Search results for 'Combinatory logic' (try it on Scholar)

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  1.  17
    Haskell B. Curry (1958). Combinatory Logic. Amsterdam, North-Holland Pub. Co..
    CHAPTER Addenda to Pure Combinatory Logic This chapter will treat various additions to, and modifications of, the subject matter of Chapters-7. ...
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  2.  4
    J. Roger Hindley (1972). Introduction to Combinatory Logic. Cambridge [Eng.]University Press.
    Introduction Combinatory logic deals with a class of formal systems designed for studying certain primitive ways in which functions can be combined to form ...
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  3.  26
    M. W. Bunder & W. J. M. Dekkers (2005). Equivalences Between Pure Type Systems and Systems of Illative Combinatory Logic. Notre Dame Journal of Formal Logic 46 (2):181-205.
    Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led (...)
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  4.  2
    Jean-Pierre Desclés, Anca Christine Pascu & Hee-Jin Ro (2014). Aspecto-Temporal Meanings Analysed by Combinatory Logic. Journal of Logic, Language and Information 23 (3):253-274.
    What is the meaning of language expressions and how to compute or calculate it? In this paper, we give an answer to this question by analysing the meanings of aspects and tenses in natural languages inside the formal model of an grammar of applicative, cognitive and enunciative operations , using the applicative formalism, functional types of categorial grammars and combinatory logic . In the enunciative theory and following , an utterance can be decomposed into two components: a modus (...)
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  5.  10
    Katalin Bombó (2005). The Church-Rosser Property in Symmetric Combinatory Logic. Journal of Symbolic Logic 70 (2):536 - 556.
    Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a much stronger theorem that no symmetric combinatory logic that contains at least two proper symmetric combinators has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic, the proof is different because symmetric combinators may form redexes in (...)
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  6.  0
    Frederic B. Fitch (1974). Elements of Combinatory Logic. New Haven,Yale University Press.
  7. Maarten Wicher Visser Bunder (1969). Set Theory Based on Combinatory Logic. Groningen, V. R. B. --Offsetdrukkerij (Kleine Der a 3-4).
     
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  8.  0
    Sören Stenlund (1971). Introduction to Combinatory Logic. Uppsala,Uppsala Universitetet, Filosofiska Föreningen Och Filosofiska Institutionen.
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  9.  8
    Katalin Bimbó (2012). Combinatory Logic: Pure, Applied, and Typed. Taylor & Francis.
    Reader-friendly without compromising the precision of exposition, the book includes many new research results not found in the available literature.
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  10.  40
    Lou Goble (2007). Combinatory Logic and the Semantics of Substructural Logics. Studia Logica 85 (2):171 - 197.
    The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant (...)
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  11.  0
    Andrea Cantini (2002). Polytime, Combinatory Logic and Positive Safe Induction. Archive for Mathematical Logic 41 (2):169-189.
    We characterize the polynomial time operations as those which are provably total in a first order system, which comprises (untyped) combinatory logic with extensionality, together with positive “safe induction” on the set of binary strings. The formalization of safe induction is inspired by Leivants idea of ramification. We also show how to replace ramification by means of modal logic.
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  12.  19
    Wil Dekkers, Martin Bunder & Henk Barendregt (1998). Completeness of the Propositions-as-Types Interpretation of Intuitionistic Logic Into Illative Combinatory Logic. Journal of Symbolic Logic 63 (3):869-890.
    Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in (...)
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  13.  18
    Henk Barendregt, Martin Bunder & Wil Dekkers (1993). Systems of Illative Combinatory Logic Complete for First-Order Propositional and Predicate Calculus. Journal of Symbolic Logic 58 (3):769-788.
    Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are (...)
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  14.  4
    Wil Dekkers, Martin Bunder & Henk Barendregt (1998). Completeness of Two Systems of Illative Combinatory Logic for First-Order Propositional and Predicate Calculus. Archive for Mathematical Logic 37 (5-6):327-341.
    Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations (...)
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  15.  4
    Patricia Johann (1994). Normal Forms in Combinatory Logic. Notre Dame Journal of Formal Logic 35 (4):573-594.
    Let $R$ be a convergent term rewriting system, and let $CR$-equality on combinatory logic terms be the equality induced by $\beta \eta R$-equality on terms of the lambda calculus under any of the standard translations between these two frameworks for higher-order reasoning. We generalize the classical notion of strong reduction to a reduction relation which generates $CR$-equality and whose irreducibles are exactly the translates of long $\beta R$-normal forms. The classical notion of strong normal form in combinatory (...)
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  16.  20
    Andrea Cantini (2003). The Axiom of Choice and Combinatory Logic. Journal of Symbolic Logic 68 (4):1091-1108.
    We combine a variety of constructive methods (including forcing, realizability, asymmetric interpretation), to obtain consistency results concerning combinatory logic with extensionality and (forms of) the axiom of choice.
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  17.  11
    Andreas Knobel (1993). Constructive Set Theoretic Models of Typed Combinatory Logic. Journal of Symbolic Logic 58 (1):99-118.
    We shall present two novel ways of deriving simply typed combinatory models. These are of interest in a constructive setting. First we look at extension models, which are certain subalgebras of full function space models. Then we shall show how the space of singletons of a combinatory model can itself be made into one. The two and the algebras in between will have many common features. We use these two constructions in proving: There is a model of constructive (...)
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  18.  1
    M. Randall Holmes (1991). Systems of Combinatory Logic Related to Quine's ‘New Foundations. Annals of Pure and Applied Logic 53 (2):103-133.
    Systems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they are (...)
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  19.  1
    Katalin Bimbó (2005). The Church-Rosser Property in Symmetric Combinatory Logic. Journal of Symbolic Logic 70 (2):536 - 556.
    Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a much stronger theorem that no symmetric combinatory logic that contains at least two proper symmetric combinators has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic, the proof is different because symmetric combinators may form redexes in (...)
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  20.  0
    M. Randall Holmes (1993). Systems of Combinatory Logic Related to Predicative and ‘Mildly Impredicative’ Fragments of Quine's ‘New Foundations. Annals of Pure and Applied Logic 59 (1):45-53.
    This paper extends the results of an earlier paper by the author . New subsystems of the combinatory logic TRC shown in that paper to be equivalent to NF are introduced; these systems are analogous to subsystems of NF with predicativity restrictions on set comprehension introduced and shown to be consistent by Crabbé. For one of these systems, an exact equivalence in consistency strength and expressive power with the analogous subsystem of NF is established.
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  21.  0
    Nikolai Krupski (2006). Typing in Reflective Combinatory Logic. Annals of Pure and Applied Logic 141 (1):243-256.
    We study Artemov’s Reflective Combinatory Logic . We provide the explicit definition of types for and prove that every well-formed term has a unique type. We establish that the typability testing and detailed type restoration can be done in polynomial time and that the derivability relation for is decidable and PSPACE-complete. These results also formalize the intended semantics of the type t:F in . Terms store the complete information about the judgment “t is a term of type F”, (...)
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  22.  24
    M. W. Bunder (1988). Arithmetic Based on the Church Numerals in Illative Combinatory Logic. Studia Logica 47 (2):129 - 143.
    In the early thirties, Church developed predicate calculus within a system based on lambda calculus. Rosser and Kleene developed Arithmetic within this system, but using a Godelization technique showed the system to be inconsistent.Alternative systems to that of Church have been developed, but so far more complex definitions of the natural numbers have had to be used. The present paper based on a system of illative combinatory logic developed previously by the author, does allow the use of the (...)
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  23. Katalin Bimbo (1999). Substructural Logics, Combinatory Logic, and Lambda-Calculus. Dissertation, Indiana University
    The dissertation deals with problems in "logic", more precisely, it deals with particular formal systems aiming at capturing patterns of valid reasoning. Sequent calculi were proposed to characterize logical connectives via introduction rules. These systems customarily also have structural rules which allow one to rearrange the set of premises and conclusions. In the "structurally free logic" of Dunn and Meyer the structural rules are replaced by combinatory rules which allow the same reshuffling of formulae, and additionally introduce (...)
     
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  24.  19
    Sabine Broda & Luís Damas (1997). Compact Bracket Abstraction in Combinatory Logic. Journal of Symbolic Logic 62 (3):729-740.
    Translations from Lambda calculi into combinatory logics can be used to avoid some implementational problems of the former systems. However, this scheme can only be efficient if the translation produces short output with a small number of combinators, in order to reduce the time and transient storage space spent during reduction of combinatory terms. In this paper we present a combinatory system and an abstraction algorithm, based on the original bracket abstraction operator of Schonfinkel [9]. The algorithm (...)
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  25.  3
    Katalin Bimbó (2003). The Church-Rosser Property in Dual Combinatory Logic. Journal of Symbolic Logic 68 (1):132-152.
    Dual combinators emerge from the aim of assigning formulas containing ← as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove (...)
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  26.  12
    Martin W. Bunder, J. Roger Hindley & Jonathan P. Seldin (1989). On Adding (Ξ) to Weak Equality in Combinatory Logic. Journal of Symbolic Logic 54 (2):590-607.
    Because the main difference between combinatory weak equality and λβ-equality is that the rule \begin{equation*}\tag{\xi} X = Y \vdash \lambda x.X = \lambda x.Y\end{equation*} is valid for the latter but not the former, it is easy to assume that another way of defining combinatory β-equality is to add rule (ξ) to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in (ξ) very carefully. If one tries to use (...)
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  27. Haskell B. Curry, J. Roger Hindley & J. P. Seldin (eds.) (1980). To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism. Academic Press.
  28. Raymond M. Smullyan (1985). To Mock a Mocking Bird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic. Knopf.
     
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  29.  4
    M. W. Bunder (1984). Category Theory Based on Combinatory Logic. Archive for Mathematical Logic 24 (1):1-16.
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  30.  4
    M. W. Bunder (1987). Some Generalizations to Two Systems of Set Theory Based on Combinatory Logic. Archive for Mathematical Logic 26 (1):5-12.
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  31.  6
    Frederic B. Fitch (1963). The System Cδ of Combinatory Logic. Journal of Symbolic Logic 28 (1):87-97.
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  32.  48
    Bruce Lercher (1967). Strong Reduction and Normal Form in Combinatory Logic. Journal of Symbolic Logic 32 (2):213-223.
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  33.  10
    M. W. Bunder (1983). A Weak Absolute Consistency Proof for Some Systems of Illative Combinatory Logic. Journal of Symbolic Logic 48 (3):771-776.
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  34.  2
    M. W. Bunder (1979). Paraconsistent Combinatory Logic,„. Bulletin of the Section of Logic 8 (4):177-180.
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  35.  30
    Roger Hindley (1967). Axioms for Strong Reduction in Combinatory Logic. Journal of Symbolic Logic 32 (2):224-236.
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  36.  15
    Frederic B. Fitch (1980). A Consistent Combinatory Logic with an Inverse to Equality. Journal of Symbolic Logic 45 (3):529-543.
  37.  10
    M. W. Bunder (1987). Some Consistency Proofs and a Characterization of Inconsistency Proofs in Illative Combinatory Logic. Journal of Symbolic Logic 52 (1):89-110.
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  38.  22
    S. Kamal Abdali (1976). An Abstraction Algorithm for Combinatory Logic. Journal of Symbolic Logic 41 (1):222-224.
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  39.  4
    M. W. Bunder (1974). Propositional and Predicate Calculuses Based on Combinatory Logic. Notre Dame Journal of Formal Logic 15 (1):25-34.
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  40.  3
    M. W. Bunder (1970). A Paradox in Illative Combinatory Logic. Notre Dame Journal of Formal Logic 11 (4):467-470.
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  41.  2
    Luis E. Sanchis (1979). Reducibilities in Two Models for Combinatory Logic. Journal of Symbolic Logic 44 (2):221-234.
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  42.  2
    Bruce Lercher (1964). Review: Frederic B. Fitch, The System $CDelta$ of Combinatory Logic. [REVIEW] Journal of Symbolic Logic 29 (4):198-199.
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  43.  1
    Beatrice Amrhein (1995). Aspects of Universal Algebra in Combinatory Logic. In Erwin Engeler (ed.), The Combinatory Programme. Birkhäuser 31--45.
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  44.  8
    M. W. Bunder (1974). Various Systems of Set Theory Based on Combinatory Logic. Notre Dame Journal of Formal Logic 15 (2):192-206.
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  45.  2
    Barkley Rosser (1941). Review: Haskell B. Curry, A Revision of the Fundamental Rules of Combinatory Logic; Haskell B. Curry, Consistency and Completeness of the Theory of Combinators. [REVIEW] Journal of Symbolic Logic 6 (4):170-170.
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  46.  8
    Nicolas D. Goodman (1972). A Simplification of Combinatory Logic. Journal of Symbolic Logic 37 (2):225-246.
  47.  8
    Haskell B. Curry (1941). A Revision of the Fundamental Rules of Combinatory Logic. Journal of Symbolic Logic 6 (2):41-53.
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  48.  5
    Remi Legrand (1988). A Basis Result in Combinatory Logic. Journal of Symbolic Logic 53 (4):1224-1226.
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  49.  5
    M. W. Bunder (1977). Consistency Notions in Illative Combinatory Logic. Journal of Symbolic Logic 42 (4):527-529.
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  50.  7
    John T. Kearns (1969). Combinatory Logic with Discriminators. Journal of Symbolic Logic 34 (4):561-575.
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