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  1. added 2018-11-26
    Defining a Halting Decidability Decider.Pete Olcott - manuscript
    In this paper we show how to define a halting decidability decider that rejects all finite string Turing machine descriptions that would otherwise make halting undecidable. All of the conventional halting problem proof counter-examples would be rejected on the basis that they specify an infinitely recursive evaluation sequence thus are malformed expressions of the language of Turing Machine descriptions.
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  2. added 2018-08-18
    On the Question of Whether the Mind Can Be Mechanized, I: From Gödel to Penrose.Peter Koellner - 2018 - Journal of Philosophy 115 (7):337-360.
    In this paper I address the question of whether the incompleteness theorems imply that “the mind cannot be mechanized,” where this is understood in the specific sense that “the mathematical outputs of the idealized human mind do not coincide with the mathematical outputs of any idealized finite machine.” Gödel argued that his incompleteness theorems implied a weaker, disjunctive conclusion to the effect that either “the mind cannot be mechanized” or “mathematical truth outstrips the idealized human mind.” Others, most notably, Lucas (...)
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  3. added 2018-05-30
    Solovay's Theorem Cannot Be Simplified.Andrew Arana - 2001 - Annals of Pure and Applied Logic 112 (1):27-41.
    In this paper we consider three potential simplifications to a result of Solovay’s concerning the Turing degrees of nonstandard models of arbitrary completions of first-order Peano Arithmetic (PA). Solovay characterized the degrees of nonstandard models of completions T of PA, showing that they are the degrees of sets X such that there is an enumeration R ≤T X of an “appropriate” Scott set and there is a family of functions (tn)n∈ω, ∆0 n(X) uniformly in n, such that lim tn(s) s→∞.
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  4. added 2018-02-26
    Halting Problem Proof From Finite Strings to Final States.Pete Olcott - manuscript
    If there truly is a proof that shows that no universal halt decider exists on the basis that certain tuples: (H, Wm, W) are undecidable, then this very same proof (implemented as a Turing machine) could be used by H to reject some of its inputs. When-so-ever the hypothetical halt decider cannot derive a formal proof from its input strings and initial state to final states corresponding the mathematical logic functions of Halts(Wm, W) or Loops(Wm, W), halting undecidability has been (...)
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  5. added 2018-02-20
    Defining a Decidability Decider for the Halting Problem.Pete Olcott - manuscript
    When we understand that every potential halt decider must derive a formal mathematical proof from its inputs to its final states previously undiscovered semantic details emerge. -/- When-so-ever the potential halt decider cannot derive a formal proof from its input strings to its final states of Halts or Loops, undecidability has been decided. -/- The formal proof involves tracing the sequence of state transitions of the input TMD as syntactic logical consequence inference steps in the formal language of Turing Machine (...)
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  6. added 2017-10-17
    On Sets X \Subseteq \Mathbb{N} for Which We Know an Algorithm That Computes a Threshold Number T(X) \in \Mathbb{N} Such That X is Infinite If and Only If X Contains an Element Greater Than T(X).Apoloniusz Tyszka - manuscript
    Let \Gamma_{n} denote (k-1)!, where n \in {3,...,16} and k \in {2} \cup [2^{2^{n-3}}+1,\infty) \cap N. For an integer n \in {3,...,16}, let \Sigma_n denote the following statement: if a system of equations S \subseteq {\Gamma_{n}(x_i)=x_k: i,k \in {1,...,n}} \cup {x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} with Gamma instead of \Gamma_n has only finitely many solutions in positive integers x_1,...,x_n, then every tuple (x_1,...,x_n) \in (N\{0})^n that solves the original system S satisfies x_1,...,x_n \leq 2^{2^{n-2}}. Our hypothesis claims that the (...)
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  7. added 2017-08-31
    Deviant Encodings and Turing’s Analysis of Computability.B. Jack Copeland & Diane Proudfoot - 2010 - Studies in History and Philosophy of Science Part A 41 (3):247-252.
    Turing’s analysis of computability has recently been challenged; it is claimed that it is circular to analyse the intuitive concept of numerical computability in terms of the Turing machine. This claim threatens the view, canonical in mathematics and cognitive science, that the concept of a systematic procedure or algorithm is to be explicated by reference to the capacities of Turing machines. We defend Turing’s analysis against the challenge of ‘deviant encodings’.Keywords: Systematic procedure; Turing machine; Church–Turing thesis; Deviant encoding; Acceptable encoding; (...)
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  8. added 2017-07-30
    Intractability and the Use of Heuristics in Psychological Explanations.Iris Rooij, Cory Wright & Todd Wareham - 2012 - Synthese 187 (2):471-487.
  9. added 2017-02-15
    Resolving the Infinitude Controversy.András Kornai - 2014 - Journal of Logic, Language and Information 23 (4):481-492.
    A simple inductive argument shows natural languages to have infinitly many sentences, but workers in the field have uncovered clear evidence of a diverse group of ‘exceptional’ languages from Proto-Uralic to Dyirbal and most recently, Pirahã, that appear to lack recursive devices entirely. We argue that in an information-theoretic setting non-recursive natural languages appear neither exceptional nor functionally inferior to the recursive majority.
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  10. added 2017-02-14
    Expression de la Recursion Primitive Dans le Calcul-XK.J. Ladriere - forthcoming - Logique Et Analyse.
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  11. added 2017-02-14
    The Impact of Starting Small on the Learnability of Recursion.Jun Lai & Fenna H. Poletiek - 2010 - In S. Ohlsson & R. Catrambone (eds.), Proceedings of the 32nd Annual Conference of the Cognitive Science Society. Cognitive Science Society.
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  12. added 2017-02-14
    SEarch Via Recursive Rejection (SERR).H. J. Müller, G. W. Humphreys & A. C. Olson - 1998 - In Richard D. Wright (ed.), Visual Attention. Oxford University Press. pp. 8--389.
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  13. added 2017-02-14
    The Contribution of Polish Logicians to Recursion Theory.Roman Murawski - 1998 - In Katarzyna Kijania-Placek & Jan Woleński (eds.), The Lvov-Warsaw School and Contemporary Philosophy. Kluwer Academic Publishers. pp. 265--282.
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  14. added 2017-02-13
    Geeks and Recursive Publics.Christopher Kelty - 2012 - In Christian Emden & David R. Midgley (eds.), Beyond Habermas: Democracy, Knowledge, and the Public Sphere. Berghahn Books. pp. 99.
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  15. added 2017-02-13
    How Much Randomness is Needed for Statistics?Bjørn Kjos-Hanssen, Antoine Taveneaux & Neil Thapen - 2012 - In S. Barry Cooper (ed.), Annals of Pure and Applied Logic. pp. 395--404.
    In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure λ, a choice needs to be made. One approach is to allow randomness tests to access the measure λ as an oracle . The other approach is the opposite one, where the randomness tests are completely effective and do not have access to the information contained in λ . While the Hippocratic approach is in general much more restrictive, there are cases where the (...)
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  16. added 2017-02-13
    Construction Theory, Self-Replication, and the Halting Problem.Hiroki Sayama - 2008 - Complexity 13 (5):16-22.
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  17. added 2017-02-13
    Computability and Computational Complexity.Patrick Doyle - 2003 - In L. Nadel (ed.), Encyclopedia of Cognitive Science. Nature Publishing Group.
  18. added 2017-02-13
    Managing Complexity by Recursion.Bernd Schiemenz - 2002 - In Robert Trappl (ed.), Cybernetics and Systems. Austrian Society for Cybernetics Studies. pp. 475--479.
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  19. added 2017-02-13
    The Continuous Functionals; Computations, Recursions and Degrees.Dag Normann - 1981 - Annals of Mathematical Logic 21 (1):1-26.
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  20. added 2017-02-13
    Recursion Theory on Algebraic Structures with Independent Sets.J. B. Remmel - 1980 - Annals of Mathematical Logic 18 (2):153-191.
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  21. added 2017-02-13
    Degrees of Functionals.Dag Normann - 1979 - Annals of Mathematical Logic 16 (3):269-304.
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  22. added 2017-02-13
    Filter Spaces and Continuous Functionals.J. M. E. Hyland - 1979 - Annals of Mathematical Logic 16 (2):101-143.
  23. added 2017-02-13
    Quelques Procedes de Definition En Topologffi Recursive.Daniel Lacombe - 1959 - In A. Heyting (ed.), Constructivity in Mathematics. Amsterdam: North-Holland Pub. Co.. pp. 24--129.
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  24. added 2017-02-13
    Quelques Procédés de Définition En Topologie Récursive.Daniel Lacombe - 1959 - In A. Heyting (ed.), Journal of Symbolic Logic. Amsterdam: North-Holland Pub. Co.. pp. 129--158.
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  25. added 2017-02-12
    Algorithmic Randomness and Measures of Complexity.George Barmpalias - forthcoming - Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.
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  26. added 2017-02-12
    Truth-Table Schnorr Randomness and Truth-Table Reducible Randomness.Kenshi Miyabe - 2011 - Mathematical Logic Quarterly 57 (3):323-338.
    Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's (...)
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  27. added 2017-02-12
    Higher Kurtz Randomness.Bjørn Kjos-Hanssen, André Nies, Frank Stephan & Liang Yu - 2010 - Annals of Pure and Applied Logic 161 (10):1280-1290.
    A real x is -Kurtz random if it is in no closed null set . We show that there is a cone of -Kurtz random hyperdegrees. We characterize lowness for -Kurtz randomness as being -dominated and -semi-traceable.
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  28. added 2017-02-12
    Ordinal Machines and Admissible Recursion Theory.Peter Koepke & Benjamin Seyfferth - 2009 - Annals of Pure and Applied Logic 160 (3):310-318.
    We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the (...)
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  29. added 2017-02-12
    Constructive Equivalence Relations on Computable Probability Measures.Laurent Bienvenu & Wolfgang Merkle - 2009 - Annals of Pure and Applied Logic 160 (3):238-254.
    A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences. In what follows, (...)
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  30. added 2017-02-12
    Effectively Closed Sets of Measures and Randomness.Jan Reimann - 2008 - Annals of Pure and Applied Logic 156 (1):170-182.
    We show that if a real x2ω is strongly Hausdorff -random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective continuous transformations and a basis theorem for -classes applied to closed sets of probability measures. We use the main result to derive a (...)
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  31. added 2017-02-12
    Connectivity Properties of Dimension Level Sets.Jack H. Lutz & Klaus Weihrauch - 2008 - Mathematical Logic Quarterly 54 (5):483-491.
    This paper initiates the study of sets in Euclidean spaces ℝn that are defined in terms of the dimensions of their elements. Specifically, given an interval I ⊆ [0, n ], we are interested in the connectivity properties of the set DIMI, consisting of all points in ℝn whose dimensions lie in I, and of its dual DIMIstr, consisting of all points whose strong dimensions lie in I. If I is [0, 1) or.
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  32. added 2017-02-12
    Uniform Domain Representations of "Lp" -Spaces.Petter K. Køber - 2007 - Mathematical Logic Quarterly 53 (2):180-205.
    The category of Scott-domains gives a computability theory for possibly uncountable topological spaces, via representations. In particular, every separable Banach-space is representable over a separable domain. A large class of topological spaces, including all Banach-spaces, is representable by domains, and in domain theory, there is a well-understood notion of parametrizations over a domain. We explore the link with parameter-dependent collections of spaces in e. g. functional analysis through a case study of "Lp" -spaces. We show that a well-known domain representation (...)
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  33. added 2017-02-12
    Randomness and the Linear Degrees of Computability.Andrew Em Lewis & George Barmpalias - 2007 - Annals of Pure and Applied Logic 145 (3):252-257.
    We show that there exists a real α such that, for all reals β, if α is linear reducible to β then β≤Tα. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no ℓ-complete Δ2 real. Upon realizing that quasi-maximality does not characterize the random reals–there exist reals which are not random but which are of quasi-maximal ℓ-degree–it is then natural to ask whether maximality could provide such a characterization. Such hopes, (...)
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  34. added 2017-02-12
    Kolmogorov–Loveland Randomness and Stochasticity.Wolfgang Merkle, Joseph S. Miller, André Nies, Jan Reimann & Frank Stephan - 2006 - Annals of Pure and Applied Logic 138 (1):183-210.
    An infinite binary sequence X is Kolmogorov–Loveland random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KL-stochastic if there is no computable non-monotonic selection rule that selects from X an infinite, biased sequence.One of the major open problems in the field of effective randomness is whether Martin-Löf randomness is the same as KL-randomness. Our first (...)
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  35. added 2017-02-12
    A Blend of Methods of Recursion Theory and Topology: A Π1 0 Tree of Shadow Points. [REVIEW]Iraj Kalantari & Larry Welch - 2004 - Archive for Mathematical Logic 43 (8):991-1008.
    This paper is a sequel to our [7]. In that paper we constructed a Π1 0 tree of avoidable points. Here we construct a Π1 0 tree of shadow points. This tree is a tree of sharp filters, where a sharp filter is a nested sequence of basic open sets converging to a point. In the construction we assign to each basic open set on the tree an address in 2<ω. One interesting fact is that while our Π1 0 tree (...)
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  36. added 2017-02-12
    Quasimaximality and Principal Filters Isomorphism Between.Rumen Dimitrov - 2004 - Archive for Mathematical Logic 43 (3):415-424.
    Let I be a quasimaximal subset of a computable basis of the fully efective vector space V ∞ . We give a necessary and sufficient condition for the existence of an isomorphism between the principal filter respectivelly. We construct both quasimaximal sets that satisfy and quasimaximal sets that do not satisfy this condition. With the latter we obtain a negative answer to Question 5.4 posed by Downey and Remmel in [3].
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  37. added 2017-02-12
    On the Complexity of Finding Paths in a Two‐Dimensional Domain I: Shortest Paths.Arthur W. Chou & Ker-I. Ko - 2004 - Mathematical Logic Quarterly 50 (6):551-572.
    The computational complexity of finding a shortest path in a two-dimensional domain is studied in the Turing machine-based computational model and in the discrete complexity theory. This problem is studied with respect to two formulations of polynomial-time computable two-dimensional domains: domains with polynomialtime computable boundaries, and polynomial-time recognizable domains with polynomial-time computable distance functions. It is proved that the shortest path problem has the polynomial-space upper bound for domains of both type and type ; and it has a polynomial-space lower (...)
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  38. added 2017-02-12
    Effective Embeddings Into Strong Degree Structures.Timothy H. McNicholl - 2003 - Mathematical Logic Quarterly 49 (3):219.
    We show that any partial order with a Σ3 enumeration can be effectively embedded into any partial order obtained by imposing a strong reducibility such as ≤tt on the c. e. sets. As a consequence, we obtain that the partial orders that result from imposing a strong reducibility on the sets in a level of the Ershov hiearchy below ω + 1 are co-embeddable.
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  39. added 2017-02-12
    Effectivity in Spaces with Admissible Multirepresentations.Matthias Schröder - 2002 - Mathematical Logic Quarterly 48 (S1):78-90.
    The property of admissibility of representations plays an important role in Type–2 Theory of Effectivity . TTE defines computability on sets with continuum cardinality via representations. Admissibility is known to be indispensable for guaranteeing reasonable effectivity properties of the used representations.The question arises whether every function that is computable with respect to arbritrary representations is also computable with respect to closely related admissible ones. We define three operators which transform representations into admissible ones in such a way that relative computability (...)
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  40. added 2017-02-12
    Non-Definability of the Ackermann Function with Type 1 Partial Primitive Recursion.Karl-Heinz Niggl - 1997 - Archive for Mathematical Logic 37 (1):1-13.
    The paper builds on a simply typed term system ${\cal PR}^\omega $ providing a notion of partial primitive recursive functional on arbitrary Scott domains $D_\sigma$ that includes a suitable concept of parallelism. Computability on the partial continuous functionals seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (SCVR) is not reducible to partial primitive recursion. So an extension ${\cal PR}^{\omega e}$ is studied that is closed under SCVR and yet stays within the realm of subrecursiveness. The twist (...)
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  41. added 2017-02-12
    The Realm of Primitive Recursion.Harold Simmons - 1988 - Archive for Mathematical Logic 27 (2):177-188.
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  42. added 2017-02-12
    Negative Results on the Reduction of the Recursion Scheme.Benedetto Intrigila - 1988 - Mathematical Logic Quarterly 34 (4):297-300.
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  43. added 2017-02-12
    Topological Size of Sets of Partial Recursive Functions.Cristian Calude - 1982 - Mathematical Logic Quarterly 28 (27‐32):455-462.
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  44. added 2017-02-12
    Inductive Inference and Computable One‐One Numberings.Rsinš Freivalds, Efim B. Kinber & Rolf Wiehagen - 1982 - Mathematical Logic Quarterly 28 (27‐32):463-479.
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  45. added 2017-02-12
    Iteration of Primitive Recursion.Paul Axt - 1965 - Mathematical Logic Quarterly 11 (3):253-255.
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  46. added 2017-02-12
    Creative Functions.J. P. Cleave - 1961 - Mathematical Logic Quarterly 7 (11‐14):205-212.
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  47. added 2017-02-11
    Reviewed Work(S): Lowness Properties and Randomness. Advances in Mathematics, Vol. 197 by André Nies; Lowness for the Class of Schnorr Random Reals. SIAM Journal on Computing, Vol. 35 by Bjørn Kjos-Hanssen; André Nies; Frank Stephan; Lowness for Kurtz Randomness. The Journal of Symbolic Logic, Vol. 74 by Noam Greenberg; Joseph S. Miller; Randomness and Lowness Notions Via Open Covers. Annals of Pure and Applied Logic, Vol. 163 by Laurent Bienvenu; Joseph S. Miller; Relativizations of Randomness and Genericity Notions. The Bulletin of the London Mathematical Society, Vol. 43 by Johanna N. Y. Franklin; Frank Stephan; Liang Yu; Randomness Notions and Partial Relativization. Israel Journal of Mathematics, Vol. 191 by George Barmpalias; Joseph S. Miller; André Nies. [REVIEW]Johanna N. Y. Franklin - forthcoming - Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    Review by: Johanna N. Y. Franklin The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 115-118, March 2013.
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  48. added 2017-02-11
    Evolutionary Scenarios for the Emergence of Recursion.Lluís Barceló-Coblijn - forthcoming - Theoria Et Historia Scientiarum 9:171-199.
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  49. added 2017-02-11
    Conference on Computability, Complexity and Randomness: Isaac Newton Institute, Cambridge, Uk July 2-6, 2012.Elvira Mayordomo & Wolfgang Merkle - forthcoming - Association for Symbolic Logic: The Bulletin of Symbolic Logic.
    Elvira Mayordomo and Wolfgang Merkle The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 135-136, March 2013.
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  50. added 2017-02-11
    Randomness and Lowness Notions Via Open Covers.Laurent Bienvenu & Joseph S. Miller - 2012 - Annals of Pure and Applied Logic 163 (5):506-518.
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