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Journal articles on the topic 'Combinatory logic'

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Published: 4 June 2021

Last updated: 25 July 2025

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1

Bimbó, Katalin. "The Church-Rosser property in symmetric combinatory logic." Journal of Symbolic Logic 70, no. 2 (2005): 536–56. http://dx.doi.org/10.2178/jsl/1120224727.

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AbstractSymmetic 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 combinatory 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 both left and right associated terms. Perhaps surprisingly, we are also able to show

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2

Legrand, Remi. "A basis result in combinatory logic." Journal of Symbolic Logic 53, no. 4 (1988): 1224–26. http://dx.doi.org/10.1017/s0022481200028048.

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The aim of this article is to show that a basis for combinatory logic [2] must contain at least one combinator with rank strictly greater than two. We use notation of [1].Let Q be a primitive combinator given by its reduction rule Qx1 … xn → C, where C is a pure combination of the variables x1,…, xn. n is called the rank of the combinator.A set {Q1,…,Qn} of combinators is a basis for combinatory logic if for every finite set {x1,…,xm} of variables and every pure combination C of these variables, there exists a pure combinator Q of Q1,…,Qn such that Qx1…xm↠C.Property. The Church-Rosser theorem

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3

Bimbó, Katalin. "The Church-Rosser property in dual combinatory logic." Journal of Symbolic Logic 68, no. 1 (2003): 132–52. http://dx.doi.org/10.2178/jsl/1045861508.

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AbstractDual 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 a general theorem stating that no dual combinatory system possesses the Church-Rosser property. Although th

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4

Broda, Sabine, and Luís Damas. "Compact bracket abstraction in combinatory logic." Journal of Symbolic Logic 62, no. 3 (1997): 729–40. http://dx.doi.org/10.2307/2275570.

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AbstractTranslations 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 Schönfinkel [9]. The algorithm introduces at most one combinator for each abstraction in the initial Lambda t

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5

Bellot, P. "A new proof for Craig's theorem." Journal of Symbolic Logic 50, no. 2 (1985): 395–96. http://dx.doi.org/10.2307/2274227.

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Craig's theorem is a result about the cardinality of a proper basis for the theory of combinators. Its proof given in [3] was shown to be incomplete by André Chauvin [2]. By using a different approach, we give a very short proof of this theorem. We use the notation of [1].Definition 1. A combinator Q is proper if there exists a natural number n such that for arbitrary variables x1,…,xn we have the following contraction rule:where C is a pure combination of the variables x1,…,xn. Q is to be understood as an abstract symbol, not as a combination of S and K's. Therefore Q comes with a contraction

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6

Fehlmann, Thomas, and Eberhard Kranich. "The Fixpoint Combinator in Combinatory Logic – A Step towards Autonomous Real-time Testing of Software?" ATHENS JOURNAL OF SCIENCES 9, no. 1 (2022): 47–64. http://dx.doi.org/10.30958/ajs.9-1-3.

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Combinatory Logic is an elegant and powerful logical theory that is used in computer science as a theoretical model for computation. Its algebraic structure supports self-application and is Turing-complete. However, contrary to Lambda Calculus, it untangles the problem of substitution, because bound variables are eliminated by inserting specific terms called Combinators. It was introduced by Schönfinkel (1924) and Curry (1930). Combinatory Logic uses just one algebraic operation, namely combining two terms, yielding another valid term of Combinatory Logic. Terms in models of Combinatory Logic

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7

Jay, Barry, and Thomas Given-Wilson. "A combinatory account of internal structure." Journal of Symbolic Logic 76, no. 3 (2011): 807–26. http://dx.doi.org/10.2178/jsl/1309952521.

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AbstractTraditional combinatory logic uses combinators S and K to represent all Turing-computable functions on natural numbers, but there are Turing-computable functions on the combinators themselves that cannot be so represented, because they access internal structure in ways that S and K cannot. Much of this expressive power is captured by adding a factorisation combinator F. The resulting SF-calculus is structure complete, in that it supports all pattern-matching functions whose patterns are in normal form, including a function that decides structural equality of arbitrary normal forms. A g

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8

Bunder, M. W. "Expedited Broda-Damas bracket abstraction." Journal of Symbolic Logic 65, no. 4 (2000): 1850–57. http://dx.doi.org/10.2307/2695081.

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AbstractA bracket abstraction algorithm is a means of translating λ-terms into combinators. Broda and Damas, in [1], introduce a new, rather natural set of combinators and a new form of bracket abstraction which introduces at most one combinator for each λ-abstraction. This leads to particularly compact combinatory terms. A disadvantage of their abstraction process is that it includes the whole Schönfinkel [4] algorithm plus two mappings which convert the Schönfinkel abstract into the new abstract. This paper shows how the new abstraction can be done more directly, in fact, using only 2n − 1 a

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9

Barendregt, Henk, Martin Bunder, and Wil Dekkers. "Systems of illative combinatory logic complete for first-order propositional and predicate calculus." Journal of Symbolic Logic 58, no. 3 (1993): 769–88. http://dx.doi.org/10.2307/2275096.

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AbstractIllative 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 not translated. Both translations are closely related in a

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10

Dekkers, Wil, Martin Bunder, and Henk Barendregt. "Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic." Journal of Symbolic Logic 63, no. 3 (1998): 869–90. http://dx.doi.org/10.2307/2586717.

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AbstractIllative 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 a more direct way, in which derivations are not translated. Both tr

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11

Jech, Thomas. "Some results on combinators in the system TRC." Journal of Symbolic Logic 64, no. 4 (1999): 1811–19. http://dx.doi.org/10.2307/2586813.

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12

Hindley, J. Roger, and David Meredith. "Principal type-schemes and condensed detachment." Journal of Symbolic Logic 55, no. 1 (1990): 90–105. http://dx.doi.org/10.2307/2274956.

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The condensed detachment rule, or ruleD, was first proposed by Carew Meredith in the 1950's for propositional logic based on implication. It is a combination of modus ponens with a “minimal” amount of substitution. We shall give a precise detailed statement of rule D. (Some attempts in the published literature to do this have been inaccurate.)The D-completeness question for a given set of logical axioms is whether every formula deducible from the axioms by modus ponens and substitution can be deduced instead by rule D alone. Under the well-known formulae-as-types correspondence between proposi

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13

DI PIERRO, ALESSANDRA, CHRIS HANKIN, and HERBERT WIKLICKY. "Reversible combinatory logic." Mathematical Structures in Computer Science 16, no. 04 (2006): 621. http://dx.doi.org/10.1017/s0960129506005391.

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14

RAJA, N., and R. K. SHYAMASUNDAR. "THE QUINE-BERNAYS COMBINATORY CALCULUS." International Journal of Foundations of Computer Science 06, no. 04 (1995): 417–30. http://dx.doi.org/10.1142/s0129054195000226.

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We develop a theory for constructing Combinatory Versions of λ-calculi. Our theory is based on a method, used by Quine and Bernays, for the general elimination of variables in formulations of first-order logic. Our Combinatory Calculus presents a significant departure from those propounded by Schönfinkel and Curry. A non-trivial extension of Quine’s technique is developed, to go beyond the realm of first-order quantification theory, and cover the entire λ-calculus. The system consists of five Combinators, powerful enough to represent λ-abstractions over arbitrary terms. The Combinatory Calculu

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15

Bergier, Hugolin. "How Combinatory Logic Can Limit Computing Complexity." EPJ Web of Conferences 244 (2020): 01009. http://dx.doi.org/10.1051/epjconf/202024401009.

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As computing capabilities are extending, the amount of source code to manage is inevitably becoming larger and more complex. No matter how hard we try, the bewildering complexity of the source code always ends up overwhelming its own creator, to the point of giving the appearance of chaos. As a solution to the cognitive complexity of source code, we are proposing to use the framework of Combinatory Logic to construct complex computational concepts that will provide a model of description of the code that is easy and intuitive to grasp. Combinatory Logic is already known as a model of computati

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16

Böhm, C., and M. Dezani-Ciancaglini. "Combinatory Logic as Monoids1." Fundamenta Informaticae 12, no. 4 (1989): 525–39. http://dx.doi.org/10.3233/fi-1989-12406.

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17

Di Pierro, Alessandra, Chris Hankin, and Herbert Wiklicky. "On Reversible Combinatory Logic." Electronic Notes in Theoretical Computer Science 135, no. 3 (2006): 25–35. http://dx.doi.org/10.1016/j.entcs.2005.09.018.

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18

Bunder, M. W. "Some improvements to Turner's algorithm for bracket abstraction." Journal of Symbolic Logic 55, no. 2 (1990): 656–69. http://dx.doi.org/10.2307/2274655.

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A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.There are several ways in which these problems can be alleviated:(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.(2) The use of an extra clause (c), appropriate to βη reduction.(3

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19

Goble, Lou. "Combinatory Logic and the Semantics of Substructural Logics." Studia Logica 85, no. 2 (2007): 171–97. http://dx.doi.org/10.1007/s11225-007-9027-z.

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20

Czajka, Łukasz. "Higher-Order Illative Combinatory Logic." Journal of Symbolic Logic 78, no. 3 (2013): 837–72. http://dx.doi.org/10.2178/jsl.7803080.

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AbstractWe show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

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21

Glickfeld, Barney, and Ross Overbeek. "A foray into combinatory logic." Journal of Automated Reasoning 2, no. 4 (1986): 419–31. http://dx.doi.org/10.1007/bf00248251.

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22

Johann, Patricia. "Normal Forms in Combinatory Logic." Notre Dame Journal of Formal Logic 35, no. 4 (1994): 573–94. http://dx.doi.org/10.1305/ndjfl/1040408614.

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23

Dezani-Ciancaglini, Mariangiola, and J. Roger Hindley. "Intersection types for combinatory logic." Theoretical Computer Science 100, no. 2 (1992): 303–24. http://dx.doi.org/10.1016/0304-3975(92)90306-z.

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24

Krupski, Nikolai. "Typing in reflective combinatory logic." Annals of Pure and Applied Logic 141, no. 1-2 (2006): 243–56. http://dx.doi.org/10.1016/j.apal.2005.11.004.

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25

Weibel, Trudy. "Extension of combinatory logic to a theory of combinatory representation." Theoretical Computer Science 97, no. 1 (1992): 157–73. http://dx.doi.org/10.1016/0304-3975(92)90392-s.

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26

Baba, Kensuke, Yukiyoshi Kameyama, and Sachio Hirokawa. "COMBINATORY LOGIC AND $ lambda $-CALCULUS FOR CLASSICAL LOGIC." Bulletin of informatics and cybernetics 32, no. 2 (2000): 105–22. http://dx.doi.org/10.5109/13496.

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27

Peña, Lorenzo. "Frege's role in the history of logic." Disputatio. Philosophical Research Bulletin 10, no. 17 (2021): 207–30. https://doi.org/10.5281/zenodo.5175948.

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While Kant's claim has been discredited — namely that logic had, by his time, neither progressed nor regressed ever since Aristotle — both the exact reason while he was wrong and the partial core of truth his assertion contained ought to be elucidated. Aristotle's was a logic of terms that ignored the calculus of statements, cultivated instead by the Stoic logicians and later Scholastics. However a unified — yet unsuccessful — logical account of terms and propositions was attempted by Leibniz. It was an anticipation of modern combinatory logic. Leibniz's suc

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28

Knobel, Andreas. "Constructive set theoretic models of typed combinatory logic." Journal of Symbolic Logic 58, no. 1 (1993): 99–118. http://dx.doi.org/10.2307/2275327.

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AbstractWe 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 set theory in which every closed extensional theory of simply typed combinatory logic is the theory of a full function space mod

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29

BOZŞAHİN, Cem. "Combinatory Logic and Natural Language Parsing." Turkish Journal of Electrical Engineering and Computer Sciences 5, no. 3 (1997): 347–57. http://dx.doi.org/10.55730/1300-0632.3759.

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30

Legrand, Remi. "A Basis Result in Combinatory Logic." Journal of Symbolic Logic 53, no. 4 (1988): 1224. http://dx.doi.org/10.2307/2274616.

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31

Redmond, Brian F. "Bounded Combinatory Logic and lower complexity." Information and Computation 248 (June 2016): 215–26. http://dx.doi.org/10.1016/j.ic.2015.12.013.

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32

Passy, Solomon, and Tinko Tinchev. "An essay in combinatory dynamic logic." Information and Computation 93, no. 2 (1991): 263–332. http://dx.doi.org/10.1016/0890-5401(91)90026-x.

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33

Ivanov, Lyubomir. "Operative vs. combinatory spaces." Journal of Symbolic Logic 55, no. 2 (1990): 561–72. http://dx.doi.org/10.2307/2274646.

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The algebraic systems of combinatory spaces [3] and operative spaces [1] have been designed to provide appropriate settings for the development of abstract recursion theory. As shown in [1, Chapter 27], these systems are closely related; namely, every combinatory space has a companion operative space with a storing operation St such that Skordev recursiveness in the former equals st-recursiveness in the latter. The problem of characterization of those operative spaces which have companion combinatory spaces was solved in [2] by introducing a class of operative spaces called Skordev spaces and

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34

Шалак, В. И. "On the Definitional Embeddability of the Combinatory Logic Theory into the First-Order Predicate Calculus." Logical Investigations 21, no. 2 (2015): 9–14. http://dx.doi.org/10.21146/2074-1472-2015-21-2-9-14.

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In this article we prove a theorem on the definitional embeddability of the combinatory logic into the first-order predicate calculus without equality. Since all efficiently computable functions can be represented in the combinatory logic, it immediately follows that they can be represented in the first-order classical predicate logic. So far mathematicians studied the computability theory as some applied theory. From our theorem it follows that the notion of computability is purely logical. This result will be of interest not only for logicians and mathematicians but also for philosophers who

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35

Kosta Došen. "Deductive Completeness." Bulletin of Symbolic Logic 2, no. 3 (1996): 243–83. http://dx.doi.org/10.2307/420991.

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AbstractThis is an exposition of Lambek's strengthening and generalization of the deduction theorem in categories related to intuitionistic propositional logic. Essential notions of category theory are introduced so as to yield a simple reformulation of Lambek's Functional Completeness Theorem, from which its main consequences can be readily drawn. The connections of the theorem with combinatory logic, and with modal and substructural logics, are briefly considered at the end.

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36

Bunder, Martin W., J. Roger Hindley та Jonathan P. Seldin. "On adding (ξ) to weak equality in combinatory logic". Journal of Symbolic Logic 54, № 2 (1989): 590–607. http://dx.doi.org/10.2307/2274872.

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AbstractBecause the main difference between combinatory weak equality and λβ-equality is that the ruleis 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 one of the more common abstraction algorithms, the result will be an equality, =ξ, that is either equivalent to βη-equality (and so strictly stronger than β-equality) or else strictly weaker than

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37

Shalack, V. I. "On First-order Theories Which Can Be Represented by Definitions." Logical Investigations 22, no. 1 (2016): 125–35. http://dx.doi.org/10.21146/2074-1472-2016-22-1-125-135.

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In the paper we consider the classical logicism program restricted to first-order logic.The main result of this paper is the proof of the theorem, which contains the necessary and sufficient conditions for a mathematical theory to be reducible to logic. Those and only those theories, which don’t impose restrictions on the size of their domains, can be reduced to pure logic. Among such theories we can mention the elementary theory of groups, the theory of combinators (combinatory logic), the elementary theory of topoi and many others. It is interesting to note that the initial formulation of th

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38

Cantini, Andrea. "The axiom of choice and combinatory logic." Journal of Symbolic Logic 68, no. 4 (2003): 1091–108. http://dx.doi.org/10.2178/jsl/1067620175.

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AbstractWe 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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39

Cantini, Andrea. "Polytime, combinatory logic and positive safe induction." Archive for Mathematical Logic 41, no. 2 (2002): 169–89. http://dx.doi.org/10.1007/s001530100105.

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40

Desclés, Jean-Pierre, Anca Christine Pascu, and Hee-Jin Ro. "Aspecto-Temporal Meanings Analysed by Combinatory Logic." Journal of Logic, Language and Information 23, no. 3 (2014): 253–74. http://dx.doi.org/10.1007/s10849-014-9189-9.

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41

Meyer, Robert K., Martin W. Bunder, and Lawrence Powers. "Implementing the ‘Fool's model’ of combinatory logic." Journal of Automated Reasoning 7, no. 4 (1991): 597–630. http://dx.doi.org/10.1007/bf01880331.

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42

Mezghiche, Mohamed. "On pseudo-cβnormal form in combinatory logic". Theoretical Computer Science 66, № 3 (1989): 323–31. http://dx.doi.org/10.1016/0304-3975(89)90157-6.

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43

DAWSON, JEREMY E., and RAJEEV GORÉ. "TERMINATION OF ABSTRACT REDUCTION SYSTEMS." International Journal of Foundations of Computer Science 20, no. 01 (2009): 57–82. http://dx.doi.org/10.1142/s0129054109006450.

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We present a general theorem capturing conditions required for the termination of abstract reduction systems. We show that our theorem generalises another similar general theorem about termination of such systems. We apply our theorem to give interesting proofs of termination for typed combinatory logic. Thus, our method can handle most path-orderings in the literature as well as the reducibility method typically used for typed combinators. Finally we show how our theorem can be used to prove termination for incrementally defined rewrite systems, including an incremental general path ordering.

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44

Piccini, Caterina. "Concetto e realtà: il metodo della logica combinatoria in Hegel." DILEF. Rivista digitale del Dipartimento di Lettere e Filosofia, no. 3 (December 13, 2023): 1–17. http://dx.doi.org/10.35948/dilef/2023.4345.

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All’interno del grande problema metodologico sul rapporto tra Logica e Realphilosophie, Hegel identifica come realmente scientifica la comprensione della relazione tra Begriff e Vorstellung, in cui si tratta della rappresentazione come strumento categoriale dove il pensiero è immerso nell’alterità. A partire dalla necessità metodologica di un nesso sistematico tra rappresentazione e concetto, costitutivo di una logica combinatoria, il presente contributo si propone di comprendere quali ragioni sistematico-concettuali hanno portato Hegel nell’Enciclopedia delle scienze filosofiche in compendio

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45

Piccini, Caterina. "Concetto e realtà: il metodo della logica combinatoria in Hegel." DILEF. Rivista digitale del Dipartimento di Lettere e Filosofia, no. 3 (December 13, 2023): 36–52. http://dx.doi.org/10.35948/dilef/2024.4345.

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All’interno del grande problema metodologico sul rapporto tra Logica e Realphilosophie, Hegel identifica come realmente scientifica la comprensione della relazione tra Begriff e Vorstellung, in cui si tratta della rappresentazione come strumento categoriale dove il pensiero è immerso nell’alterità. A partire dalla necessità metodologica di un nesso sistematico tra rappresentazione e concetto, costitutivo di una logica combinatoria, il presente contributo si propone di comprendere quali ragioni sistematico-concettuali hanno portato Hegel nell’Enciclopedia delle scienze filosofiche in compendio

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46

Intrigila, Benedetto, and Richard Statman. "Solution to the Range Problem for Combinatory Logic." Fundamenta Informaticae 111, no. 2 (2011): 203–22. http://dx.doi.org/10.3233/fi-2011-560.

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47

Jech, Thomas. "OTTER experiments in a system of combinatory logic." Journal of Automated Reasoning 14, no. 3 (1995): 413–26. http://dx.doi.org/10.1007/bf00881715.

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48

Bendkowski, Maciej, Katarzyna Grygiel, and Marek Zaionc. "On the likelihood of normalization in combinatory logic." Journal of Logic and Computation 27, no. 7 (2017): 2251–69. http://dx.doi.org/10.1093/logcom/exx005.

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49

Piperno, Adolfo. "Abstraction problems in combinatory logic: A compositive approach." Theoretical Computer Science 66, no. 1 (1989): 27–43. http://dx.doi.org/10.1016/0304-3975(89)90143-6.

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50

BISKRI, ISMAÏL, JEAN-PIERRE DESCLÉS, and BOUCIF AMAR BENSABER. "COORDINATION AND APPLICATIVE CATEGORIAL TYPE LOGIC." International Journal on Artificial Intelligence Tools 15, no. 06 (2006): 1007–19. http://dx.doi.org/10.1142/s0218213006003089.

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Despite extensive theoretical work carried out on coordination during the last three decades, much research is still devoted to proving the merits of one model over another. In our article, we will show how the coordination process in French (with using the conjunction et (and)) can be explained through Applicative and Combinatory Categorial Grammar (ACCG).

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