Search results for 'Computability' (try it on Scholar)

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  1. Giorgi Japaridze (2010). Towards Applied Theories Based on Computability Logic. Journal of Symbolic Logic 75 (2):565-601.score: 18.0
    Computability logic (CL) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, "truth" means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves its soundness and (...)
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  2. Wenyan Xu & Sanyang Liu (2013). The Parallel Versus Branching Recurrences in Computability Logic. Notre Dame Journal of Formal Logic 54 (1):61-78.score: 18.0
    This paper shows that the basic logic induced by the parallel recurrence $\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}}$ of computability logic (i.e., the one in the signature $\{\neg,$\wedge$,\vee,\hspace {-2pt}\mbox {\raisebox {-0.01pt}{\@setfontsize \small {7}{8}$\wedge$}\hspace {-3.55pt}\raisebox {4.5pt}{\tiny $\mid$}\hspace {2pt}},\hspace {-2pt}\mbox {\raisebox {0.12cm}{\@setfontsize \small {7}{8}$\vee$}\hspace {-3.6pt}\raisebox {0.02cm}{\tiny $\mid$}\hspace {2pt}}\}$ ) is a proper superset of the basic logic induced by the branching recurrence $\mbox {\raisebox {-0.05cm}{$\circ$}\hspace {-0.11cm}\raisebox {3.1pt}{\tiny $\mid$}\hspace {2pt}}$ (i.e., the one in the signature $\{\neg,$\wedge$,\vee,\mbox {\raisebox {-0.05cm}{$\circ$}\hspace (...)
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  3. Giorgi Japaridze (2013). The Taming of Recurrences in Computability Logic Through Cirquent Calculus, Part II. Archive for Mathematical Logic 52 (1-2):213-259.score: 18.0
    This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ${{\neg}}$ , parallel conjunction ${{\wedge}}$ , parallel disjunction ${{\vee}}$ , branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (the previous) Part I containing preliminaries and a soundness proof, and (the present) Part II containing a completeness proof.
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  4. Giorgi Japaridze (2013). The Taming of Recurrences in Computability Logic Through Cirquent Calculus, Part I. Archive for Mathematical Logic 52 (1-2):173-212.score: 18.0
    This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ${\neg}$ , parallel conjunction ${\wedge}$ , parallel disjunction ${\vee}$ , branching recurrence ⫰, and branching corecurrence ⫯. The article is published in two parts, with (the present) Part I containing preliminaries and a soundness proof, and (the forthcoming) Part II containing a completeness proof.
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  5. N. D. Jones (1997). Computability and Complexity: From a Programming Perspective Vol. 21. Mit Press.score: 18.0
    This makes his book especially valuable." -- Yuri Gurevich, Professor of Computer Science, University of Michigan Computability and complexity theory should be of central concern to practitioners as well as theorists.
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  6. Yongcheng Wu & Decheng Ding (2006). Computability of Measurable Sets Via Effective Topologies. Archive for Mathematical Logic 45 (3):365-379.score: 18.0
    We investigate in the frame of TTE the computability of functions of the measurable sets from an infinite computable measure space such as the measure and the four kinds of set operations. We first present a series of undecidability and incomputability results about measurable sets. Then we construct several examples of computable topological spaces from the abstract infinite computable measure space, and analyze the computability of the considered functions via respectively each of the standard representations of the computable (...)
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  7. Robert F. Hadley (2008). Consistency, Turing Computability and Gödel's First Incompleteness Theorem. Minds and Machines 18 (1):1-15.score: 16.0
    It is well understood and appreciated that Gödel’s Incompleteness Theorems apply to sufficiently strong, formal deductive systems. In particular, the theorems apply to systems which are adequate for conventional number theory. Less well known is that there exist algorithms which can be applied to such a system to generate a gödel-sentence for that system. Although the generation of a sentence is not equivalent to proving its truth, the present paper argues that the existence of these algorithms, when conjoined with Gödel’s (...)
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  8. Eli Dresner (2008). Turing-, Human- and Physical Computability: An Unasked Question. [REVIEW] Minds and Machines 18 (3):349-355.score: 15.0
  9. F. Dahlgren (2004). Computability and Continuity in Computable Metric Partial Algebras Equipped with Computability Structures. Mathematical Logic Quarterly 50 (4):486.score: 15.0
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  10. Armin Hemmerling (1998). Computability of String Functions Over Algebraic Structures Armin Hemmerling. Mathematical Logic Quarterly 44 (1):1-44.score: 15.0
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  11. Armin Hemmerling (1998). Computability Over Structures of Infinite Signature. Mathematical Logic Quarterly 44 (3):394-416.score: 15.0
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  12. Hiroyasu Kamo & Kiko Kawamura (1999). Computability of Self‐Similar Sets. Mathematical Logic Quarterly 45 (1):23-30.score: 15.0
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  13. Stéphane Le Roux & Martin Ziegler (2008). Singular Coverings and Non‐Uniform Notions of Closed Set Computability. Mathematical Logic Quarterly 54 (5):545-560.score: 15.0
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  14. Yongcheng Wu & Decheng Ding (2005). Computability of Measurable Sets Via Effective Metrics. Mathematical Logic Quarterly 51 (6):543-559.score: 15.0
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  15. Ning Zhong & Bing‐Yu Zhang (1999). Lp‐Computability. Mathematical Logic Quarterly 45 (4):449-456.score: 15.0
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  16. M. Ziegler (2002). Computability on Regular Subsets of Euclidean Space. Mathematical Logic Quarterly 48 (S1):157-181.score: 15.0
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  17. George Boolos, John Burgess, Richard P. & C. Jeffrey (2007). Computability and Logic. Cambridge University Press.score: 14.0
    Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel’s incompleteness theorems, but also a large number of optional topics, from Turing’s theory of computability to Ramsey’s theorem. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a new and simpler treatment of the representability of recursive (...)
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  18. Nachum Dershowitz & Yuri Gurevich (2008). A Natural Axiomatization of Computability and Proof of Church's Thesis. Bulletin of Symbolic Logic 14 (3):299-350.score: 14.0
    Church's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turing-computable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of (...)
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  19. Michael Rescorla (2007). Church's Thesis and the Conceptual Analysis of Computability. Notre Dame Journal of Formal Logic 48 (2):253-280.score: 14.0
    Church's thesis asserts that a number-theoretic function is intuitively computable if and only if it is recursive. A related thesis asserts that Turing's work yields a conceptual analysis of the intuitive notion of numerical computability. I endorse Church's thesis, but I argue against the related thesis. I argue that purported conceptual analyses based upon Turing's work involve a subtle but persistent circularity. Turing machines manipulate syntactic entities. To specify which number-theoretic function a Turing machine computes, we must correlate these (...)
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  20. Martin Davis (1958/1982). Computability & Unsolvability. Dover.score: 14.0
    Classic text considersgeneral theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, computable functionals, classification of unsolvable decision problems and more.
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  21. Nigel Cutland (1980). Computability, an Introduction to Recursive Function Theory. Cambridge University Press.score: 14.0
    What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). Dr (...)
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  22. E. Börger (1989). Computability, Complexity, Logic. New York, N.Y., U.S.A.Elsevier Science Pub. Co..score: 14.0
    The theme of this book is formed by a pair of concepts: the concept of formal language as carrier of the precise expression of meaning, facts and problems, and the concept of algorithm or calculus, i.e. a formally operating procedure for the solution of precisely described questions and problems. The book is a unified introduction to the modern theory of these concepts, to the way in which they developed first in mathematical logic and computability theory and later in automata (...)
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  23. 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.score: 14.0
    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 (...)
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  24. Ayda I. Arruda, Newton C. A. Costdaa & R. Chuaqui (eds.) (1977). Non-Classical Logics, Model Theory, and Computability: Proceedings of the Third Latin-American Symposium on Mathematical Logic, Campinas, Brazil, July 11-17, 1976. [REVIEW] Sale Distributors for the U.S.A. And Canada, Elsevier/North-Holland.score: 14.0
  25. Vasco Brattka (2008). Borel Complexity and Computability of the Hahn–Banach Theorem. Archive for Mathematical Logic 46 (7-8):547-564.score: 14.0
    The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the (...)
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  26. Melvin Fitting (1987). Computability Theory, Semantics, and Logic Programming. Clarendon Press.score: 14.0
    This book describes computability theory and provides an extensive treatment of data structures and program correctness. It makes accessible some of the author's work on generalized recursion theory, particularly the material on the logic programming language PROLOG, which is currently of great interest. Fitting considers the relation of PROLOG logic programming to the LISP type of language.
     
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  27. Shawn Hedman (2004). A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity. Oxford University Press.score: 12.0
    The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, (...)
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  28. B. Maclennan (2003). Transcending Turing Computability. Minds and Machines 13 (1):3-22.score: 12.0
    It has been argued that neural networks and other forms of analog computation may transcend the limits of Turing-machine computation; proofs have been offered on both sides, subject to differing assumptions. In this article I argue that the important comparisons between the two models of computation are not so much mathematical as epistemological. The Turing-machine model makes assumptions about information representation and processing that are badly matched to the realities of natural computation (information representation and processing in or inspired by (...)
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  29. Mark Silcox & Jon Cogburn (2006). Computability Theory and Literary Competence. British Journal of Aesthetics 46 (4):369-386.score: 12.0
    criticism defend the idea that an individual reader's understanding of a text can be a factor in determining the meaning of what is written in that text, and hence must play a part in determining the very identity conditions of works of literary art. We examine some accounts that have been given of the type of readerly ‘competence’ that a reader must have in order for her responses to a text to play this sort of constitutive role. We argue that (...)
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  30. Matthew W. Parker (2009). Computing the Uncomputable; or, the Discrete Charm of Second-Order Simulacra. Synthese 169 (3):447 - 463.score: 12.0
    We examine a case in which non-computable behavior in a model is revealed by computer simulation. This is possible due to differing notions of computability for sets in a continuous space. The argument originally given for the validity of the simulation involves a simpler simulation of the simulation , still further simulations thereof, and a universality conjecture. There are difficulties with that argument, but there are other, heuristic arguments supporting the qualitative results. It is urged, using this example, that (...)
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  31. Edward R. Griffor (ed.) (1999). Handbook of Computability Theory. Elsevier.score: 12.0
    The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of (...)
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  32. Oron Shagrir (2002). Effective Computation by Humans and Machines. Minds and Machines 12 (2):221-240.score: 12.0
    There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly (...)
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  33. Daniele Mundici & Wilfried Sieg, Computability Theory.score: 12.0
    Daniele Mundici and Wilfred Sieg. Computability Theory.
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  34. S. B. Cooper, T. A. Slaman & S. S. Wainer (eds.) (1996). Computability, Enumerability, Unsolvability: Directions in Recursion Theory. Cambridge University Press.score: 12.0
    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 (...)
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  35. Wilfried Sieg, Church Without Dogma: Axioms for Computability.score: 12.0
    Church's and Turing's theses dogmatically assert that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses. To investigate effective calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting (...)
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  36. Nir Fresco (2011). Concrete Digital Computation: What Does It Take for a Physical System to Compute? [REVIEW] Journal of Logic, Language and Information 20 (4):513-537.score: 12.0
    This paper deals with the question: what are the key requirements for a physical system to perform digital computation? Time and again cognitive scientists are quick to employ the notion of computation simpliciter when asserting basically that cognitive activities are computational. They employ this notion as if there was or is a consensus on just what it takes for a physical system to perform computation, and in particular digital computation. Some cognitive scientists in referring to digital computation simply adhere to (...)
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  37. Robert I. Soare (1996). Computability and Recursion. Bulletin of Symbolic Logic 2 (3):284-321.score: 12.0
    We consider the informal concept of "computability" or "effective calculability" and two of the formalisms commonly used to define it, "(Turing) computability" and "(general) recursiveness". We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. (...)
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  38. Steve Awodey, Lars Birkedal & Dana Scott, Local Realizability Toposes and a Modal Logic for Computability.score: 12.0
    This work is a step toward the development of a logic for types and computation that includes not only the usual spaces of mathematics and constructions, but also spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes that we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we first study axiomatically, and then by deriving a modal calculus as its internal logic. The (...)
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  39. Herbert B. Enderton (2011). Computability Theory: An Introduction to Recursion Theory. Academic Press.score: 12.0
    Machine generated contents note: 1. The Computability Concept;2. General Recursive Functions;3. Programs and Machines;4. Recursive Enumerability;5. Connections to Logic;6. Degrees of Unsolvability;7. Polynomial-Time Computability;Appendix: Mathspeak;Appendix: Countability;Appendix: Decadic Notation;.
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  40. Dale Jacquette (forthcoming). Computable Diagonalizations and Turing's Cardinality Paradox. Journal for General Philosophy of Science:1-24.score: 12.0
    A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of (...)
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  41. Gunther Mainhardt (2004). P Versus Np and Computability Theoretic Constructions in Complexity Theory Over Algebraic Structures. Journal of Symbolic Logic 69 (1):39-64.score: 12.0
    We show that there is a structure of countably infinite signature with $P = N_{2}P$ and a structure of finite signature with $P = N_{1}P$ and $N_{1}P \neq N_{2}P$ . We give a further example of a structure of finite signature with $P \neq N_{1}P$ and $N_{1}P \neq N_{2}P$ . Together with a result from [10] this implies that for each possibility of P versus NP over structures there is an example of countably infinite signature. Then we show that for (...)
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  42. Stewart Shapiro (1983). Remarks on the Development of Computability. History and Philosophy of Logic 4 (1-2):203-220.score: 12.0
    The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized and the theory of (...)
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  43. W. Sieg (2006). Godel on Computability. Philosophia Mathematica 14 (2):189-207.score: 12.0
    The identification of an informal concept of ‘effective calculability’ with a rigorous mathematical notion like ‘recursiveness’ or ‘Turing computability’ is still viewed as problematic, and I think rightly so. I analyze three different and conflicting perspectives Gödel articulated in the three decades from 1934 to 1964. The significant shifts in Gödel's position underline the difficulties of the methodological issues surrounding the Church-Turing Thesis.
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  44. Mike Stannett (2003). Computation and Hypercomputation. Minds and Machines 13 (1):115-153.score: 12.0
    Does Nature permit the implementation of behaviours that cannot be simulated computationally? We consider the meaning of physical computation in some detail, and present arguments in favour of physical hypercomputation: for example, modern scientific method does not allow the specification of any experiment capable of refuting hypercomputation. We consider the implications of relativistic algorithms capable of solving the (Turing) Halting Problem. We also reject as a fallacy the argument that hypercomputation has no relevance because non-computable values are indistinguishable from sufficiently (...)
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  45. Robert I. Soare (2004). Computability Theory and Differential Geometry. Bulletin of Symbolic Logic 10 (4):457-486.score: 12.0
    Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature | K | ≤ 1. Let Met (M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met (M) such as the diameter. They showed that for every Turing machine T e , e ∈ ω, there is a sequence (uniformly effective in e) of homology (...)
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  46. Jennifer Chubb, Jeffry L. Hirst & Timothy H. McNicholl (2009). Reverse Mathematics, Computability, and Partitions of Trees. Journal of Symbolic Logic 74 (1):201-215.score: 12.0
    We examine the reverse mathematics and computability theory of a form of Ramsey's theorem in which the linear n-tuples of a binary tree are colored.
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  47. S. B. Cooper & Andrea Sorbi (eds.) (2011). Computability in Context: Computation and Logic in the Real World. World Scientific.score: 12.0
    Recent new paradigms of computation, based on biological and physical models, address in a radically new way questions of efficiency and challenge assumptions ...
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  48. Fernando Ferreira (ed.) (2010). Programs, Proofs, Processes: 6th Conference on Computability in Europe, Cie, 2010, Ponta Delgada, Azores, Portugal, June 30-July 4, 2010 ; Proceedings. [REVIEW] Springer.score: 12.0
    The LNCS series reports state-of-the-art results in computer science research, development, and education, at a high level and in both printed and electronic form.
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  49. James Cain (1999). The Theory of Computability Developed in Terms of Satisfaction. Notre Dame Journal of Formal Logic 40 (4):515-532.score: 12.0
    The notion of computability is developed through the study of the behavior of a set of languages interpreted over the natural numbers which contain their own fully defined satisfaction predicate and whose only other vocabulary is limited to0, individual variables, the successor function, the identity relation and operators for disjunction, conjunction, and existential quantification.
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  50. Dag Normann (2006). Computing with Functionals: Computability Theory or Computer Science? Bulletin of Symbolic Logic 12 (1):43-59.score: 12.0
    We review some of the history of the computability theory of functionals of higher types, and we will demonstrate how contributions from logic and theoretical computer science have shaped this still active subject.
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