Results for 'Rod G. Downey'

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  1.  1
    Exact Pairs for the Ideal of the K-Trivial Sequences in the Turing Degrees.George Barmpalias & Rod G. Downey - 2014 - Journal of Symbolic Logic 79 (3):676-692.
    TheK-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists aK-trivial degreedsuch that for all (...)
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  2.  27
    Index Sets and Parametric Reductions.Rod G. Downey & Michael R. Fellows - 2001 - Archive for Mathematical Logic 40 (5):329-348.
    We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities introduced by the authors. We solve a question of Peter Cholakand the first author by proving the fundamental index sets associated with a computable set A, {e : W e ≤ q u A} for q∈ {m, T} are Σ4 0 complete. We also show hat FPT(≤ q n ), that is {e : W e computable and ≡ q n ?}, is Σ4 (...)
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  3.  13
    On Δ 2 0 -Categoricity of Equivalence Relations.Rod Downey, Alexander G. Melnikov & Keng Meng Ng - 2015 - Annals of Pure and Applied Logic 166 (9):851-880.
  4.  7
    Difference Sets and Computability Theory.Rod Downey, Zoltán Füredi, Carl G. Jockusch & Lee A. Rubel - 1998 - Annals of Pure and Applied Logic 93 (1-3):63-72.
    For a set A of non-negative integers, let D be the set of non-negative differences of elements of A. Clearly, if A is computable, then D is computably enumerable. We show that every simple set which contains 0 is the difference set of some computable set and that every computably enumerable set is computably isomorphic to the difference set of some computable set. Also, we prove that there is a computable set which is the difference set of the complement of (...)
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  5.  51
    Computability, Enumerability, Unsolvability, Directions in Recursion Theory, Edited by S. B. Cooper, T. A. Slaman, and S. S. Wainer, London Mathematical Society Lecture Note Series, No. 224, Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1996, Vii + 347 Pp. - Leo Harrington and Robert I. Soare, Dynamic Properties of Computably Enumerable Sets, Pp. 105–121. - Eberhard Herrmann, On the ∀∃-Theory of the Factor Lattice by the Major Subset Relation, Pp. 139–166. - Manuel Lerman, Embeddings Into the Recursively Enumerable Degrees, Pp. 185–204. - Xiaoding Yi, Extension of Embeddings on the Recursively Enumerable Degrees Modulo the Cappable Degrees, Pp. 313–331. - André Nies, Relativization of Structures Arising From Computability Theory. Pp. 219–232. - Klaus Ambos-Spies, Resource-Bounded Genericity. Pp. 1–59. - Rod Downey, Carl G. Jockusch, and Michael Stob. Array Nonrecursive Degrees and Genericity, Pp. 93–104. - Masahiro Kumabe, Degrees of Generic Sets, Pp. 167–183. [REVIEW]C. T. Chong - 1999 - Journal of Symbolic Logic 64 (3):1362-1365.
  6.  63
    Calibrating Randomness.Rod Downey, Denis R. Hirschfeldt, André Nies & Sebastiaan A. Terwijn - 2006 - Bulletin of Symbolic Logic 12 (3):411-491.
    We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; (...)
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  7.  71
    Lowness and Π₂⁰ Nullsets.Rod Downey, Andre Nies, Rebecca Weber & Liang Yu - 2006 - Journal of Symbolic Logic 71 (3):1044-1052.
    We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.
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  8.  17
    On $\Pi^0_1$ Classes and Their Ranked Points.Rod Downey - 1991 - Notre Dame Journal of Formal Logic 32 (4):499-512.
  9.  74
    Totally Ω-Computably Enumerable Degrees and Bounding Critical Triples.Rod Downey, Noam Greenberg & Rebecca Weber - 2007 - Journal of Mathematical Logic 7 (2):145-171.
    We characterize the class of c.e. degrees that bound a critical triple as those degrees that compute a function that has no ω-c.e. approximation.
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  10.  7
    [Omnibus Review].Rod Downey - 1997 - Journal of Symbolic Logic 62 (3):1048-1055.
    Robert I. Soare, Automorphisms of the Lattice of Recursively Enumerable Sets. Part I: Maximal Sets.Manuel Lerman, Robert I. Soare, $d$-Simple Sets, Small Sets, and Degree Classes.Peter Cholak, Automorphisms of the Lattice of Recursively Enumerable Sets.Leo Harrington, Robert I. Soare, The $\Delta^0_3$-Automorphism Method and Noninvariant Classes of Degrees.
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  11.  22
    On Schnorr and Computable Randomness, Martingales, and Machines.Rod Downey, Evan Griffiths & Geoffrey Laforte - 2004 - Mathematical Logic Quarterly 50 (6):613-627.
    We examine the randomness and triviality of reals using notions arising from martingales and prefix-free machines.
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  12.  11
    Relativizing Chaitin's Halting Probability.Rod Downey, Denis R. Hirschfeldt, Joseph S. Miller & André Nies - 2005 - Journal of Mathematical Logic 5 (02):167-192.
    As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory (...)
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  13.  11
    Splitting Theorems in Recursion Theory.Rod Downey & Michael Stob - 1993 - Annals of Pure and Applied Logic 65 (1):1-106.
    A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results about (...)
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  14.  32
    A Δ20 Set with No Infinite Low Subset in Either It or its Complement.Rod Downey, Denis R. Hirschfeldt, Steffen Lempp & Reed Solomon - 2001 - Journal of Symbolic Logic 66 (3):1371-1381.
    We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and effective content of versions of Ramsey's Theorem. In particular, our result implies that every ω-model of RCA 0 + SRT 2 2 must contain a nonlow set.
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  15.  30
    Jumps of Hemimaximal Sets.Rod Downey & Mike Stob - 1991 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 37 (8):113-120.
  16.  21
    Jumps of Hemimaximal Sets.Rod Downey & Mike Stob - 1991 - Mathematical Logic Quarterly 37 (8):113-120.
  17.  10
    Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, 2010, Xxviii + 855 Pp. [REVIEW]Laurent Bienvenu - 2012 - Bulletin of Symbolic Logic 18 (1):126-128.
  18.  35
    R. G. Downey and M. R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, New York, Berlin, and Heidelberg, 1999, Xv + 533 Pp. [REVIEW]Jörg Flum - 2002 - Bulletin of Symbolic Logic 8 (4):528-529.
  19.  11
    Asymptotic Density and the Ershov Hierarchy.Rod Downey, Carl Jockusch, Timothy H. McNicholl & Paul Schupp - 2015 - Mathematical Logic Quarterly 61 (3):189-195.
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  20.  2
    Lattice Nonembeddings and Initial Segments of the Recursively Enumerable Degrees.Rod Downey - 1990 - Annals of Pure and Applied Logic 49 (2):97-119.
  21.  8
    On a Question of A. Retzlaff.Rod Downey - 1983 - Mathematical Logic Quarterly 29 (6):379-384.
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  22.  8
    A Hierarchy of Computably Enumerable Degrees.Rod Downey & Noam Greenberg - 2018 - Bulletin of Symbolic Logic 24 (1):53-89.
    We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees and the array noncomputable degrees. The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.
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  23.  31
    On the Complexity of the Successivity Relation in Computable Linear Orderings.Rod Downey, Steffen Lempp & Guohua Wu - 2010 - Journal of Mathematical Logic 10 (1):83-99.
    In this paper, we solve a long-standing open question, about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering [Formula: see text] 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 [Formula: see text]-isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is (...)
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  24.  47
    Degree Theoretic Definitions of the Low2 Recursively Enumerable Sets.Rod Downey & Richard A. Shore - 1995 - Journal of Symbolic Logic 60 (3):727 - 756.
  25.  22
    Friedberg Splittings of Recursively Enumerable Sets.Rod Downey & Michael Stob - 1993 - Annals of Pure and Applied Logic 59 (3):175-199.
    A splitting A1A2 = A of an r.e. set A is called a Friedberg splitting if for any r.e. set W with W — A not r.e., W — Ai≠0 for I = 1,2. In an earlier paper, the authors investigated Friedberg splittings of maximal sets and showed that they formed an orbit with very interesting degree-theoretical properties. In the present paper we continue our investigations, this time analyzing Friedberg splittings and in particular their orbits and degrees for various classes (...)
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  26.  10
    Every Recursive Boolean Algebra is Isomorphic to One with Incomplete Atoms.Rod Downey - 1993 - Annals of Pure and Applied Logic 60 (3):193-206.
    The theorem of the title is proven, solving an old question of Remmel. The method of proof uses an algebraic technique of Remmel-Vaught combined with a complex tree of strategies argument where the true path is needed to figure out the final isomorphism.
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  27.  20
    Effective Packing Dimension and Traceability.Rod Downey & Keng Meng Ng - 2010 - Notre Dame Journal of Formal Logic 51 (2):279-290.
    We study the Turing degrees which contain a real of effective packing dimension one. Downey and Greenberg showed that a c.e. degree has effective packing dimension one if and only if it is not c.e. traceable. In this paper, we show that this characterization fails in general. We construct a real $A\leq_T\emptyset''$ which is hyperimmune-free and not c.e. traceable such that every real $\alpha\leq_T A$ has effective packing dimension 0. We construct a real $B\leq_T\emptyset'$ which is not c.e. traceable (...)
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  28.  3
    There is No Fat Orbit.Rod Downey & Leo Harrington - 1996 - Annals of Pure and Applied Logic 80 (3):277-289.
    We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.
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  29.  6
    Correction to “Undecidability of L(F∞) and Other Lattices of R.E. Substructures”.Rod Downey - 1990 - Annals of Pure and Applied Logic 48 (3):299-301.
  30.  16
    A Contiguous Nonbranching Degree.Rod Downey - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (4):375-383.
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  31.  10
    A Contiguous Nonbranching Degree.Rod Downey - 1989 - Mathematical Logic Quarterly 35 (4):375-383.
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  32.  30
    Arithmetical Sacks Forcing.Rod Downey & Liang Yu - 2006 - Archive for Mathematical Logic 45 (6):715-720.
    We answer a question of Jockusch by constructing a hyperimmune-free minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented.
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  33.  4
    The Upward Closure of a Perfect Thin Class.Rod Downey, Noam Greenberg & Joseph S. Miller - 2008 - Annals of Pure and Applied Logic 156 (1):51-58.
    There is a perfect thin class whose upward closure in the Turing degrees has full measure . Thus, in the Muchnik lattice of classes, the degree of 2-random reals is comparable with the degree of some perfect thin class. This solves a question of Simpson [S. Simpson, Mass problems and randomness, Bulletin of Symbolic Logic 11 1–27].
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  34.  17
    On Co-Simple Isols and Their Intersection Types.Rod Downey & Theodore A. Slaman - 1992 - Annals of Pure and Applied Logic 56 (1-3):221-237.
    We solve a question of McLaughlin by showing that if A is a regressive co-simple isol, there is a co-simple regressive isol B such that the intersection type of A and B is trivial. The proof is a nonuniform 0 priority argument that can be viewed as the execution of a single strategy from a 0-argument. We establish some limit on the properties of such pairs by showing that if AxB has low degree, then the intersection type of A and (...)
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  35.  17
    Review: R. G. Downey, M. R. Fellows, Parameterized Complexity. [REVIEW]Jörg Flum - 2002 - Bulletin of Symbolic Logic 8 (4):528-529.
  36.  40
    Limits on Jump Inversion for Strong Reducibilities.Barbara F. Csima, Rod Downey & Keng Meng Ng - 2011 - Journal of Symbolic Logic 76 (4):1287-1296.
    We show that Sacks' and Shoenfield's analogs of jump inversion fail for both tt- and wtt-reducibilities in a strong way. In particular we show that there is a ${\mathrm{\Delta }}_{2}^{0}$ set B > tt ∅′ such that there is no c.e. set A with A′ ≡ wtt B. We also show that there is a ${\mathrm{\Sigma }}_{2}^{0}$ set C > tt ∅′ such that there is no ${\mathrm{\Delta }}_{2}^{0}$ set D with D′ ≡ wtt C.
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  37.  11
    Characterizing Lowness for Demuth Randomness.Laurent Bienvenu, Rod Downey, Noam Greenberg, André Nies & Dan Turetsky - 2014 - Journal of Symbolic Logic 79 (2):526-560.
    We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15]. We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.
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  38.  29
    Some Orbits for E.Peter Cholak, Rod Downey & Eberhard Herrmann - 2001 - Annals of Pure and Applied Logic 107 (1-3):193-226.
    In this article we establish the existence of a number of new orbits in the automorphism group of the computably enumerable sets. The degree theoretical aspects of these orbits also are examined.
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  39.  25
    On the Cantor-Bendixon Rank of Recursively Enumerable Sets.Peter Cholak & Rod Downey - 1993 - Journal of Symbolic Logic 58 (2):629-640.
    The main result of this paper is to show that for every recursive ordinal α ≠ 0 and for every nonrecursive r.e. degree d there is a r.e. set of rank α and degree d.
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  40.  18
    Schnorr Randomness.Rodney G. Downey & Evan J. Griffiths - 2004 - Journal of Symbolic Logic 69 (2):533 - 554.
    Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new (...)
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  41.  18
    Jump Inversions Inside Effectively Closed Sets and Applications to Randomness.George Barmpalias, Rod Downey & Keng Meng Ng - 2011 - Journal of Symbolic Logic 76 (2):491 - 518.
    We study inversions of the jump operator on ${\mathrm{\Pi }}_{1}^{0}$ classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0′ sets which are not 2-random. Both of the classes coincide with the degrees above 0′ which are not 0′-dominated. A (...)
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  42.  12
    Lattice Nonembeddings and Intervals of the Recursively Enumerable Degrees.Peter Cholak & Rod Downey - 1993 - Annals of Pure and Applied Logic 61 (3):195-221.
    Let b and c be r.e. Turing degrees such that b>c. We show that there is an r.e. degree a such that b>a>c and all lattices containing a critical triple, including the lattice M5, cannot be embedded into the interval [c, a].
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  43.  18
    On the Universal Splitting Property.Rod Downey - 1997 - Mathematical Logic Quarterly 43 (3):311-320.
    We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low2. This solves a question from Ambos-Spies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed.
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  44.  42
    There Are No Maximal Low D.C.E. Degrees.Liang Yu & Rod Downey - 2004 - Notre Dame Journal of Formal Logic 45 (3):147-159.
  45.  21
    Asymptotic Density and Computably Enumerable Sets.Rodney G. Downey, Carl G. Jockusch & Paul E. Schupp - 2013 - Journal of Mathematical Logic 13 (2):1350005.
    We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: The degrees of such sets A are precisely the nonlow c.e. degrees. There is a c.e. set A of density 1 with no computable subset of nonzero density. There is a (...)
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  46.  5
    A $Delta^02$ Set with Barely $Sigma^02$ Degree.Rod Downey, Geoffrey Laforte & Steffen Lempp - 1999 - Journal of Symbolic Logic 64 (4):1700-1718.
    We construct a $\Delta^0_2$ degree which fails to be computably enumerable in any computably enumerable set strictly below $\emptyset'$.
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  47.  3
    Avoiding Effective Packing Dimension 1 Below Array Noncomputable C.E. Degrees.Rod Downey & Jonathan Stephenson - 2018 - Journal of Symbolic Logic 83 (2):717-739.
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  48.  20
    A Δ02 Set with Barely Σ02 Degree.Rod Downey, Geoffrey Laforte & Steffen Lempp - 1999 - Journal of Symbolic Logic 64 (4):1700 - 1718.
    We construct a Δ 0 2 degree which fails to be computably enumerable in any computably enumerable set strictly below $\emptyset'$.
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  49.  4
    A Set with Barely Degree.Rod Downey, Geoffrey Laforte & Steffen Lempp - 1999 - Journal of Symbolic Logic 64 (4):1700-1718.
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  50.  7
    Bounded Fixed-Parameter Tractability and Reducibility.Rod Downey, Jörg Flum, Martin Grohe & Mark Weyer - 2007 - Annals of Pure and Applied Logic 148 (1):1-19.
    We study a refined framework of parameterized complexity theory where the parameter dependence of fixed-parameter tractable algorithms is not arbitrary, but restricted by a function in some family . For every family of functions, this yields a notion of -fixed-parameter tractability. If is the class of all polynomially bounded functions, then -fixed-parameter tractability coincides with polynomial time decidability and if is the class of all computable functions, -fixed-parameter tractability coincides with the standard notion of fixed-parameter tractability. There are interesting choices (...)
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