Results for ' Medvedev degrees'

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  1.  32
    Characterizing the Join-Irreducible Medvedev Degrees.Paul Shafer - 2011 - Notre Dame Journal of Formal Logic 52 (1):21-38.
    We characterize the join-irreducible Medvedev degrees as the degrees of complements of Turing ideals, thereby solving a problem posed by Sorbi. We use this characterization to prove that there are Medvedev degrees above the second-least degree that do not bound any join-irreducible degrees above this second-least degree. This solves a problem posed by Sorbi and Terwijn. Finally, we prove that the filter generated by the degrees of closed sets is not prime. This solves (...)
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  2.  22
    The ∀∃-theory of the effectively closed Medvedev degrees is decidable.Joshua A. Cole & Takayuki Kihara - 2010 - Archive for Mathematical Logic 49 (1):1-16.
    We show that there is a computable procedure which, given an ∀∃-sentence ${\varphi}$ in the language of the partially ordered sets with a top element 1 and a bottom element 0, computes whether ${\varphi}$ is true in the Medvedev degrees of ${\Pi^0_1}$ classes in Cantor space, sometimes denoted by ${\mathcal{P}_s}$.
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  3.  34
    A survey of Mučnik and Medvedev degrees.Peter G. Hinman - 2012 - Bulletin of Symbolic Logic 18 (2):161-229.
    We survey the theory of Mucnik and Medvedev degrees of subsets of $^{\omega}{\omega}$with particular attention to the degrees of $\Pi_{1}^{0}$ subsets of $^{\omega}2$. Sections 1-6 present the major definitions and results in a uniform notation. Sections 7-6 present proofs, some more complete than others, of the major results of the subject together with much of the required background material.
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  4.  17
    Comparing the degrees of enumerability and the closed Medvedev degrees.Paul Shafer & Andrea Sorbi - 2019 - Archive for Mathematical Logic 58 (5-6):527-542.
    We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the (...)
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  5.  8
    On New Notions of Algorithmic Dimension, Immunity, and Medvedev Degree.David J. Webb - 2022 - Bulletin of Symbolic Logic 28 (4):532-533.
    We prove various results connected together by the common thread of computability theory.First, we investigate a new notion of algorithmic dimension, the inescapable dimension, which lies between the effective Hausdorff and packing dimensions. We also study its generalizations, obtaining an embedding of the Turing degrees into notions of dimension.We then investigate a new notion of computability theoretic immunity that arose in the course of the previous study, that of a set of natural numbers with no co-enumerable subsets. We demonstrate (...)
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  6.  17
    Kripke Models, Distributive Lattices, and Medvedev Degrees.Sebastiaan A. Terwijn - 2007 - Studia Logica 85 (3):319-332.
    We define a variant of the standard Kripke semantics for intuitionistic logic, motivated by the connection between constructive logic and the Medvedev lattice. We show that while the new semantics is still complete, it gives a simple and direct correspondence between Kripke models and algebraic structures such as factors of the Medvedev lattice.
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  7. Review: U. T. Medvedev, Degrees of Difficulty of Mass Problems. [REVIEW]Andrzej Mostowski - 1956 - Journal of Symbolic Logic 21 (3):320-321.
  8.  39
    Classifying the Branching Degrees in the Medvedev Lattice of $\Pi^0_1$ Classes.Christopher P. Alfeld - 2008 - Notre Dame Journal of Formal Logic 49 (3):227-243.
    A $\Pi^0_1$ class can be defined as the set of infinite paths through a computable tree. For classes $P$ and $Q$, say that $P$ is Medvedev reducible to $Q$, $P \leq_M Q$, if there is a computably continuous functional mapping $Q$ into $P$. Let $\mathcal{L}_M$ be the lattice of degrees formed by $\Pi^0_1$ subclasses of $2^\omega$ under the Medvedev reducibility. In "Non-branching degrees in the Medvedev lattice of $\Pi \sp{0}\sb{1} classes," I provided a characterization of (...)
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  9.  13
    Coding true arithmetic in the Medvedev degrees of classes.Paul Shafer - 2012 - Annals of Pure and Applied Logic 163 (3):321-337.
  10.  22
    A Note on Closed Degrees of Difficulty of the Medvedev Lattice.Caterina Bianchini & Andrea Sorbi - 1996 - Mathematical Logic Quarterly 42 (1):127-133.
    We consider some nonprincipal filters of the Medvedev lattice. We prove that the filter generated by the nonzero closed degrees of difficulty is not principal and we compare this filter, with respect to inclusion, with some other filters of the lattice. All the filters considered in this paper are disjoint from the prime ideal generated by the dense degrees of difficulty.
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  11.  36
    The Medvedev lattice of computably closed sets.Sebastiaan A. Terwijn - 2006 - Archive for Mathematical Logic 45 (2):179-190.
    Simpson introduced the lattice of Π0 1 classes under Medvedev reducibility. Questions regarding completeness in are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π0 1 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of does not allow an implication operator (i.e. that is not a (...)
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  12.  47
    Topological aspects of the Medvedev lattice.Andrew Em Lewis, Richard A. Shore & Andrea Sorbi - 2011 - Archive for Mathematical Logic 50 (3-4):319-340.
    We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this (...)
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  13.  15
    The Medvedev Lattice of Degrees of Difficulty.Andrea Sorbi - 1996 - In S. B. Cooper, T. A. Slaman & S. S. Wainer (eds.), Computability, enumerability, unsolvability: directions in recursion theory. New York: Cambridge University Press. pp. 224--289.
  14.  29
    Non-Branching Degrees in the Medvedev Lattice of [image] Classes.Christopher P. Alfeld - 2007 - Journal of Symbolic Logic 72 (1):81 - 97.
    A $\Pi _{1}^{0}$ class is the set of paths through a computable tree. Given classes P and Q, P is Medvedev reducible to Q, P ≤M Q, if there is a computably continuous functional mapping Q into P. We look at the lattice formed by $\Pi _{1}^{0}$ subclasses of 2ω under this reduction. It is known that the degree of a splitting class of c.e. sets is non-branching. We further characterize non-branching degrees, providing two additional properties which guarantee (...)
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  15.  23
    Coding true arithmetic in the Medvedev and Muchnik degrees.Paul Shafer - 2011 - Journal of Symbolic Logic 76 (1):267 - 288.
    We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are pairwise recursively isomorphic: the first-order theory of the closed Medvedev degrees, the first-order theory of the compact Medvedev degrees, (...)
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  16.  37
    Embeddings into the Medvedev and Muchnik lattices of Π0 1 classes.Stephen Binns & Stephen G. Simpson - 2004 - Archive for Mathematical Logic 43 (3):399-414.
    Let w and M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 0 subsets of 2ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of w . We show that many countable distributive lattices are lattice-embeddable below any non-zero element of M.
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  17.  41
    Degrees of difficulty of generalized r.e. separating classes.Douglas Cenzer & Peter G. Hinman - 2008 - Archive for Mathematical Logic 46 (7-8):629-647.
    Important examples of $\Pi^0_1$ classes of functions $f \in {}^\omega\omega$ are the classes of sets (elements of ω 2) which separate a given pair of disjoint r.e. sets: ${\mathsf S}_2(A_0, A_1) := \{f \in{}^\omega2 : (\forall i < 2)(\forall x \in A_i)f(x) \neq i\}$ . A wider class consists of the classes of functions f ∈ ω k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: ${\mathsf S}_k(A_0,\ldots,A_k-1) := (...)
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  18.  42
    Constructive Logic and the Medvedev Lattice.Sebastiaan A. Terwijn - 2006 - Notre Dame Journal of Formal Logic 47 (1):73-82.
    We study the connection between factors of the Medvedev lattice and constructive logic. The algebraic properties of these factors determine logics lying in between intuitionistic propositional logic and the logic of the weak law of the excluded middle (also known as De Morgan, or Jankov, logic). We discuss the relation between the weak law of the excluded middle and the algebraic notion of join-reducibility. Finally we discuss autoreducible degrees.
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  19.  14
    Density of the Medvedev lattice of Π0 1 classes.Douglas Cenzer & Peter G. Hinman - 2003 - Archive for Mathematical Logic 42 (6):583-600.
    The partial ordering of Medvedev reducibility restricted to the family of Π0 1 classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a Π0 1 class, which we call a ``c.e. separating class''. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes.
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  20.  38
    First-Order Logic in the Medvedev Lattice.Rutger Kuyper - 2015 - Studia Logica 103 (6):1185-1224.
    Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic, using the notion of a first-order hyperdoctrine from categorical logic, to a structure which we will (...)
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  21.  32
    A splitting theorem for the Medvedev and Muchnik lattices.Stephen Binns - 2003 - Mathematical Logic Quarterly 49 (4):327.
    This is a contribution to the study of the Muchnik and Medvedev lattices of non-empty Π01 subsets of 2ω. In both these lattices, any non-minimum element can be split, i. e. it is the non-trivial join of two other elements. In fact, in the Medvedev case, ifP > MQ, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices (...)
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  22.  20
    Inside the Muchnik degrees I: Discontinuity, learnability and constructivism.K. Higuchi & T. Kihara - 2014 - Annals of Pure and Applied Logic 165 (5):1058-1114.
    Every computable function has to be continuous. To develop computability theory of discontinuous functions, we study low levels of the arithmetical hierarchy of nonuniformly computable functions on Baire space. First, we classify nonuniformly computable functions on Baire space from the viewpoint of learning theory and piecewise computability. For instance, we show that mind-change-bounded learnability is equivalent to finite View the MathML source2-piecewise computability 2 denotes the difference of two View the MathML sourceΠ10 sets), error-bounded learnability is equivalent to finite View (...)
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  23.  25
    Inside the Muchnik degrees II: The degree structures induced by the arithmetical hierarchy of countably continuous functions.K. Higuchi & T. Kihara - 2014 - Annals of Pure and Applied Logic 165 (6):1201-1241.
    It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial Π10 subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty Π10 subsets of Cantor space, we show the existence of a finite-Δ20-piecewise degree containing infinitely many finite-2-piecewise degrees, and a finite-2-piecewise degree containing infinitely many finite-Δ20-piecewise degrees 2 denotes the difference of two Πn0 sets), (...)
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  24.  30
    Weihrauch degrees, omniscience principles and weak computability.Vasco Brattka & Guido Gherardi - 2011 - Journal of Symbolic Logic 76 (1):143 - 176.
    In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees (...)
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  25.  67
    Some remarks on the algebraic structure of the Medvedev lattice.Andrea Sorbi - 1990 - Journal of Symbolic Logic 55 (2):831-853.
    This paper investigates the algebraic structure of the Medvedev lattice M. We prove that M is not a Heyting algebra. We point out some relations between M and the Dyment lattice and the Mucnik lattice. Some properties of the degrees of enumerability are considered. We give also a result on embedding countable distributive lattices in the Medvedev lattice.
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  26.  13
    On some filters and ideals of the Medvedev lattice.Andrea Sorbi - 1990 - Archive for Mathematical Logic 30 (1):29-48.
    Let $\mathfrak{M}$ be the Medvedev lattice: this paper investigates some filters and ideals (most of them already introduced by Dyment, [4]) of $\mathfrak{M}$ . If $\mathfrak{G}$ is any of the filters or ideals considered, the questions concerning $\mathfrak{G}$ which we try to answer are: (1) is $\mathfrak{G}$ prime? What is the cardinality of ${\mathfrak{M} \mathord{\left/ {\vphantom {\mathfrak{M} \mathfrak{G}}} \right. \kern-0em} \mathfrak{G}}$ ? Occasionally, we point out some general facts on theT-degrees or the partial degrees, by which these (...)
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  27.  26
    Muchnik degrees and cardinal characteristics.Benoit Monin & André Nies - 2021 - Journal of Symbolic Logic 86 (2):471-498.
    A mass problem is a set of functions $\omega \to \omega $. For mass problems ${\mathcal {C}}, {\mathcal {D}}$, one says that ${\mathcal {C}}$ is Muchnik reducible to ${\mathcal {D}}$ if each function in ${\mathcal {C}}$ is computed by a function in ${\mathcal {D}}$. In this paper we study some highness properties of Turing oracles, which we view as mass problems. We compare them with respect to Muchnik reducibility and its uniform strengthening, Medvedev reducibility.For $p \in [0,1]$ let ${\mathcal (...)
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  28.  11
    Levels of Uniformity.Rutger Kuyper - 2019 - Notre Dame Journal of Formal Logic 60 (1):119-138.
    We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of nonuniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses how uniform a reduction is. We study this notion for several well-known reductions from algorithmic randomness. Furthermore, since our new structures are Brouwer algebras, we study their propositional theories. Finally, we study if our new structures are elementarily equivalent to each other.
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  29.  39
    Generalizations of the Weak Law of the Excluded Middle.Andrea Sorbi & Sebastiaan A. Terwijn - 2015 - Notre Dame Journal of Formal Logic 56 (2):321-331.
    We study a class of formulas generalizing the weak law of the excluded middle and provide a characterization of these formulas in terms of Kripke frames and Brouwer algebras. We use these formulas to separate logics corresponding to factors of the Medvedev lattice.
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  30.  5
    A comparison of various analytic choice principles.Paul-Elliot Anglès D’Auriac & Takayuki Kihara - 2021 - Journal of Symbolic Logic 86 (4):1452-1485.
    We investigate computability theoretic and descriptive set theoretic contents of various kinds of analytic choice principles by performing a detailed analysis of the Medvedev lattice of $\Sigma ^1_1$ -closed sets. Among others, we solve an open problem on the Weihrauch degree of the parallelization of the $\Sigma ^1_1$ -choice principle on the integers. Harrington’s unpublished result on a jump hierarchy along a pseudo-well-ordering plays a key role in solving this problem.
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  31.  32
    Small Π0 1 Classes.Stephen Binns - 2005 - Archive for Mathematical Logic 45 (4):393-410.
    The property of smallness for Π0 1 classes is introduced and is investigated with respect to Medvedev and Muchnik degree. It is shown that the property of containing a small Π0 1 class depends only on the Muchnik degree of a Π0 1 class. A comparison is made with the idea of thinness for Π0 1 classesmsthm.
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  32.  5
    Small Π01 Classes.Stephen Binns - 2006 - Archive for Mathematical Logic 45 (4):393-410.
    The property of smallness for Π01 classes is introduced and is investigated with respect to Medvedev and Muchnik degree. It is shown that the property of containing a small Π01 class depends only on the Muchnik degree of a Π01 class. A comparison is made with the idea of thinness for Π01 classesmsthm.
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  33.  21
    Hyperimmunity in 2\sp ℕ.Stephen Binns - 2007 - Notre Dame Journal of Formal Logic 48 (2):293-316.
    We investigate the notion of hyperimmunity with respect to how it can be applied to Π{\sp 0}{\sb 1} classes and their Muchnik degrees. We show that hyperimmunity is a strong enough concept to prove the existence of Π{\sp 0}{\sb 1} classes with intermediate Muchnik degree—in contrast to Post's attempts to construct intermediate c.e. degrees.
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  34.  49
    Embedding FD(ω) into {mathcal{P}_s} densely.Joshua A. Cole - 2008 - Archive for Mathematical Logic 46 (7-8):649-664.
    Let ${\mathcal{P}_s}$ be the lattice of degrees of non-empty ${\Pi_1^0}$ subsets of 2 ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in ${\mathcal{P}_s}$ . Cenzer and Hinman proved that ${\mathcal{P}_s}$ is dense, by adapting the Sacks Preservation and Sacks Coding Strategies used in the proof of the density of the c.e. Turing degrees. With a construction that is a modification of the (...)
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  35.  7
    Embedding FD(ω) into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_s}$$\end{document} densely. [REVIEW]Joshua A. Cole - 2008 - Archive for Mathematical Logic 46 (7-8):649-664.
    Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_s}$$\end{document} be the lattice of degrees of non-empty \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^0}$$\end{document} subsets of 2ω under Medvedev reducibility. Binns and Simpson proved that FD(ω), the free distributive lattice on countably many generators, is lattice-embeddable below any non-zero element in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_s}$$\end{document}. Cenzer and Hinman proved that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...)
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  36. Chelovek i ego otrazhenie v religii.M. I. Medvedev - 1983 - Minsk: Izd-vo BGU im. V. Lenina.
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  37.  3
    Ėkologicheskoe soznanie.V. I. Medvedev - 2001 - Moskva: Logos. Edited by A. A. Aldasheva.
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  38.  7
    Osmyslenie dukhovnoi tselostnosti: sbornik statei.A. V. Medvedev (ed.) - 1992 - Ekaterinburg: Izd-vo Uralʹskogo universiteta.
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  39. Vizantiĭskiĭ gumanizm chetyrnadt︠s︡atogo-pi︠a︡tnadt︠s︡atogo vv.Igorʹ Pavlovich Medvedev - 1976 - Edited by Geōrgios Gemistos Plēthōn.
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  40.  69
    An invitation to model-theoretic galois theory.Alice Medvedev & Ramin Takloo-Bighash - 2010 - Bulletin of Symbolic Logic 16 (2):261 - 269.
    We carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage (...)
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  41.  18
    Filosofii︠a︡ i︠a︡zyka: ocherki istorii.Vladimir Ivanovich Medvedev - 2012 - Sankt-Peterburg: Izdatelʹstvo RKhGA.
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  42. Filosofii︠a︡ kak dei︠a︡telʹnostʹ: idei Li︠u︡dviga Vitgenshteĭna.N. V. Medvedev - 1999 - Tambov: Tambovskiĭ gos. universitet im. G.R. Derzhavina.
     
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  43. Obʺi︠a︡snenie, ponimanie, i︠a︡zyk.V. I. Medvedev - 1997 - Sankt-Peterburg: Stupeni.
     
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  44. Russian Imago 2000: issledovanii︠a︡ po psikhoanalizu kulʹtury: sbornik stateĭ.V. A. Medvedev (ed.) - 2001 - Sankt-Peterburg: Izd-vo "Aleteĭi︠a︡".
  45. Russian Imago 2001: issledovanii︠a︡ po psikhoanalizu kulʹtury: sbornik stateĭ.V. A. Medvedev (ed.) - 2002 - Sankt-Peterburg: Izd-vo "Aleteĭi︠a︡".
  46.  38
    Grouplike minimal sets in ACFA and in T A.Alice Medvedev - 2010 - Journal of Symbolic Logic 75 (4):1462-1488.
    This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in T A . The thesis concerns minimal formulae of the form x ∈ A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f) that fall into each of the three cases (...)
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  47.  17
    A polifonia do Círculo.Iuri Pavlovich Medvedev, Daria Aleksandrovna Medvedeva & David Shepherd - 2016 - Bakhtiniana 11 (1):99-144.
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  48.  22
    Der neugefundene Text eines Briefes von Maximos Katelianos: noch eine Fälschung von Karl Benedikt Hase.Igor P. Medvedev - 2016 - Byzantinische Zeitschrift 109 (2):821-836.
    Name der Zeitschrift: Byzantinische Zeitschrift Jahrgang: 109 Heft: 2 Seiten: 821-836.
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  49.  6
    Экологическое сознание : учебное пособие по педагогическим, психологическим направлениям и специальностям.Vsevolod Ivanovich Medvedev, A. A. Aldasheva & Federal§Naëiìa Ëtìselevaëiìa Programma "Gosudarstvennaëiìa Podderzhka Integraëtìsii Vysshego Obrazov - 2001 - Moskva: Logos. Edited by A. A. Aldasheva.
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  50. Vizantiĭskiĭ gumanizm XIV-XV vv.I. P. Medvedev & George Gemistus Plethon - 1997 - Sankt-Peterburg: Aleteĭi︠a︡.
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