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Rod Downey [70]Rodney G. Downey [20]Rodney Downey [10]Rod G. Downey [2]
Rod& Nies Downey [1]Rodney& Griffiths Downey [1]
  1.  24
    Calibrating Randomness.Rod Downey, Denis R. Hirschfeldt, André Nies & Sebastiaan A. Terwijn - 2006 - Bulletin of Symbolic Logic 12 (3):411-491.
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  2. Bases of Supermaximal Subspaces and Steinitz Systems. I.Rod Downey - 1984 - Journal of Symbolic Logic 49 (4):1146-1159.
  3.  5
    Relativizing Chaitin's Halting Probability.Rod Downey, Denis R. Hirschfeldt, Joseph S. Miller & André Nies - 2005 - Journal of Mathematical Logic 5 (02):167-192.
  4.  3
    Asymptotic Density and Computably Enumerable Sets.Rodney G. Downey, Carl G. Jockusch & Paul E. Schupp - 2013 - Journal of Mathematical Logic 13 (2):1350005.
  5.  15
    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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  6.  45
    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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  7.  2
    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.
  8. Lattice Nonembeddings and Initial Segments of the Recursively Enumerable Degrees.Rod Downey - 1990 - Annals of Pure and Applied Logic 49 (2):97-119.
  9. Decomposition and Infima in the Computably Enumerable Degrees.Rodney G. Downey, Geoffrey L. Laforte & Richard A. Shore - 2003 - Journal of Symbolic Logic 68 (2):551-579.
    Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a ⋃ c) ⋂ (b ⋃ c), a ⋃ c | b ⋃ c, and c < a ⋃ b.
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  10.  1
    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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  11.  15
    Jumps of Hemimaximal Sets.Rod Downey & Mike Stob - 1991 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 37 (8):113-120.
  12.  37
    Totally Ω-Computably Enumerable Degrees and Bounding Critical Triples.Rod Downey, Noam Greenberg & Rebecca Weber - 2007 - Journal of Mathematical Logic 7 (2):145-171.
  13.  12
    Decidability and Computability of Certain Torsion-Free Abelian Groups.Rodney G. Downey, Sergei S. Goncharov, Asher M. Kach, Julia F. Knight, Oleg V. Kudinov, Alexander G. Melnikov & Daniel Turetsky - 2010 - Notre Dame Journal of Formal Logic 51 (1):85-96.
    We study completely decomposable torsion-free abelian groups of the form $\mathcal{G}_S := \oplus_{n \in S} \mathbb{Q}_{p_n}$ for sets $S \subseteq \omega$. We show that $\mathcal{G}_S$has a decidable copy if and only if S is $\Sigma^0_2$and has a computable copy if and only if S is $\Sigma^0_3$.
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  14. Characterizing Lowness for Demuth Randomness.Laurent Bienvenu, Rod Downey, Noam Greenberg, André Nies & Dan Turetsky - 2014 - Journal of Symbolic Logic 79 (2):526-560.
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  15.  6
    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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  16.  6
    Highness and Bounding Minimal Pairs.Rodney G. Downey, Steffen Lempp & Richard A. Shore - 1993 - Mathematical Logic Quarterly 39 (1):475-491.
  17.  11
    On $\Pi^0_1$ Classes and Their Ranked Points.Rod Downey - 1991 - Notre Dame Journal of Formal Logic 32 (4):499-512.
  18.  10
    A Δ02 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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  19.  1
    Jumps of Hemimaximal Sets.Rod Downey & Mike Stob - 1991 - Mathematical Logic Quarterly 37 (8):113-120.
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  20.  2
    On a Question of A. Retzlaff.Rod Downey - 1983 - Mathematical Logic Quarterly 29 (6):379-384.
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  21.  3
    Countable Thin Π01 Classes.Douglas Cenzer, Rodney Downey, Carl Jockusch & Richard A. Shore - 1993 - Annals of Pure and Applied Logic 59 (2):79-139.
    Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that no (...)
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  22. Computability Theory and Linear Orders.Rod Downey - 1998 - In I͡Uriĭ Leonidovich Ershov (ed.), Handbook of Recursive Mathematics. Elsevier. pp. 138--823.
     
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  23.  20
    Degree Theoretic Definitions of the Low2 Recursively Enumerable Sets.Rod Downey & Richard A. Shore - 1995 - Journal of Symbolic Logic 60 (3):727 - 756.
  24.  15
    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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  25.  5
    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.  3
    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.  23
    Contiguity and Distributivity in the Enumerable Turing Degrees.Rodney G. Downey & Steffen Lempp - 1997 - Journal of Symbolic Logic 62 (4):1215-1240.
    We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. This settles a twenty-year old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no m-topped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.
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  28.  8
    On the Complexity of the Successivity Relation in Computable Linear Orderings.Rod Downey, Steffen Lempp & Guohua Wu - 2010 - Journal of Mathematical Logic 10 (01n02):83-99.
  29.  13
    The Complexity of Orbits of Computably Enumerable Sets.Peter A. Cholak, Rodney Downey & Leo A. Harrington - 2008 - Bulletin of Symbolic Logic 14 (1):69 - 87.
    The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is ${\Sigma _1^1 }$ -complete. This result and proof have a number of nice corollaries: the Scott rank of ε is $\omega _1^{{\rm{CK}}}$ + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly $\Delta (...)
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  30.  6
    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 such (...)
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  31.  3
    Fixed-Parameter Tractability and Completeness IV: On Completeness for W[P] and PSPACE Analogues.Karl A. Abrahamson, Rodney G. Downey & Michael R. Fellows - 1995 - Annals of Pure and Applied Logic 73 (3):235-276.
    We describe new results in parametrized complexity theory. In particular, we prove a number of concrete hardness results for W[P], the top level of the hardness hierarchy introduced by Downey and Fellows in a series of earlier papers. We also study the parametrized complexity of analogues of PSPACE via certain natural problems concerning k-move games. Finally, we examine several aspects of the structural complexity of W [P] and related classes. For instance, we show that W[P] can be characterized in terms (...)
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  32. 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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  33.  2
    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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  34.  14
    A Contiguous Nonbranching Degree.Rod Downey - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (4):375-383.
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  35.  14
    On Choice Sets and Strongly Non-Trivial Self-Embeddings of Recursive Linear Orders.Rodney G. Downey & Michael F. Moses - 1989 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 35 (3):237-246.
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  36.  3
    On Self-Embeddings of Computable Linear Orderings.Rodney G. Downey, Carl Jockusch & Joseph S. Miller - 2006 - Annals of Pure and Applied Logic 138 (1):52-76.
    The Dushnik–Miller Theorem states that every infinite countable linear ordering has a nontrivial self-embedding. We examine computability-theoretical aspects of this classical theorem.
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  37.  9
    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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  38.  18
    Embedding Lattices Into the Wtt-Degrees Below 0'.Rod Downey & Christine Haught - 1994 - Journal of Symbolic Logic 59 (4):1360-1382.
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  39.  27
    On Computable Self-Embeddings of Computable Linear Orderings.Rodney G. Downey, Bart Kastermans & Steffen Lempp - 2009 - Journal of Symbolic Logic 74 (4):1352 - 1366.
    We solve a longstanding question of Rosenstein, and make progress toward solving a longstanding open problem in the area of computable linear orderings by showing that every computable ƞ-like linear ordering without an infinite strongly ƞ-like interval has a computable copy without nontrivial computable self-embedding. The precise characterization of those computable linear orderings which have computable copies without nontrivial computable self-embedding remains open.
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  40.  15
    Degrees of D. C. E. Reals.Rod Downey, Guohua Wu & Xizhong Zheng - 2004 - Mathematical Logic Quarterly 50 (45):345-350.
    A real α is called a c. e. real if it is the halting probability of a prefix free Turing machine. Equivalently, α is c. e. if it is left computable in the sense that L = {q ∈ ℚ : q ≤ α} is a computably enumerable set. The natural field formed by the c. e. reals turns out to be the field formed by the collection of the d. c. e. reals, which are of the form α—β, where (...)
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  41.  1
    The Kolmogorov Complexity of Random Reals.Liang Yu, Decheng Ding & Rodney Downey - 2004 - Annals of Pure and Applied Logic 129 (1-3):163-180.
    We investigate the initial segment complexity of random reals. Let K denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K and K. It is well-known that a real α is 1-random iff there is a constant c such that for all n, Kn−c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour of K (...)
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  42.  1
    Correction to “Undecidability of L and Other Lattices of R.E. Substructures”.Rod Downey - 1990 - Annals of Pure and Applied Logic 48 (3):299-301.
  43.  21
    Space Complexity of Abelian Groups.Douglas Cenzer, Rodney G. Downey, Jeffrey B. Remmel & Zia Uddin - 2009 - Archive for Mathematical Logic 48 (1):115-140.
    We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups ${\mathbb {Z}, Z(p^{\infty})}$ , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. For (...)
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  44.  6
    Abelian P -Groups and the Halting Problem.Rodney Downey, Alexander G. Melnikov & Keng Meng Ng - 2016 - Annals of Pure and Applied Logic 167 (11):1123-1138.
  45.  15
    On the Parameterized Complexity of Short Computation and Factorization.Liming Cai, Jianer Chen, Rodney G. Downey & Michael R. Fellows - 1997 - Archive for Mathematical Logic 36 (4-5):321-337.
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  46.  20
    Every 1-Generic Computes a Properly 1-Generic.Barbara F. Csima, Rod Downey, Noam Greenberg, Denis R. Hirschfeldt & Joseph S. Miller - 2006 - Journal of Symbolic Logic 71 (4):1385 - 1393.
    A real is called properly n-generic if it is n-generic but not n+1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.
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  47.  14
    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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  48.  13
    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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  49.  6
    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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  50.  6
    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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