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André Nies [35]Andreé Nies [2]
  1. Laurent Bienvenu, Rupert Hölzl, Joseph S. Miller & André Nies (2014). Denjoy, Demuth and Density. Journal of Mathematical Logic 14 (1):1450004.
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  2. George Barmpalias, Vasco Brattka, Adam Day, Rod Downey, John Hitchcock, Michal Koucký, Andy Lewis, Jack Lutz, André Nies & Alexander Shen (2013). Isaac Newton Institute, Cambridge, UK July 2–6, 2012. Bulletin of Symbolic Logic 19 (1).
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  3. André Nies (2012). Computably Enumerable Sets Below Random Sets. 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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  4. George Barmpalias & André Nies (2011). Upper Bounds on Ideals in the Computably Enumerable Turing Degrees. 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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  5. Noam Greenberg & André Nies (2011). Benign Cost Functions and Lowness Properties. 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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  6. Greg Hjorth & André Nies (2011). Borel Structures and Borel Theories. 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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  7. Antonín Kučera & André Nies (2011). Demuth Randomness and Computational Complexity. 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. Bjørn Kjos-Hanssen, André Nies, Frank Stephan & Liang Yu (2010). Higher Kurtz Randomness. 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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  9. Patricia Blanchette, Kit Fine, Heike Mildenberger, André Nies, Anand Pillay, Alexander Razborov, Alexandra Shlapentokh, John R. Steel & Boris Zilber (2009). Notre Dame, Indiana May 20–May 23, 2009. Bulletin of Symbolic Logic 15 (4).
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  10. Bjørn Kjos-Hanssen & Andrée Nies (2009). Superhighness. 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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  11. André Nies & Pavel Semukhin (2009). Finite Automata Presentable Abelian Groups. 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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  12. Santiago Figueira, André Nies & Frank Stephan (2008). Lowness Properties and Approximations of the Jump. 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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  13. André Nies (2007). Describing Groups. 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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  14. Rod Downey, Denis R. Hirschfeldt, André Nies & Sebastiaan A. Terwijn (2006). Calibrating Randomness. Bulletin of Symbolic Logic 12 (3):411-491.
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  15. Rod Downey, Andre Nies, Rebecca Weber & Liang Yu (2006). Lowness and Π₂⁰ Nullsets. 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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  16. Wolfgang Merkle, Joseph S. Miller, André Nies, Jan Reimann & Frank Stephan (2006). Kolmogorov–Loveland Randomness and Stochasticity. 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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  17. Joseph S. Miller & André Nies (2006). Randomness and Computability: Open Questions. Bulletin of Symbolic Logic 12 (3):390-410.
  18. Verónica Becher, Santiago Figueira, André Nies & Silvana Picchi (2005). Program Size Complexity for Possibly Infinite Computations. 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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  19. Rod Downey, Denis R. Hirschfeldt, Joseph S. Miller & André Nies (2005). Relativizing Chaitin's Halting Probability. Journal of Mathematical Logic 5 (02):167-192.
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  20. Joos Heintz, Antonın Kucera, Joseph Miller, André Nies, Jan Reimann, Theodore Slaman, Diego Vaggione, Paul Vitányi & Verónica Becher (2005). Córdoba, Argentina September 20–24, 2004. Bulletin of Symbolic Logic 11 (4).
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  21. André Nies, Frank Stephan & Sebastiaan A. Terwijn (2005). Randomness, Relativization and Turing Degrees. Journal of Symbolic Logic 70 (2):515 - 535.
    We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to θ(n−1). We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) ≥ |x| − c. The 'only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove (...)
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  22. André Nies (2003). Parameter Definability in the Recursively Enumerable Degrees. Journal of Mathematical Logic 3 (01):37-65.
  23. Douglas Cenzer & Andre Nies (2001). Initial Segments of the Lattice of Π01 Classes. 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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  24. Steffen Lempp, André Nies & D. Reed Solomon (2001). On the Filter of Computably Enumerable Supersets of an R-Maximal Set. 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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  25. André Nies (2001). Interpreting in the Computably Enumerable Weak Truth Table Degrees. 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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  26. André Nies & Andrea Sorbi (2000). Structural Properties and Σ02 Enumeration Degrees. 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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  27. André Nies (1999). A New Spectrum of Recursive Models. 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. Bakhadyr Khoussainov, Andre Nies & Richard A. Shore (1997). Computable Models of Theories with Few Models. 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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  29. André Nies, Richard A. Shore & Theodore A. Slaman (1996). Definability in the Recursively Enumerable Degrees. Bulletin of Symbolic Logic 2 (4):392-404.
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  30. Steffen Lempp & André Nies (1995). The Undecidability of the II4 Theory for the R. E. Wtt and Turing Degrees. 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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  31. Steffen Lempp & Andre Nies (1995). The Undecidability of the II$^_4$ Theory for the R. E. Wtt and Turing Degrees. 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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  32. André Nies & Richard A. Shore (1995). Interpreting True Arithmetic in the Theory of the R.E. Truth Table Degrees. 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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  33. André Nies (1994). Recursively Enumerable Equivalence Relations Modulo Finite Differences. 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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  34. Klaus Ambos-Spies & André Nies (1992). Cappable Recursively Enumerable Degrees and Post's Program. 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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  35. Klaus Ambos-Spies, André Nies & Richard A. Shore (1992). The Theory of the Recursively Enumerable Weak Truth-Table Degrees is Undecidable. 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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