 (3): 1227-1247. 2002.
  •  28
    Effectively and Noneffectively Nowhere Simple Sets
    Mathematical Logic Quarterly 42 (1): 241-248. 1996.
    R. Shore proved that every recursively enumerable set can be split into two nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice ϵ of all r. e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r. e. sets which can be split into two effectively nowhere simple sets, and r. e. sets which can be split into two r. e. non-nowhere simple sets.…Read more
  •  43
    Enumerations in computable structure theory
    with Sergey Goncharov, Julia Knight, Charles McCoy, Russell Miller, and Reed Solomon
    Annals of Pure and Applied Logic 136 (3): 219-246. 2005.
    We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a com…Read more
  •  43
    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
  •  15
    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
  •  14
    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 ]
  •  15
    Preface
    with Douglas Cenzer, David Marker, and Carol Wood
    Archive for Mathematical Logic 48 (1): 1-6. 2009.
  •  28
    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
  •  43
    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
  •  208
    Frequency computations and the cardinality theorem
    with Martin Kummer and Jim Owings
    Journal of Symbolic Logic 57 (2): 682-687. 1992.