27 found
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  1.  40
    Computability Theory.S. Barry Cooper - 2003 - Chapman & Hall.
    Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary (...)
  2. On minimal pairs of enumeration degrees.Kevin McEvoy & S. Barry Cooper - 1985 - Journal of Symbolic Logic 50 (4):983-1001.
  3.  20
    Cupping and noncupping in the enumeration degrees of ∑20 sets.S. Barry Cooper, Andrea Sorbi & Xiaoding Yi - 1996 - Annals of Pure and Applied Logic 82 (3):317-342.
    We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.
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  4.  6
    Enumeration Reducibility Using Bounded Information: Counting Minimal Covers.S. Barry Cooper - 1987 - Mathematical Logic Quarterly 33 (6):537-560.
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  5.  22
    Enumeration Reducibility Using Bounded Information: Counting Minimal Covers.S. Barry Cooper - 1987 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (6):537-560.
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  6.  49
    The density of the low2 n-r.e. degrees.S. Barry Cooper - 1991 - Archive for Mathematical Logic 31 (1):19-24.
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  7.  33
    How Enumeration Reducibility Yields Extended Harrington Non-Splitting.Mariya I. Soskova & S. Barry Cooper - 2008 - Journal of Symbolic Logic 73 (2):634 - 655.
  8. There is No Low Maximal D.C.E. Degree.Marat Arslanov, S. Barry Cooper & Angsheng Li - 2000 - Mathematical Logic Quarterly 46 (3):409-416.
    We show that for any computably enumerable set A and any equation image set L, if L is low and equation image, then there is a c.e. splitting equation image such that equation image. In Particular, if L is low and n-c.e., then equation image is n-c.e. and hence there is no low maximal n-c.e. degree.
     
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  9.  22
    Cupping and noncupping in the enumeration degrees of∑< sub> 2< sup> 0 sets.S. Barry Cooper, Andrea Sorbi & Xiaoding Yi - 1996 - Annals of Pure and Applied Logic 82 (3):317-342.
  10.  8
    Complementing below recursively enumerable degrees.S. Barry Cooper & Richard L. Epstein - 1987 - Annals of Pure and Applied Logic 34 (1):15-32.
  11.  13
    On a Conjecture of Kleene and Post.S. Barry Cooper - 2001 - Mathematical Logic Quarterly 47 (1):3-34.
    A proof is given that 0′ is definable in the structure of the degrees of unsolvability. This answers a long-standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.
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  12.  27
    On the distribution of Lachlan nonsplitting bases.S. Barry Cooper, Angsheng Li & Xiaoding Yi - 2002 - Archive for Mathematical Logic 41 (5):455-482.
    We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w,v ≤ a, if a ≤ w or; v or; b then either a ≤ w or; b or a ≤ v or; b. In this paper we investigate the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high /low hierarchy. We prove that there (...)
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  13.  16
    Automorphisms of η-like computable linear orderings and Kierstead's conjecture.Charles M. Harris, Kyung Il Lee & S. Barry Cooper - 2016 - Mathematical Logic Quarterly 62 (6):481-506.
    We develop an approach to the longstanding conjecture of Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η-like computable linear ordering, such that has no interval of order type η, and such that the order type of is determined by a -limitwise monotonic maximal block function, there exists computable such that has no nontrivial automorphism.
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  14.  41
    The machine as data: a computational view of emergence and definability.S. Barry Cooper - 2015 - Synthese 192 (7):1955-1988.
    Turing’s paper on computable numbers has played its role in underpinning different perspectives on the world of information. On the one hand, it encourages a digital ontology, with a perceived flatness of computational structure comprehensively hosting causality at the physical level and beyond. On the other, it can give an insight into the way in which higher order information arises and leads to loss of computational control—while demonstrating how the control can be re-established, in special circumstances, via suitable type reductions. (...)
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  15.  36
    Bounding and Nonbounding Minimal Pairs in the Enumeration Degrees.S. Barry Cooper, Angsheng Li, Andrea Sorbi & Yue Yang - 2005 - Journal of Symbolic Logic 70 (3):741 - 766.
    We show that every nonzero $\Delta _{2}^{0}$ e-degree bounds a minimal pair. On the other hand, there exist $\Sigma _{2}^{0}$ e-degrees which bound no minimal pair.
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  16.  8
    Preface.S. Barry Cooper, Herman Geuvers, Anand Pillay & Jouko Väänänen - 2008 - Annals of Pure and Applied Logic 156 (1):1-2.
  17.  88
    Splitting and nonsplitting II: A low {\sb 2$} C.E. degree about which ${\bf 0}'$ is not splittable.S. Barry Cooper & Angsheng Li - 2002 - Journal of Symbolic Logic 67 (4):1391-1430.
    It is shown that there exists a low2 Harrington non-splitting base-that is, a low2 computably enumerable (c.e.) degree a such that for any c.e. degrees x, y, if $0' = x \vee y$ , then either $0' = x \vee a$ or $0' = y \vee a$ . Contrary to prior expectations, the standard Harrington non-splitting construction is incompatible with the $low_{2}-ness$ requirements to be satisfied, and the proof given involves new techniques with potentially wider application.
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  18.  25
    There is no low maximal d. c. e. degree– Corrigendum.Marat Arslanov, S. Barry Cooper & Angsheng Li - 2004 - Mathematical Logic Quarterly 50 (6):628-636.
    We give a corrected proof of an extension of the Robinson Splitting Theorem for the d. c. e. degrees.
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  19.  5
    Preface.Samuel R. Buss, S. Barry Cooper, Benedikt Löwe & Andrea Sorbi - 2009 - Annals of Pure and Applied Logic 160 (3):229-230.
  20.  9
    Bounding and nonbounding minimal pairs in the enumeration degrees.S. Barry Cooper, Angsheng Li, Andrea Sorbi & Yue Yang - 2005 - Journal of Symbolic Logic 70 (3):741-766.
    We show that every nonzero Δ20, e-degree bounds a minimal pair. On the other hand, there exist Σ20, e-degrees which bound no minimal pair.
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  21. How the World Computes.S. Barry Cooper (ed.) - 2012
     
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  22.  71
    On Lachlan’s major sub-degree problem.S. Barry Cooper & Angsheng Li - 2008 - Archive for Mathematical Logic 47 (4):341-434.
    The Major Sub-degree Problem of A. H. Lachlan (first posed in 1967) has become a long-standing open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, ${{\bf 0' = b \lor x}}$ if and only if (...)
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  23. Sets and Proofs.S. Barry Cooper & John K. Truss - 2001 - Studia Logica 69 (3):446-448.
  24.  12
    Turing Centenary Conference: How the World Computes.S. Barry Cooper, Anuj Dawar, Martin Hyland & Benedikt Löwe - 2014 - Annals of Pure and Applied Logic 165 (9):1353-1354.
  25.  14
    The discontinuity of splitting in the recursively enumerable degrees.S. Barry Cooper & Xiaoding Yi - 1995 - Archive for Mathematical Logic 34 (4):247-256.
    In this paper we examine a class of pairs of recursively enumerable degrees, which is related to the Slaman-Soare Phenomenon.
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  26.  10
    What Makes A Computation Unconventional?S. Barry Cooper - 2013 - In Gordana Dodig-Crnkovic Raffaela Giovagnoli (ed.), Computing Nature. pp. 255--269.
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  27.  25
    Noncappable enumeration degrees below 0'e. [REVIEW]S. Barry Cooper & Andrea Sorbi - 1996 - Journal of Symbolic Logic 61 (4):1347 - 1363.
    We prove that there exists a noncappable enumeration degree strictly below 0' e.
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