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André Nies [49]Andrée Nies [1]
  1. Structural Properties and Σ20 Enumeration Degrees.André Nies & Andrea Sorbi - 2000 - Journal of Symbolic Logic 65 (1):285-292.
    We prove that each Σ 0 2 set which is hypersimple relative to $\emptyset$ ' is noncuppable in the structure of the Σ 0 2 enumeration degrees. This gives a connection between properties of Σ 0 2 sets under inclusion and and the Σ 0 2 enumeration degrees. We also prove that some low non-computably enumerable enumeration degree contains no set which is simple relative to $\emptyset$ '.
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  2. Computability and Randomness.André Nies - 2008 - Oxford University Press UK.
    The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts (...)
     
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  3.  62
    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]; higher level randomness notions (...)
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  4.  41
    Randomness and Computability: Open Questions.Joseph S. Miller & André Nies - 2006 - Bulletin of Symbolic Logic 12 (3):390-410.
  5.  25
    Lowness Properties and Approximations of the Jump.Santiago Figueira, André Nies & Frank Stephan - 2008 - Annals of Pure and Applied Logic 152 (1):51-66.
    We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informally, a set A is strongly jump-traceable if for each order function h, for each input e one may effectively enumerate a set Te of possible values for the jump JA, and the number of values enumerated is at most h. A′ (...)
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  6.  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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  7.  32
    Demuth Randomness and Computational Complexity.Antonín Kučera & André Nies - 2011 - Annals of Pure and Applied Logic 162 (7):504-513.
    Demuth tests generalize Martin-Löf tests in that one can exchange the m-th component a computably bounded number of times. A set fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that GmGm+1 for each m, we have weak Demuth randomness.We show that a weakly Demuth random set can be high and , yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable.We (...)
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  8.  28
    Benign Cost Functions and Lowness Properties.Noam Greenberg & André Nies - 2011 - Journal of Symbolic Logic 76 (1):289 - 312.
    We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost functions, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of (...)
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  9.  26
    Computable Models of Theories with Few Models.Bakhadyr Khoussainov, Andre Nies & Richard A. Shore - 1997 - Notre Dame Journal of Formal Logic 38 (2):165-178.
    In this paper we investigate computable models of -categorical theories and Ehrenfeucht theories. For instance, we give an example of an -categorical but not -categorical theory such that all the countable models of except its prime model have computable presentations. We also show that there exists an -categorical but not -categorical theory such that all the countable models of except the saturated model, have computable presentations.
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  10.  12
    Computing K-Trivial Sets by Incomplete Random Sets.Laurent Bienvenu, Adam R. Day, Noam Greenberg, Antonín Kučera, Joseph S. Miller, André Nies & Dan Turetsky - 2014 - Bulletin of Symbolic Logic 20 (1):80-90.
    EveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.
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  11.  20
    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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  12.  8
    Denjoy, Demuth and Density.Laurent Bienvenu, Rupert Hölzl, Joseph S. Miller & André Nies - 2014 - Journal of Mathematical Logic 14 (1):1450004.
    We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class.
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  13.  68
    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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  14.  47
    Describing Groups.André Nies - 2007 - Bulletin of Symbolic Logic 13 (3):305-339.
    Two ways of describing a group are considered. 1. A group is finite-automaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of the (...)
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  15.  16
    Demuth’s Path to Randomness.Antonín Kučera, André Nies & Christopher P. Porter - 2015 - Bulletin of Symbolic Logic 21 (3):270-305.
    Osvald Demuth studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later.In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the (...)
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  16.  21
    Parameter Definability in the Recursively Enumerable Degrees.André Nies - 2003 - Journal of Mathematical Logic 3 (01):37-65.
    The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the [Formula: see text] relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k ≥ 7, the [Formula: see text] relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that Low 1 is parameter definable, and we provide methods that lead to a new example of a ∅-definable ideal. Moreover, we prove that (...)
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  17. Definability in the Recursively Enumerable Degrees.André Nies, Richard A. Shore & Theodore A. Slaman - 1996 - Bulletin of Symbolic Logic 2 (4):392-404.
  18.  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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  19. On the Filter of Computably Enumerable Supersets of an R-Maximal Set.Steffen Lempp, André Nies & D. Reed Solomon - 2001 - Archive for Mathematical Logic 40 (6):415-423.
    We study the filter ℒ*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ*(A) is still Σ0 3-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ*(A).
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  20.  33
    The Theory of the Recursively Enumerable Weak Truth-Table Degrees is Undecidable.Klaus Ambos-Spies, André Nies & Richard A. Shore - 1992 - Journal of Symbolic Logic 57 (3):864-874.
    We show that the partial order of Σ0 3-sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.
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  21. 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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  22.  32
    Initial Segments of the Lattice of Π10 Classes.Douglas Cenzer & Andre Nies - 2001 - Journal of Symbolic Logic 66 (4):1749-1765.
    We show that in the lattice E Π of Π 0 1 classes there are initial segments [ $\emptyset$ , P] = L(P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a Π 0 1 class P such that L is isomorphic to the lattice L(P)*, which is L(P), modulo finite differences. For the 2-element lattice, we obtain (...)
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  23.  32
    Upper Bounds on Ideals in the Computably Enumerable Turing Degrees.George Barmpalias & André Nies - 2011 - Annals of Pure and Applied Logic 162 (6):465-473.
    We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no prime ideal in the c.e. Turing degrees. This answers a question of Calhoun [2]. Every proper ideal in the c.e. Turing degrees has a low2 upper bound. Furthermore, the partial order of ideals under inclusion is dense.
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  24.  2
    Muchnik Degrees and Cardinal Characteristics.Benoit Monin & André Nies - forthcoming - Journal of Symbolic Logic:1-28.
  25. Program Size Complexity for Possibly Infinite Computations.Verónica Becher, Santiago Figueira, André Nies & Silvana Picchi - 2005 - Notre Dame Journal of Formal Logic 46 (1):51-64.
    We define a program size complexity function $H^\infty$ as a variant of the prefix-free Kolmogorov complexity, based on Turing monotone machines performing possibly unending computations. We consider definitions of randomness and triviality for sequences in ${\{0,1\}}^\omega$ relative to the $H^\infty$ complexity. We prove that the classes of Martin-Löf random sequences and $H^\infty$-random sequences coincide and that the $H^\infty$-trivial sequences are exactly the recursive ones. We also study some properties of $H^\infty$ and compare it with other complexity functions. In particular, $H^\infty$ (...)
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  26.  7
    Computably Enumerable Sets Below Random Sets.André Nies - 2012 - Annals of Pure and Applied Logic 163 (11):1596-1610.
    We use Demuth randomness to study strong lowness properties of computably enumerable sets, and sometimes of Δ20 sets. A set A⊆N is called a base for Demuth randomness if some set Y Turing above A is Demuth random relative to A. We show that there is an incomputable, computably enumerable base for Demuth randomness, and that each base for Demuth randomness is strongly jump-traceable. We obtain new proofs that each computably enumerable set below all superlow Martin-Löf random sets is strongly (...)
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  27.  8
    A New Spectrum of Recursive Models.André Nies - 1999 - Notre Dame Journal of Formal Logic 40 (3):307-314.
    We describe a strongly minimal theory S in an effective language such that, in the chain of countable models of S, only the second model has a computable presentation. Thus there is a spectrum of an -categorical theory which is neither upward nor downward closed. We also give an upper bound on the complexity of spectra.
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  28.  13
    Finite Automata Presentable Abelian Groups.André Nies & Pavel Semukhin - 2009 - Annals of Pure and Applied Logic 161 (3):458-467.
    We give new examples of FA presentable torsion-free abelian groups. Namely, for every n2, we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group in which every nontrivial cyclic subgroup is not FA recognizable.
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  29.  39
    The Undecidability of the II4 Theory for the R. E. Wtt and Turing Degrees.Steffen Lempp & André Nies - 1995 - Journal of Symbolic Logic 60 (4):1118 - 1136.
    We show that the Π 4 -theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.
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  30.  35
    The Undecidability of the II$^_4$ Theory for the R. E. Wtt and Turing Degrees.Steffen Lempp & André Nies - 1995 - Journal of Symbolic Logic 60 (4):1118-1136.
    We show that the $\Pi_4$-theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.
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  31.  32
    Superhighness.Bjørn Kjos-Hanssen & Andrée Nies - 2009 - Notre Dame Journal of Formal Logic 50 (4):445-452.
    We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class $superhigh^\diamond$ and show that it contains some, but not all, of the noncomputable K-trivial sets.
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  32.  19
    Recursively Enumerable Equivalence Relations Modulo Finite Differences.André Nies - 1994 - Mathematical Logic Quarterly 40 (4):490-518.
    We investigate the upper semilattice Eq* of recursively enumerable equivalence relations modulo finite differences. Several natural subclasses are shown to be first-order definable in Eq*. Building on this we define a copy of the structure of recursively enumerable many-one degrees in Eq*, thereby showing that Th has the same computational complexity as the true first-order arithmetic.
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  33.  13
    Cappable Recursively Enumerable Degrees and Post's Program.Klaus Ambos-Spies & André Nies - 1992 - Archive for Mathematical Logic 32 (1):51-56.
    We give a simple structural property which characterizes the r.e. sets whose (Turing) degrees are cappable. Since cappable degrees are incomplete, this may be viewed as a solution of Post's program, which asks for a simple structural property of nonrecursive r.e. sets which ensures incompleteness.
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  34.  12
    Interpreting N in the Computably Enumerable Weak Truth Table Degrees.André Nies - 2001 - Annals of Pure and Applied Logic 107 (1-3):35-48.
    We give a first-order coding without parameters of a copy of in the computably enumerable weak truth table degrees. As a tool, we develop a theory of parameter definable subsets.
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  35.  8
    Interpreting True Arithmetic in the Theory of the R.E. Truth Table Degrees.André Nies & Richard A. Shore - 1995 - Annals of Pure and Applied Logic 75 (3):269-311.
    We show that the elementary theory of the recursively enumerable tt-degrees has the same computational complexity as true first-order arithmetic. As auxiliary results, we prove theorems about exact pairs and initial segments in the tt-degrees.
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  36.  6
    Randomness Notions and Reverse Mathematics.André Nies & Paul Shafer - 2020 - Journal of Symbolic Logic 85 (1):271-299.
    We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent (...)
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  37.  5
    Martin-Löf Randomness Implies Multiple Recurrence in Effectively Closed Sets.Rodney G. Downey, Satyadev Nandakumar & André Nies - 2019 - Notre Dame Journal of Formal Logic 60 (3):491-502.
    This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space {0,1}N with the uniform measure and the usual shift. We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets P of positive measure holds for iterations starting at the point. This means that for each k∈N there is an n such (...)
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  38.  10
    The Complexity of Topological Group Isomorphism.Alexander S. Kechris, André Nies & Katrin Tent - 2018 - Journal of Symbolic Logic 83 (3):1190-1203.
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  39.  15
    Complexity of Equivalence Relations and Preorders From Computability Theory.Egor Ianovski, Russell Miller, Keng Meng Ng & André Nies - 2014 - Journal of Symbolic Logic 79 (3):859-881.
    We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relationsR,S, a componentwise reducibility is defined byR≤S⇔ ∃f∀x, y[x R y↔fS f].Here,fis taken from a suitable class of effective functions. For us the relations will be on natural numbers, andfmust be computable. We show that there is a${\rm{\Pi }}_1^0$-complete equivalence relation, but no${\rm{\Pi }}_k^0$-complete fork≥ 2. We show that${\rm{\Sigma }}_k^0$preorders arising naturally in the above-mentioned areas are${\rm{\Sigma }}_k^0$-complete. This includes polynomial timem-reducibility on (...)
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  40.  3
    Computing From Projections of Random Points.Noam Greenberg, Joseph S. Miller & André Nies - 2019 - Journal of Mathematical Logic 20 (1):1950014.
    We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call 1/2-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the K-trivial sets. We characterize 1/2-bases as the sets computable from both halves of Chaitin’s Ω, and as the sets that obey the cost function c(x,s)=Ωs−Ωx−−−−−−−√. Generalizing these results yields a dense hierarchy of (...)
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  41. Notre Dame, Indiana May 20–May 23, 2009.Patricia Blanchette, Heike Mildenberger, André Nies, Anand Pillay, Alexander Razborov, Alexandra Shlapentokh, John R. Steel & Boris Zilber - 2009 - Bulletin of Symbolic Logic 15 (4).
     
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  42.  9
    Borel Structures and Borel Theories.Greg Hjorth & André Nies - 2011 - Journal of Symbolic Logic 76 (2):461 - 476.
    We show that there is a complete, consistent Borel theory which has no "Borel model" in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel. We also investigate Borel isomorphisms between Borel structures.
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  43.  7
    Córdoba, Argentina September 20–24, 2004.Joos Heintz, Antonın Kucera, Joseph Miller, André Nies, Jan Reimann, Theodore Slaman, Diego Vaggione, Paul Vitányi & Verónica Becher - 2005 - Bulletin of Symbolic Logic 11 (4).
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  44.  5
    Isaac Newton Institute, Cambridge, UK July 2–6, 2012.George Barmpalias, Vasco Brattka, Adam Day, Rod Downey, John Hitchcock, Michal Koucký, Andy Lewis, Jack Lutz, André Nies & Alexander Shen - 2013 - Bulletin of Symbolic Logic 19 (1).
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