Search results for 'Recursively defined function' (try it on Scholar)

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  1. Laureano Luna (2010). Ungrounded Causal Chains and Beginningless Time. Logic and Logical Philosophy 18 (3-4):297-307.score: 270.0
    We use two logical resources, namely, the notion of recursively defined function and the Benardete-Yablo paradox, together with some inherent features of causality and time, as usually conceived, to derive two results: that no ungrounded causal chain exists and that time has a beginning.
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  2. Laureano Luna (2014). No Successfull Infinite Regress. Logic and Logical Philosophy 23 (2).score: 238.0
    We model infinite regress structures -not arguments- by means of ungrounded recursively defined functions in order to show that no such structure can perform the task of providing determination to the items composing it, that is, that no determination process containing an infinite regress structure is successful.
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  3. Laureano Luna (forthcoming). No Successful Infinite Regress. Logic and Logical Philosophy.score: 238.0
    We model infinite regress structures -not arguments- by means of ungrounded recursively defined functions in order to show that no such structure can perform the task of providing determination to the items composing it, that is, that no determination process containing an infinite regress structure is successful.
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  4. Hugues Leblanc & Peter Roeper (1992). Probability Functions: The Matter of Their Recursive Definability. Philosophy of Science 59 (3):372-388.score: 116.0
    This paper studies the extent to which probability functions are recursively definable. It proves, in particular, that the (absolute) probability of a statement A is recursively definable from a certain point on, to wit: from the (absolute) probabilities of certain atomic components and conjunctions of atomic components of A on, but to no further extent. And it proves that, generally, the probability of a statement A relative to a statement B is recursively definable from a certain point (...)
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  5. Karl-Heinz Niggl (1997). Non-Definability of the Ackermann Function with Type 1 Partial Primitive Recursion. Archive for Mathematical Logic 37 (1):1-13.score: 116.0
    The paper builds on a simply typed term system ${\cal PR}^\omega $ providing a notion of partial primitive recursive functional on arbitrary Scott domains $D_\sigma$ that includes a suitable concept of parallelism. Computability on the partial continuous functionals seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (SCVR) is not reducible to partial primitive recursion. So an extension ${\cal PR}^{\omega e}$ is studied that is closed under SCVR and yet stays within the realm of subrecursiveness. The twist (...)
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  6. Jesse B. Wright (1972). Characterization of Recursively Enumerable Sets. Journal of Symbolic Logic 37 (3):507-511.score: 110.0
    Let N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets $P, Q, R \subseteq N^2$ operations of transposition, composition, and bracketing are defined as follows: $P^\cup = \{\langle x, y\rangle | \langle y, x\rangle \epsilon P\}, PQ = \{\langle x, z\rangle| \exists y\langle x, y\rangle \epsilon P & \langle y, z\rangle \epsilon Q\}$ , and [ P, Q, R] = (...)
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  7. Damian Rössler (2013). Infinitely $P$-Divisible Points on Abelian Varieties Defined Over Function Fields of Characteristic $Pgt 0$. Notre Dame Journal of Formal Logic 54 (3-4):579-589.score: 108.0
    In this article we consider some questions raised by F. Benoist, E. Bouscaren, and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is $\mathbb{Z}$, then there are no infinitely $p$-divisible points of order a power of $p$.
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  8. Oded Ghitza (2013). The Theta-Syllable: A Unit of Speech Information Defined by Cortical Function. Frontiers in Psychology 4.score: 96.0
    A recent commentary (Oscillators and syllables: a cautionary note. Cummins, 2012) questions the validity of a class of speech perception models inspired by the possible role of neuronal oscillations in decoding speech (e.g., Ghitza 2011, Giraud & Poeppel 2012). In arguing against the approach, Cummins raises a cautionary flag “from a phonetician’s point of view.” Here we respond to his arguments from an auditory processing viewpoint, referring to a phenomenological model of Ghitza (2011) taken as a representative of the criticized (...)
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  9. Charles Morgan, Local Connectedness and Distance Functions.score: 90.0
    Local connectedness functions for (κ, 1)-simplified morasses, localisations of the coupling function c studied in [M96, §1], are defined and their elementary properties discussed. Several different, useful, canonical ways of arriving at the functions are examined. This analysis is then used to give explicit formulae for generalisations of the local distance functions which were defined recursively in [K00], leading to simple proofs of the principal properties of those functions. It is then extended to the properties of (...)
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  10. Vincent W. J. Van Gerven Oei (2012). Cumposition: Theses on Philosophy's Etymology. Continent 2 (1).score: 87.0
    continent. 2.1 (2012): 44–55. Philosophers are sperm, poetry erupts sperm and dribbles, philosopher recodes term, to terminate, —A. Staley Groves 1 There is, in the relation of human languages to that of things, something that can be approximately described as “overnaming”—the deepest linguistic reason for all melancholy and (from the point of view of the thing) for all deliberate muteness. Overnaming as the linguistic being of melancholy points to another curious relation of language: the overprecision that obtains in the tragic (...)
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  11. W. W. Tait (1965). Functionals Defined by Transfinite Recursion. Journal of Symbolic Logic 30 (2):155-174.score: 81.0
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  12. Richard Beigel, William Gasarch, Martin Kummer, Georgia Martin, Timothy McNicholl & Frank Stephan (2000). The Complexity of Oddan. Journal of Symbolic Logic 65 (1):1 - 18.score: 81.0
    For a fixed set A, the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A: $ODD^A_n = \{(x_1, \dots ,x_n): {\tt\#}^A_n(x_1, \dots, x_n) \text{is odd}\}$ and, for m ≥ 2, $\text{MOD}m^A_n = \{(x_1, \dots ,x_n):{\tt\#}^A_n(x_1, \dots ,x_n) \not\equiv 0 (\text{mod} m)\},$ where ${\tt\#}^A_n(x_1, \dots ,x_n) = A(x_1)+\cdots+A(x_n)$ . (We identify A(x) with χ A (x), where χ A (...)
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  13. J. W. Addison (2004). Tarski's Theory of Definability: Common Themes in Descriptive Set Theory, Recursive Function Theory, Classical Pure Logic, and Finite-Universe Logic. Annals of Pure and Applied Logic 126 (1-3):77-92.score: 81.0
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  14. J. C. E. Dekker (1990). An Isolic Generalization of Cauchy's Theorem for Finite Groups. Archive for Mathematical Logic 29 (4):231-236.score: 81.0
    In his note [5] Hausner states a simple combinatorial principle, namely: $$(H)\left\{ {\begin{array}{*{20}c} {if f is a function a non - empty finite set \sigma into itself, p a} \\ {prime, f^p = i_\sigma and \sigma _0 the set of fixed points of f, then } \\ {\left| \sigma \right| \equiv \left| {\sigma _0 } \right|(mod p).} \\\end{array}} \right.$$ .He then shows how this principle can be used to prove:Fermat's little theorem,Cauchy's theorem for finite groups,Lucas' theorem for binomial numbers.Letε=(0,1, (...)
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  15. Luis Elpidio Sanchis (1967). Functionals Defined by Recursion. Notre Dame Journal of Formal Logic 8 (3):161-174.score: 81.0
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  16. R. E. Vesley (1966). Review: W. W. Tait, Functionals Defined by Transfinite Recursion. [REVIEW] Journal of Symbolic Logic 31 (3):509-510.score: 81.0
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  17. R. E. Vesley (1996). Tait WW. Functionals Defined by Transfinite Recursion. Journal of Symbolic Logic 31 (3):509-510.score: 81.0
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  18. Peter Smith, Expressing and Capturing the Primitive Recursive Functions.score: 79.0
    The last Episode wasn’t about logic or formal theories at all: it was about common-or-garden arithmetic and the informal notion of computability. We noted that addition can be defined in terms of repeated applications of the successor function. Multiplication can be defined in terms of repeated applications of addition. The exponential and factorial functions can be defined, in different ways, in terms of repeated applications of multiplication. There’s already a pattern emerging here! The main task in (...)
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  19. Nigel Cutland (1980). Computability, an Introduction to Recursive Function Theory. Cambridge University Press.score: 76.0
    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). (...)
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  20. Manuel L. Campagnolo & Kerry Ojakian (2008). The Elementary Computable Functions Over the Real Numbers: Applying Two New Techniques. [REVIEW] Archive for Mathematical Logic 46 (7-8):593-627.score: 76.0
    The basic motivation behind this work is to tie together various computational complexity classes, whether over different domains such as the naturals or the reals, or whether defined in different manners, via function algebras (Real Recursive Functions) or via Turing Machines (Computable Analysis). We provide general tools for investigating these issues, using two techniques we call approximation and lifting. We use these methods to obtain two main theorems. First, we provide an alternative proof of the result from Campagnolo (...)
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  21. Michael Rathjen (1994). Collapsing Functions Based on Recursively Large Ordinals: A Well-Ordering Proof for KPM. [REVIEW] Archive for Mathematical Logic 33 (1):35-55.score: 76.0
    It is shown how the strong ordinal notation systems that figure in proof theory and have been previously defined by employing large cardinals, can be developed directly on the basis of their recursively large counterparts. Thereby we provide a completely new approach to well-ordering proofs as will be exemplified by determining the proof-theoretic ordinal of the systemKPM of [R91].
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  22. Cristian Calude, Gabriel Istrate & Marius Zimand (1992). Recursive Baire Classification and Speedable Functions. Mathematical Logic Quarterly 38 (1):169-178.score: 72.0
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  23. Mary Jane Dinardo & Thomas C. Toppino (1984). Formation of Ill-Defined Concepts as a Function of Category Size and Category Exposure. Bulletin of the Psychonomic Society 22 (4):317-320.score: 72.0
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  24. Efrain C. Azmitia & Particia M. Whitaker-Azmitia (1986). Searching for an Ill-Defined Brain Function Results in an Uneasy Reconciliation. Behavioral and Brain Sciences 9 (2):335.score: 72.0
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  25. F. Bagemihl (1971). A Property of the Function Ψ(Α) Defined by 2ℵα = ℵα+Ψ(Α). Mathematical Logic Quarterly 17 (1):23-24.score: 72.0
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  26. Qing Zhou (1996). Computable Real‐Valued Functions on Recursive Open and Closed Subsets of Euclidean Space. Mathematical Logic Quarterly 42 (1):379-409.score: 72.0
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  27. Michael Rathjen (1992). A Proof-Theoretic Characterization of the Primitive Recursive Set Functions. Journal of Symbolic Logic 57 (3):954-969.score: 69.0
    Let KP- be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V:= universe of sets) be a ▵0-definable set function, i.e. there is a ▵0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and $V \models \forall x \exists!y\varphi (x, y)$ . In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with the collection (...)
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  28. Wolfgang Burr & Volker Hartung (1998). A Characterization of the Sigma_1-Definable Functions of KPomega + (Uniform; AC). Archive for Mathematical Logic 37 (3):199-214.score: 69.0
    The subject of this paper is a characterization of the $\Sigma_1$ -definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: $KP\omega+(uniform\;AC)$ . This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and $x\mapsto\omega$ . This goal is achieved by a Gödel Dialectica-style functional interpretation of $KP\omega+(uniform\;AC)$ and a computability proof for the involved functionals.
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  29. László Á Kóczy (2007). A Recursive Core for Partition Function Form Games. Theory and Decision 63 (1):41-51.score: 69.0
    We present a well-defined generalisation of the core to coalitional games with externalities, where the value of a deviation is given by an endogenous response, the solution (if nonempty: the core) of the residual game.
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  30. William R. Stirton (2012). How to Assign Ordinal Numbers to Combinatory Terms with Polymorphic Types. Archive for Mathematical Logic 51 (5-6):475-501.score: 67.0
    The article investigates a system of polymorphically typed combinatory logic which is equivalent to Gödel’s T. A notion of (strong) reduction is defined over terms of this system and it is proved that the class of well-formed terms is closed under both bracket abstraction and reduction. The main new result is that the number of contractions needed to reduce a term to normal form is computed by an ε 0-recursive function. The ordinal assignments used to obtain this result (...)
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  31. William R. Stirton (2013). A Decidable Theory of Type Assignment. Archive for Mathematical Logic 52 (5-6):631-658.score: 67.0
    This article investigates a theory of type assignment (assigning types to lambda terms) called ETA which is intermediate in strength between the simple theory of type assignment and strong polymorphic theories like Girard’s F (Proofs and types. Cambridge University Press, Cambridge, 1989). It is like the simple theory and unlike F in that the typability and type-checking problems are solvable with respect to ETA. This is proved in the article along with three other main results: (1) all primitive recursive functionals (...)
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  32. Farzad Didehvar (1999). On a Class of Recursively Enumerable Sets. Mathematical Logic Quarterly 45 (4):467-470.score: 66.0
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  33. G. Longo & E. Moggi (1984). The Hereditary Partial Effective Functionals and Recursion Theory in Higher Types. Journal of Symbolic Logic 49 (4):1319-1332.score: 63.0
    A type-structure of partial effective functionals over the natural numbers, based on a canonical enumeration of the partial recursive functions, is developed. These partial functionals, defined by a direct elementary technique, turn out to be the computable elements of the hereditary continuous partial objects; moreover, there is a commutative system of enumerations of any given type by any type below (relative numberings). By this and by results in [1] and [2], the Kleene-Kreisel countable functionals and the hereditary effective operations (...)
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  34. Paulo Oliva & Thomas Powell (2012). On Spector's Bar Recursion. Mathematical Logic Quarterly 58 (4‐5):356-265.score: 63.0
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  35. Stanley S. Wainer (1999). Accessible Recursive Functions. Bulletin of Symbolic Logic 5 (3):367-388.score: 63.0
    The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from "within". On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been "coded" at previous levels. The question is: how can (...)
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  36. Simon Thompson (1985). Axiomatic Recursion Theory and the Continuous Functionals. Journal of Symbolic Logic 50 (2):442-450.score: 63.0
    We define, in the spirit of Fenstad [2], a higher type computation theory, and show that countable recursion over the continuous functionals forms such a theory. We also discuss Hyland's proposal from [4] for a scheme with which to supplement S1-S9, and show that this augmented set of schemes fails to generate countable recursion. We make another proposal to which the methods of this section do not apply.
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  37. Karl-Heinz Niggl (1998). A Restricted Computation Model on Scott Domains and its Partial Primitive Recursive Functionals. Archive for Mathematical Logic 37 (7):443-481.score: 63.0
    The paper builds on both a simply typed term system ${\cal PR}^\omega$ and a computation model on Scott domains via so-called parallel typed while programs (PTWP). The former provides a notion of partial primitive recursive functional on Scott domains $D_\rho$ supporting a suitable concept of parallelism. Computability on Scott domains seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (scvr) is not reducible to partial primitive recursion. So extensions ${\cal PR}^{\omega e}$ and PTWP $^e$ are studied that (...)
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  38. Vincent Astier (2008). Elementary Equivalence of Some Rings of Definable Functions. Archive for Mathematical Logic 47 (4):327-340.score: 56.0
    We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R n to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition (...)
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  39. Bence Nanay (2010). A Modal Theory of Function. Journal of Philosophy 107 (8):412-431.score: 54.0
    The function of a trait token is usually defined in terms of some properties of other (past, present, future) tokens of the same trait type. I argue that this strategy is problematic, as trait types are (at least partly) individuated by their functional properties, which would lead to circularity. In order to avoid this problem, I suggest a way to define the function of a trait token in terms of the properties of the very same trait token. (...)
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  40. Ron Amundson & George V. Lauder (1994). Function Without Purpose. Biology and Philosophy 9 (4):443-469.score: 54.0
    Philosophers of evolutionary biology favor the so-called etiological concept of function according to which the function of a trait is its evolutionary purpose, defined as the effect for which that trait was favored by natural selection. We term this the selected effect (SE) analysis of function. An alternative account of function was introduced by Robert Cummins in a non-evolutionary and non-purposive context. Cummins''s account has received attention but little support from philosophers of biology. This paper (...)
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  41. Richard Martin Pagni (2009). The Origin and Development of the Acidity Function. Foundations of Chemistry 11 (1):43-50.score: 54.0
    The acidity function is a thermodynamic quantitative measure of acid strength for non-aqueous and concentrated aqueous Brønsted acids, with acid strength being defined as the extent to which the acid protonates a base of known basicity. The acidity function, which was developed, both theoretically and experimentally, by Louis P. Hammett of Columbia University during the 1930s, has proven useful in the area of physical organic chemistry where it has been used to correlate rates of acid-catalyzed reactions and (...)
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  42. Shan Gao, The Wave Function and Particle Ontology.score: 54.0
    In quantum mechanics, the wave function of a N-body system is a mathematical function defined in a 3N-dimensional configuration space. We argue that wave function realism implies particle ontology when assuming: (1) the wave function of a N-body system describes N physical entities; (2) each triple of the 3N coordinates of a point in configuration space that relates to one physical entity represents a point in ordinary three-dimensional space. Moreover, the motion of particles is random (...)
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  43. M. Revzen (2006). The Wigner Function as Distribution Function. Foundations of Physics 36 (4):546-562.score: 54.0
    Some entangled states have nonnegative Wigner representative function. The latter allow being viewed as a distribution function of local hidden variables. It is argued herewith that the interpretation of expectation values using such distribution functions as local hidden variable theory requires restrictions pertaining to the observables under study. The reasoning lead to support the view that violation of Bell’s inequalities that is always possible for entangled states hinges not only on the states involved but also whether the dynamical (...)
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  44. Véronique Izard, Pierre Pica, Elizabeth S. Spelke & Stanislas Dehaene (2008). Exact Equality and Successor Function: Two Key Concepts on the Path Towards Understanding Exact Numbers. Philosophical Psychology 21 (4):491 – 505.score: 54.0
    Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...)
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  45. Dingmar van Eck (forthcoming). Validating Function-Based Design Methods: An Explanationist Perspective. Philosophy and Technology:1-21.score: 54.0
    Analysis of the adequacy of engineering design methods, as well as analysis of the utility of concepts of function often invoked in these methods, is a neglected topic in both philosophy of technology and in engineering proper. In this paper, I present an approach—dubbed an explanationist perspective—for assessing the adequacy of function-based design methods. Engineering design is often intertwined with explanation, for instance, in reverse engineering and subsequent redesign, knowledge base-assisted designing, and diagnostic reasoning. I argue that the (...)
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  46. Robert I. Soare (1996). Computability and Recursion. Bulletin of Symbolic Logic 2 (3):284-321.score: 52.0
    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. After a careful (...)
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  47. Jeremy Avigad (2002). An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations. Journal of Mathematical Logic 2 (01):91-112.score: 52.0
    The notion of a function from ℕ to ℕ defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke–Platek admissible set theory with an axiom of infinity. Formalizing the argument provides (...)
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  48. Paolo Casalegno (1985). On the T-Degrees of Partial Functions. Journal of Symbolic Logic 50 (3):580-588.score: 52.0
    Let $\langle\mathscr{T},\leq\rangle$ be the usual structure of the degrees of unsolvability and $\langle\mathscr{D},\leq\rangle$ the structure of the T-degrees of partial functions defined in [7]. We prove that every countable distributive lattice with a least element can be isomorphically embedded as an initial segment of $\langle\mathscr{D},\leq\rangle$ : as a corollary, the first order theory of $\langle\mathscr{D},\leq\rangle$ is recursively isomorphic to that of $\langle\mathscr{T},\leq\rangle$ . We also show that $\langle\mathscr{D},\leq\rangle$ and $\langle\mathscr{T},\leq\rangle$ are not elementarily equivalent.
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  49. Gian Aldo Antonelli (1996). What's in a Function? Synthese 107 (2):167 - 204.score: 52.0
    In this paper we argue that Revision Rules, introduced by Anil Gupta and Nuel Belnap as a tool for the analysis of the concept of truth, also provide a useful tool for defining computable functions. This also makes good on Gupta's and Belnap's claim that Revision Rules provide a general theory of definition, a claim for which they supply only the example of truth. In particular we show how Revision Rules arise naturally from relaxing and generalizing a classical construction due (...)
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  50. Peter Dybjer (2000). A General Formulation of Simultaneous Inductive-Recursive Definitions in Type Theory. Journal of Symbolic Logic 65 (2):525-549.score: 52.0
    The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löf's universe á la Tarski. A set U 0 of codes for small sets is generated inductively at the same time as a function T 0 , which maps a code to the corresponding small set, is defined by recursion on the way the elements of U 0 are generated. In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive definition (...)
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