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

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  1. Va Vardanyan, On Provability Resembling Computability, Proving Aa Voronkov & Constructive Logic (1989). Section 2. Model Theory. In Jens Erik Fenstad, Ivan Timofeevich Frolov & Risto Hilpinen (eds.), Logic, Methodology, and Philosophy of Science Viii: Proceedings of the Eighth International Congress of Logic, Methodology, and Philosophy of Science, Moscow, 1987. Sole Distributors for the U.S.A. And Canada, Elsevier Science
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  2. Matthias Jenny (2016). Counterpossibles in Science: The Case of Relative Computability. Noûs.
    I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s more, (...)
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  3. George Boolos, John Burgess, Richard P. & C. Jeffrey (2007). Computability and Logic. Cambridge University Press.
    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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  4.  23
    Andrew Arana (2015). Review of Computability: Turing, Gödel, Church, and Beyond. [REVIEW] Notre Dame Philosophical Reviews 3 (20).
    A review of Computability: Turing, Gödel, Church, and Beyond by Copeland, Posy and Shagrir.
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  5.  7
    Giorgi Japaridze (2010). Towards Applied Theories Based on Computability Logic. Journal of Symbolic Logic 75 (2):565-601.
    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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  6.  7
    Armin Hemmerling (1998). Computability of String Functions Over Algebraic Structures Armin Hemmerling. Mathematical Logic Quarterly 44 (1):1-44.
    We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in (...)
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  7.  10
    Ning Zhong & Bing‐Yu Zhang (1999). Lp‐Computability. Mathematical Logic Quarterly 45 (4):449-456.
    In this paper we investigate conditions for Lp-computability which are in accordance with the classical Grzegorczyk notion of computability for a continuous function. For a given computable real number p ≥ 1 and a compact computable rectangle I ⊂ ℝq, we show that an Lp function f ∈ Lp is LP-computable if and only if f is sequentially computable as a linear functional and the Lp-modulus function of f is effectively continuous at the origin of ℝq.
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  8.  9
    N. D. Jones (1997). Computability and Complexity: From a Programming Perspective Vol. 21. MIT Press.
    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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  9.  14
    M. Ziegler (2002). Computability on Regular Subsets of Euclidean Space. Mathematical Logic Quarterly 48 (S1):157-181.
    For the computability of subsets of real numbers, several reasonable notions have been suggested in the literature. We compare these notions in a systematic way by relating them to pairs of ‘basic’ ones. They turn out to coincide for full-dimensional convex sets; but on the more general class of regular sets, they reveal rather interesting ‘weaker/stronger’ relations. This is in contrast to single real numbers and vectors where all ‘reasonable’ notions coincide.
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  10. Armin Hemmerling (1998). Computability Over Structures of Infinite Signature. Mathematical Logic Quarterly 44 (3):394-416.
    Continuing the paper [7], in which the Blum-Shub-Smale approach to computability over the reals has been generalized to arbitrary algebraic structures, this paper deals with computability and recognizability over structures of infinite signature. It begins with discussing related properties of the linear and scalar real structures and of their discrete counterparts over the natural numbers. Then the existence of universal functions is shown to be equivalent to the effective encodability of the underlying structure. Such structures even have universal (...)
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  11.  10
    Yongcheng Wu & Decheng Ding (2005). Computability of Measurable Sets Via Effective Metrics. Mathematical Logic Quarterly 51 (6):543-559.
    We consider how to represent the measurable sets in an infinite measure space. We use sequences of simple measurable sets converging under metrics to represent general measurable sets. Then we study the computability of the measure and the set operators of measurable sets with respect to such representations.
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  12.  5
    Giorgi Japaridze (2013). The Taming of Recurrences in Computability Logic Through Cirquent Calculus, Part II. Archive for Mathematical Logic 52 (1-2):213-259.
    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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  13.  6
    Yongcheng Wu & Decheng Ding (2006). Computability of Measurable Sets Via Effective Topologies. Archive for Mathematical Logic 45 (3):365-379.
    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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  14.  7
    Hiroyasu Kamo & Kiko Kawamura (1999). Computability of Self‐Similar Sets. Mathematical Logic Quarterly 45 (1):23-30.
    We investigate computability of a self-similar set on a Euclidean space. A nonempty compact subset of a Euclidean space is called a self-similar set if it equals to the union of the images of itself by some set of contractions. The main result in this paper is that if all of the contractions are computable, then the self-similar set is a recursive compact set. A further result on the case that the self-similar set forms a curve is also discussed.
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  15.  7
    Giorgi Japaridze (2013). The Taming of Recurrences in Computability Logic Through Cirquent Calculus, Part I. Archive for Mathematical Logic 52 (1-2):173-212.
    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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  16.  8
    Wenyan Xu & Sanyang Liu (2013). The Parallel Versus Branching Recurrences in Computability Logic. Notre Dame Journal of Formal Logic 54 (1):61-78.
    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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  17.  2
    F. Dahlgren (2004). Computability and Continuity in Computable Metric Partial Algebras Equipped with Computability Structures. Mathematical Logic Quarterly 50 (4):486.
    In this paper we give an axiomatisation of the concept of a computability structure with partial sequences on a many-sorted metric partial algebra, thus extending the axiomatisation given by Pour-El and Richards in [9] for Banach spaces. We show that every Banach-Mazur computable partial function from an effectively separable computable metric partial Σ-algebra A to a computable metric partial Σ-algebra B must be continuous, and conversely, that every effectively continuous partial function with semidecidable domain and which preserves the (...) of a computably enumerable dense set must be computable. Finally, as an application of these results we give an alternative proof of the first main theorem for Banach spaces first proved by Pour-El and Richards. (shrink)
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  18.  12
    S. Barry Cooper (2004). Computability Theory. Chapman & Hall.
  19.  12
    Stéphane Le Roux & Martin Ziegler (2008). Singular Coverings and Non‐Uniform Notions of Closed Set Computability. Mathematical Logic Quarterly 54 (5):545-560.
    The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe have asserted the existence of non-empty co-r. e. closed sets devoid of computable points: sets which are even “large” in the sense of positive Lebesgue measure.This leads us to investigate for various classes of computable real subsets whether they always contain a computable point.
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  20.  73
    Robert F. Hadley (2008). Consistency, Turing Computability and Gödel's First Incompleteness Theorem. Minds and Machines 18 (1):1-15.
    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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  21.  57
    Eli Dresner (2008). Turing-, Human- and Physical Computability: An Unasked Question. [REVIEW] Minds and Machines 18 (3):349-355.
  22.  13
    H. Rogers (1987). Theory of Recursive Functions and Effective Computability. MIT Press.
  23.  86
    Martin Davis (1958). Computability & Unsolvability. Dover.
    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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  24.  71
    Mathieu Hoyrup (2013). Computability of the Ergodic Decomposition. Annals of Pure and Applied Logic 164 (5):542-549.
    The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than non-ergodic ones. In a previous paper we studied the extent to which non-ergodic measures inherit the computability properties of ergodic ones, and introduced the notion of an effectively decomposable measure. We asked the (...)
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  25.  4
    Giorgi Japaridze (2003). Introduction to Computability Logic. Annals of Pure and Applied Logic 123 (1-3):1-99.
    This work is an attempt to lay foundations for a theory of interactive computation and bring logic and theory of computing closer together. It semantically introduces a logic of computability and sets a program for studying various aspects of that logic. The intuitive notion of computational problems is formalized as a certain new, procedural-rule-free sort of games between the machine and the environment, and computability is understood as existence of an interactive Turing machine that wins the game against (...)
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  26. Nachum Dershowitz & Yuri Gurevich (2008). A Natural Axiomatization of Computability and Proof of Church's Thesis. Bulletin of Symbolic Logic 14 (3):299-350.
    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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  27.  31
    Robert I. Soare (1996). Computability and Recursion. Bulletin of Symbolic Logic 2 (3):284-321.
    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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  28.  38
    B. Jack Copeland & Diane Proudfoot (2010). Deviant Encodings and Turing's Analysis of Computability. Studies in History and Philosophy of Science Part A 41 (3):247-252.
    Turing’s analysis of computability has recently been challenged; it is claimed that it is circular to analyse the intuitive concept of numerical computability in terms of the Turing machine. This claim threatens the view, canonical in mathematics and cognitive science, that the concept of a systematic procedure or algorithm is to be explicated by reference to the capacities of Turing machines. We defend Turing’s analysis against the challenge of ‘deviant encodings’.Keywords: Systematic procedure; Turing machine; Church–Turing thesis; Deviant encoding; (...)
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  29.  35
    Robert I. Soare (2004). Computability Theory and Differential Geometry. Bulletin of Symbolic Logic 10 (4):457-486.
    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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  30. Michael Rescorla (2007). Church's Thesis and the Conceptual Analysis of Computability. Notre Dame Journal of Formal Logic 48 (2):253-280.
    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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  31.  84
    Wayne C. Myrvold (1995). Computability in Quantum Mechanics. In Werner De Pauli-Schimanovich, Eckehart Köhler & Friedrich Stadler (eds.), Vienna Circle Institute Yearbook. Kluwer Academic Publishers 33-46.
    In this paper, the issues of computability and constructivity in the mathematics of physics are discussed. The sorts of questions to be addressed are those which might be expressed, roughly, as: Are the mathematical foundations of our current theories unavoidably non-constructive: or, Are the laws of physics computable?
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  32.  26
    W. Sieg (2006). Godel on Computability. Philosophia Mathematica 14 (2):189-207.
    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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  33.  1
    Robert I. Soare (2009). Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory. Annals of Pure and Applied Logic 160 (3):368-399.
    We begin with the history of the discovery of computability in the 1930’s, the roles of Gödel, Church, and Turing, and the formalisms of recursive functions and Turing automatic machines . To whom did Gödel credit the definition of a computable function? We present Turing’s notion [1939, §4] of an oracle machine and Post’s development of it in [1944, §11], [1948], and finally Kleene-Post [1954] into its present form. A number of topics arose from Turing functionals including continuous functionals (...)
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  34.  6
    Vasco Brattka (2008). Borel Complexity and Computability of the Hahn–Banach Theorem. Archive for Mathematical Logic 46 (7-8):547-564.
    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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  35.  39
    Mark Hogarth (1994). Non-Turing Computers and Non-Turing Computability. Psa 1994:126--138.
    A true Turing machine (TM) requires an infinitely long paper tape. Thus a TM can be housed in the infinite world of Newtonian spacetime (the spacetime of common sense), but not necessarily in our world, because our world-at least according to our best spacetime theory, general relativity-may be finite. All the same, one can argue for the "existence" of a TM on the basis that there is no such housing problem in some other relativistic worlds that are similar ("close") to (...)
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  36. Dag Normann (2006). Computing with Functionals: Computability Theory or Computer Science? Bulletin of Symbolic Logic 12 (1):43-59.
    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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  37.  45
    Nigel Cutland (1980). Computability, an Introduction to Recursive Function Theory. Cambridge University Press.
    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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  38.  12
    Karen Lange & Robert I. Soare (2007). Computability of Homogeneous Models. Notre Dame Journal of Formal Logic 48 (1):143-170.
    In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous models," (...)
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  39. Gunther Mainhardt (2004). P Versus Np and Computability Theoretic Constructions in Complexity Theory Over Algebraic Structures. Journal of Symbolic Logic 69 (1):39-64.
    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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  40.  34
    Michael Rescorla (2012). Copeland and Proudfoot on Computability. Studies in History and Philosophy of Science Part A 43 (1):199-202.
    Many philosophers contend that Turing’s work provides a conceptual analysis of numerical computability. In (Rescorla, 2007), I dissented. I argued that the problem of deviant notations stymies existing attempts at conceptual analysis. Copeland and Proudfoot respond to my critique. I argue that their putative solution does not succeed. We are still awaiting a genuine conceptual analysis.
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  41.  1
    Giorgi Japaridze (2007). The Intuitionistic Fragment of Computability Logic at the Propositional Level. Annals of Pure and Applied Logic 147 (3):187-227.
    This paper presents a soundness and completeness proof for propositional intuitionistic calculus with respect to the semantics of computability logic. The latter interprets formulas as interactive computational problems, formalized as games between a machine and its environment. Intuitionistic implication is understood as algorithmic reduction in the weakest possible — and hence most natural — sense, disjunction and conjunction as deterministic-choice combinations of problems , and “absurd” as a computational problem of universal strength.
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  42.  24
    Wilfried Sieg, Church Without Dogma: Axioms for Computability.
    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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  43.  15
    Ayda I. Arruda, Newton C. A. Costa & 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.
  44.  18
    E. Börger (1989). Computability, Complexity, Logic. New York, N.Y., U.S.A.Elsevier Science Pub. Co..
    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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  45.  10
    Richard L. Epstein & Walter A. Carnielli (1989). Computability Computable Functions, Logic, and the Foundations of Mathematics. Monograph Collection (Matt - Pseudo).
    This book is dedicated to a classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient (...)
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  46.  12
    Jennifer Chubb, Jeffry L. Hirst & Timothy H. McNicholl (2009). Reverse Mathematics, Computability, and Partitions of Trees. Journal of Symbolic Logic 74 (1):201-215.
    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.  37
    Mark Silcox & Jon Cogburn (2006). Computability Theory and Literary Competence. British Journal of Aesthetics 46 (4):369-386.
    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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  48.  15
    Barbara F. Csima & Robert I. Soare (2006). Computability Results Used in Differential Geometry. Journal of Symbolic Logic 71 (4):1394 - 1410.
    Topologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their results depended on the existence of certain sequences of c.e. sets, constructed at their request by Csima and Soare, whose settling times had the necessary dominating properties. (...)
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  49. Shawn Hedman (2004). A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity. Oxford University Press.
    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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  50. Shawn Hedman (2004). A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity. Oxford University Press Uk.
    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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