45 found
Order:
Disambiguations:
S. Barry Cooper [23]S. B. Cooper [22]
  1.  9
    S. Barry Cooper (2004). Computability Theory. Chapman & Hall.
  2. S. B. Cooper (1984). Partial Degrees and the Density Problem. Part 2: The Enumeration Degrees of the ∑2 Sets Are Dense. Journal of Symbolic Logic 49 (2):503 - 513.
    Direct download (5 more)  
     
    Export citation  
     
    My bibliography   5 citations  
  3. Kevin McEvoy & S. Barry Cooper (1985). On Minimal Pairs of Enumeration Degrees. Journal of Symbolic Logic 50 (4):983-1001.
    Direct download (6 more)  
     
    Export citation  
     
    My bibliography   4 citations  
  4.  2
    Mariya I. Soskova & S. Barry Cooper (2008). How Enumeration Reducibility Yields Extended Harrington Non-Splitting. Journal of Symbolic Logic 73 (2):634 - 655.
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  5. Thomas Strahm, S. Barry Cooper & John K. Truss (2002). First Steps Into Metapredicativity in Explicit Mathematics. Bulletin of Symbolic Logic 8 (4):535-536.
     
    Export citation  
     
    My bibliography   4 citations  
  6.  5
    S. B. Cooper (1973). Minimal Degrees and the Jump Operator. Journal of Symbolic Logic 38 (2):249-271.
  7.  5
    S. B. Cooper (1974). Minimal Pairs and High Recursively Enumerable Degrees. Journal of Symbolic Logic 39 (4):655-660.
    Direct download (5 more)  
     
    Export citation  
     
    My bibliography   8 citations  
  8.  13
    S. Barry Cooper & Angsheng Li (2008). On Lachlan's Major Sub-Degree Problem. 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 (...)
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  9.  6
    S. Barry Cooper (1991). The Density of the Low2 N-R.E. Degrees. Archive for Mathematical Logic 31 (1):19-24.
    No categories
    Direct download (5 more)  
     
    Export citation  
     
    My bibliography   4 citations  
  10.  6
    S. B. Cooper, Benedikt Löwe & Andrea Sorbi (eds.) (2007). New Computational Paradigms: Changing Conceptions of What is Computable. Springer.
    Logicians and theoretical physicists will also benefit from this book.
    Direct download  
     
    Export citation  
     
    My bibliography   1 citation  
  11.  2
    S. B. Cooper (1989). A Jump Class of Noncappable Degrees. Journal of Symbolic Logic 54 (2):324-353.
    Direct download (5 more)  
     
    Export citation  
     
    My bibliography   4 citations  
  12.  11
    M. Arslanov & S. B. Cooper (2004). There is No Low Maximal D.C.E. Degree - Corrigendum. Mathematical Logic Quarterly 50 (6):628.
    We give a corrected proof of an extension of the Robinson Splitting Theorem for the d. c. e. degrees.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  13.  4
    S. Barry Cooper (1987). Enumeration Reducibility Using Bounded Information: Counting Minimal Covers. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 33 (6):537-560.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  14.  4
    S. B. Cooper & C. S. Copestake (1988). Properly Σ2 Enumeration Degrees. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 34 (6):491-522.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  15.  8
    S. B. Cooper (1989). The Strong Anticupping Property for Recursively Enumerable Degrees. Journal of Symbolic Logic 54 (2):527-539.
    Direct download (6 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  16.  10
    M. Arslanov, S. B. Cooper & A. Li (2000). There is No Low Maximal D.C.E. Degree. 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.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  17.  7
    S. Barry Cooper, Angsheng Li, Andrea Sorbi & Yue Yang (2005). Bounding and Nonbounding Minimal Pairs in the Enumeration Degrees. 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.
    Direct download (3 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  18. S. Barry Cooper (1987). Enumeration Reducibility Using Bounded Information: Counting Minimal Covers. Mathematical Logic Quarterly 33 (6):537-560.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  19.  4
    S. B. Cooper (1982). Partial Degrees and the Density Problem. Journal of Symbolic Logic 47 (4):854-859.
    Direct download (6 more)  
     
    Export citation  
     
    My bibliography   3 citations  
  20. S. B. Cooper & C. S. Copestake (1988). Properly Σ2 Enumeration Degrees. Mathematical Logic Quarterly 34 (6):491-522.
    Direct download (3 more)  
     
    Export citation  
     
    My bibliography   2 citations  
  21.  7
    S. Barry Cooper & Andrea Sorbi (1996). Noncappable Enumeration Degrees Below 0'e. [REVIEW] Journal of Symbolic Logic 61 (4):1347 - 1363.
    We prove that there exists a noncappable enumeration degree strictly below 0' e.
    Direct download (5 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  22.  8
    S. Barry Cooper, Angsheng Li & Xiaoding Yi (2002). On the Distribution of Lachlan Nonsplitting Bases. 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 (...)
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  23.  2
    S. Barry Cooper (2015). The Machine as Data: A Computational View of Emergence and Definability. 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. (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  24.  4
    S. B. Cooper (1972). Jump Equivalence of the ? 0/2 Hyperhyperimmune Sets. Journal of Symbolic Logic 37 (3):598-600.
  25. S. Barry Cooper, Andrea Sorbi & Xiaoding Yi (1996). Cupping and Noncupping in the Enumeration Degrees of ∑20 Sets. 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.
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography   1 citation  
  26.  17
    S. Barry Cooper & Angsheng Li (2002). Splitting and Nonsplitting, II: A $Low_2$ C.E. Degree Above Which 0' is Not Splittable. 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.
    Direct download (6 more)  
     
    Export citation  
     
    My bibliography  
  27.  4
    S. Barry Cooper & Xiaoding Yi (1995). The Discontinuity of Splitting in the Recursively Enumerable Degrees. 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.
    Direct download (4 more)  
     
    Export citation  
     
    My bibliography  
  28.  3
    S. Barry Cooper & Richard L. Epstein (1987). Complementing Below Recursively Enumerable Degrees. Annals of Pure and Applied Logic 34 (1):15-32.
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  29.  2
    S. Barry Cooper, Andrea Sorbi & Xiaoding Yi (1996). Cupping and Noncupping in the Enumeration Degrees of∑< Sub> 2< Sup> 0 Sets. Annals of Pure and Applied Logic 82 (3):317-342.
    Direct download  
     
    Export citation  
     
    My bibliography  
  30.  2
    S. Barry Cooper, Herman Geuvers, Anand Pillay & Jouko Väänänen (2008). Preface. Annals of Pure and Applied Logic 156 (1):1-2.
    Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  31.  1
    S. B. Cooper & Andrew E. M. Lewis (2005). "Properly Sigma~2 Minimal Degrees and 0" Complementation. Mathematical Logic Quarterly 51 (3):274.
    We show that there exists a properly Σ2 minimal degree b, and moreover that b can be chosen to join with 0′ to 0″ – so that b is a 0″ complement for every degree a such that 0′ ≤ a < 0″.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  32.  1
    S. B. Cooper (1972). Degrees of Unsolvability Complementary Between Recursively Enumerable Degrees, Part 1. Annals of Mathematical Logic 4 (1):31-73.
  33. Marat Arslanov, S. Barry Cooper & Angsheng Li (2004). There is No Low Maximal D. C. E. Degree– Corrigendum. Mathematical Logic Quarterly 50 (6):628-636.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  34. M. M. Arslanov, C. T. Chong, S. B. Cooper & Y. Yang (2005). The Minimal E-Degree Problem in Fragments of Peano Arithmetic. Annals of Pure and Applied Logic 131 (1-3):159-175.
    We study the minimal enumeration degree problem in models of fragments of Peano arithmetic () and prove the following results: in any model M of Σ2 induction, there is a minimal enumeration degree if and only if M is a nonstandard model. Furthermore, any cut in such a model has minimal e-degree. By contrast, this phenomenon fails in the absence of Σ2 induction. In fact, whether every Σ2 cut has minimal e-degree is independent of the Σ2 bounding principle.
    Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  35. Samuel R. Buss, S. Barry Cooper, Benedikt Löwe & Andrea Sorbi (2009). Preface. Annals of Pure and Applied Logic 160 (3):229-230.
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  36.  19
    S. B. Cooper, T. A. Slaman & S. S. Wainer (eds.) (1996). Computability, Enumerability, Unsolvability: Directions in Recursion Theory. Cambridge University Press.
    The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it is hoped will (...)
    Direct download  
     
    Export citation  
     
    My bibliography  
  37.  8
    S. B. Cooper & Andrea Sorbi (eds.) (2011). Computability in Context: Computation and Logic in the Real World. World Scientific.
    Recent new paradigms of computation, based on biological and physical models, address in a radically new way questions of efficiency and challenge assumptions ...
    Direct download  
     
    Export citation  
     
    My bibliography  
  38. S. B. Cooper (1975). Degrees of Unsolvability Complementary Between Recursively Enumerable Degrees, Part I. Journal of Symbolic Logic 40 (1):86-86.
    Direct download  
     
    Export citation  
     
    My bibliography  
  39. S. B. Cooper (1998). 1997 European Summer Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 4 (1):55-117.
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  40. S. Barry Cooper (ed.) (2012). How the World Computes.
    No categories
     
    Export citation  
     
    My bibliography  
  41.  10
    S. B. Cooper & J. K. Truss (eds.) (1999). Models and Computability: Invited Papers From Logic Colloquium '97, European Meeting of the Association for Symbolic Logic, Leeds, July 1997. Cambridge University Press.
    Together, Models and Computability and its sister volume Sets and Proofs will provide readers with a comprehensive guide to the current state of mathematical logic. All the authors are leaders in their fields and are drawn from the invited speakers at 'Logic Colloquium '97' (the major international meeting of the Association of Symbolic Logic). It is expected that the breadth and timeliness of these two volumes will prove an invaluable and unique resource for specialists, post-graduate researchers, and the informed and (...)
    Direct download  
     
    Export citation  
     
    My bibliography  
  42. S. B. Cooper (2001). On a Conjecture of Kleene and Post. 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.
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    My bibliography  
  43. S. B. Cooper (1973). Review: J. Myhill, Category Methods in Recursion Theory. [REVIEW] Journal of Symbolic Logic 38 (4):654-654.
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography  
  44. S. Barry Cooper & John K. Truss (2001). Sets and Proofs. Studia Logica 69 (3):446-448.
     
    Export citation  
     
    My bibliography  
  45. S. Barry Cooper (2013). What Makes A Computation Unconventional? In Gordana Dodig-Crnkovic Raffaela Giovagnoli (ed.), Computing Nature. 255--269.
    Direct download (2 more)  
     
    Export citation  
     
    My bibliography