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Journal articles on the topic 'Y Combinator'

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Published: 4 June 2021

Last updated: 31 January 2023

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1

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 and the (quasi-)normalization theorem are valid for the combinatory reduction system under consideration.Proof. See [4], [5], and [6].Any basis must contain at least one combinator with rank strictly greater than two.Let us assume a basis B with combinators of rank strictly less than 3; then there exists a pure combination X of the combinators in B such that: XABC ↠ CAB.(*) First of all, we remark that if XABC ↠ M ↠ CAB, M must contain at least one occurrence of each of the variables A, B and C.Notation. We denote by E[X1,…,Xn] expressions that contain at least one occurrence of every term X1,…,Xn.

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2

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 that certain symmetric combinatory logics that include just one particular constant are not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property.

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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 the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared.

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4

Goble, Lou. "Combinator Logics." Studia Logica 76, no. 1 (2004): 17–66. http://dx.doi.org/10.1023/b:stud.0000027466.68014.52.

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5

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 general characterisation of the structure complete, confluent combinatory calculi is given along with some examples. These are able to represent all their Turing-computable functions whose domain is limited to normal forms. The combinator F can be typed using an existential type to represent internal type information.

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6

OKASAKI, CHRIS. "THEORETICAL PEARLS." Journal of Functional Programming 13, no. 4 (2003): 815–22. http://dx.doi.org/10.1017/s0956796802004483.

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A combinator expression is flat if it can be written without parentheses, that is, if all applications nest to the left, never to the right. This note explores a simple method for flattening combinator expressions involving arbitrary combinators.

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7

Joy, M. S., V. J. Rayward-Smith, and F. W. Burton. "Efficient combinator code." Computer Languages 10, no. 3-4 (1985): 211–24. http://dx.doi.org/10.1016/0096-0551(85)90017-7.

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8

Steedman, Mark. "The Lost Combinator." Computational Linguistics 44, no. 4 (2018): 613–29. http://dx.doi.org/10.1162/coli_a_00328.

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9

Waldmann, Johannes. "The Combinator S." Information and Computation 159, no. 1-2 (2000): 2–21. http://dx.doi.org/10.1006/inco.2000.2874.

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10

Parfitt, Eric James. "Patterns in Combinator Evolution." Complex Systems 26, no. 2 (2017): 119–34. http://dx.doi.org/10.25088/complexsystems.26.2.119.

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11

BRODA, SABINE, and LUÍS DAMAS. "On combinatory complete sets of proper combinators." Journal of Functional Programming 7, no. 6 (1997): 593–612. http://dx.doi.org/10.1017/s0956796897002888.

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A combinatory system (or equivalently the set of its basic combinators) is called combinatorially complete for a functional system, if any member of the latter can be defined by an entity of the former system. In this paper the decision problem of combinatory completeness for finite sets of proper combinators is studied for three subsystems of the pure lambda calculus. Precise characterizations of proper combinator bases for the linear and the affine λ-calculus are given, and the respective decision problems are shown to be decidable. Furthermore, it is determined which extensions with proper combinators of bases for the linear λ-calculus are combinatorially complete for the λI-calculus.

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12

Waite, Martin, Bret Giddings, and Simon Lavington. "Parallel associative combinator evaluation II." Future Generation Computer Systems 8, no. 4 (1992): 303–19. http://dx.doi.org/10.1016/0167-739x(92)90065-j.

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13

Siu, Chapman. "Ramble: Parser Combinator for R." Journal of Open Source Software 2, no. 11 (2017): 209. http://dx.doi.org/10.21105/joss.00209.

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14

Hill, Steve. "Combinators for parsing expressions." Journal of Functional Programming 6, no. 3 (1996): 445–64. http://dx.doi.org/10.1017/s0956796800001799.

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AbstractThis paper describes a scheme for constructing parsers based on the top-down combinator approach. In particular, it describes a set of combinators for parsing expressions described by ambiguous grammars with precedence and associativity rules. The new combinators embody the mechanical grammar manipulations typically employed to remove left-recursion and hence help to avoid the possibility of a non-terminating parser. A number of approaches to the problem are described—the most elegant and efficient method is based on continuation passing. As a practical demonstration, a parser for the expression part of the C programming language is presented. The expression combinators are general, and may be constructed from any suitable set of top-down combinators. A comparison with parser generators shows that the combinator approach is most applicable for rapid development.

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15

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 rule.Definition 2. A set (Q1,…, Qm} of combinators is a basis for combinatory logic if for every finite set {x1,…, xk} of variables and every pure combination C of these variables, there exists a pure combination Q of Q1,…, Qm such that Qx1 … xk ↠ C.Craig's Theorem. Every basis for combinatory logic containing only proper combinators contains at least two elements.Proof. Let {Q} be a singleton basis for combinatory logic, and let us show that we cannot have combinatory completeness. This is an easy consequence of the next two lemmas.Lemma 1. Q is a projection. That is, Qx1 … xn ↠ xj, for some j.Proof. Let I be a proper combination of Q such that Ix ↠ x for a variable x, and let M be a term such that Ix ↠ M → x and M → x is a nontrivial contraction.

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16

Camarão, Carlos, Lucilia Figueiredo, and Hermann Rodrigues. "Mímico: a monadic combinator parser generator." Journal of the Brazilian Computer Society 9, no. 1 (2003): 27–40. http://dx.doi.org/10.1590/s0104-65002003000200004.

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17

Thompson, S., and R. Lins. "The Categorical Multi-Combinator Machine: CMCM." Computer Journal 35, no. 2 (1992): 170–76. http://dx.doi.org/10.1093/comjnl/35.2.170.

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18

Koopman, Philip J., Peter Lee, and Daniel P. Siewiorek. "Cache behavior of combinator graph reduction." ACM Transactions on Programming Languages and Systems 14, no. 2 (1992): 265–97. http://dx.doi.org/10.1145/128861.128867.

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19

Swierstra, S. D. "Combinator Parsers: From Toys to Tools." Electronic Notes in Theoretical Computer Science 41, no. 1 (2001): 38–59. http://dx.doi.org/10.1016/s1571-0661(05)80545-6.

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20

Hirokawa, Sachio. "Complexity of the combinator reduction machine." Theoretical Computer Science 41 (1985): 289–303. http://dx.doi.org/10.1016/0304-3975(85)90076-3.

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21

Hartel, Pieter H. "Performance of lazy combinator graph reduction." Software: Practice and Experience 21, no. 3 (1991): 299–329. http://dx.doi.org/10.1002/spe.4380210306.

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22

DONNELLY, KEVIN, and MATTHEW FLUET. "Transactional events." Journal of Functional Programming 18, no. 5-6 (2008): 649–706. http://dx.doi.org/10.1017/s0956796808006916.

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AbstractConcurrent programs require high-level abstractions in order to manage complexity and enable compositional reasoning. In this paper, we introduce a novel concurrency abstraction, dubbed transactional events, which combines first-class synchronous message passing events with all-or-nothing transactions. This combination enables simple solutions to interesting problems in concurrent programming. For example, guarded synchronous receive can be implemented as an abstract transactional event, whereas in other languages it requires a non-abstract, non-modular protocol. As another example, three-way rendezvous can be implemented as an abstract transactional event, which is impossible using first-class events alone. Both solutions are easy to code and easy to reason about.The expressive power of transactional events arises from a sequencing combinator whose semantics enforces an all-or-nothing transactional property – either both of the constituent events synchronize in sequence or neither of them synchronizes. This sequencing combinator, along with a non-deterministic choice combinator, gives transactional events the compositional structure of a monad-with-plus. We provide a formal semantics for transactional events and give a detailed account of an implementation.

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23

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 term. This avoids explosive term growth during successive abstractions and makes the system suitable for practical applications. We prove the correctness of the algorithm and establish some relations between the combinatory system and the Lambda calculus.

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24

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 algorithm steps if there are n occurrences of the variable to be abstracted in the term. Some properties of the Broda-Damas combinators are also considered.

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25

Lester, Martin. "Control Flow Analysis for SF Combinator Calculus." Electronic Proceedings in Theoretical Computer Science 199 (December 7, 2015): 51–67. http://dx.doi.org/10.4204/eptcs.199.4.

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26

Birkimsher, P. C. "Combinator Reduction in a Shared-memory Multiprocessor." Computer Journal 30, no. 3 (1987): 214–22. http://dx.doi.org/10.1093/comjnl/30.3.214.

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27

Koopman, P. J., and P. Lee. "A fresh look at combinator graph reduction." ACM SIGPLAN Notices 24, no. 7 (1989): 110–19. http://dx.doi.org/10.1145/74818.74828.

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28

Fu, Yuxi, and Zhenrong Yang. "Understanding the mismatch combinator in chi calculus." Theoretical Computer Science 290, no. 1 (2003): 779–830. http://dx.doi.org/10.1016/s0304-3975(02)00373-0.

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29

van Bakel, Steffen, and Maribel Fernández. "Normalization, approximation, and semantics for combinator systems." Theoretical Computer Science 290, no. 1 (2003): 975–1019. http://dx.doi.org/10.1016/s0304-3975(02)00548-0.

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30

Musicante, Martín A., and Rafael D. Lins. "GM-C: A graph multi-combinator machine." Microprocessing and Microprogramming 31, no. 1-5 (1991): 81–84. http://dx.doi.org/10.1016/s0165-6074(08)80048-8.

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31

Joy, M. S., and V. J. Rayward-Smith. "NP-Completeness of a Combinator Optimization Problem." Notre Dame Journal of Formal Logic 36, no. 2 (1995): 319–35. http://dx.doi.org/10.1305/ndjfl/1040248462.

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32

Hankin, Chris, Geoffrey Burn, and Simon Peyton Jones. "A safe approach to parallel combinator reduction." Theoretical Computer Science 56, no. 1 (1988): 17–36. http://dx.doi.org/10.1016/0304-3975(86)90004-6.

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33

Statman, Rick. "On sets of solutions to combinator equations." Theoretical Computer Science 66, no. 1 (1989): 99–104. http://dx.doi.org/10.1016/0304-3975(89)90148-5.

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34

KENNEDY, ANDREW J. "FUNCTIONAL PEARL Pickler combinators." Journal of Functional Programming 14, no. 6 (2004): 727–39. http://dx.doi.org/10.1017/s0956796804005209.

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The tedium of writing pickling and unpickling functions by hand is relieved using a combinator library similar in spirit to the well-known parser combinators. Picklers for primitive types are combined to support tupling, alternation, recursion, and structure sharing. Code is presented in Haskell; an alternative implementation in ML is discussed.

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35

STATMAN, RICK. "On the existence of n but not n + 1 easy combinators." Mathematical Structures in Computer Science 9, no. 4 (1999): 361–65. http://dx.doi.org/10.1017/s0960129598002825.

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Recall that M is easy if it is consistent with every combinator. We say that M is m-easy if there is no proof with < m + 1 steps that M is inconsistent with any combinator. Obviously, if M is easy, it is m-easy for each m. Here we shall show that for infinitely many m there are m but not m + 1 easy terms.

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36

Partridge, Andrew, and David Wright. "Predictive parser combinators need four values to report errors." Journal of Functional Programming 6, no. 2 (1996): 355–64. http://dx.doi.org/10.1017/s0956796800001714.

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AbstractA combinator-based parser is a parser constructed directly from a BNF grammar, using higher-order functions (combinators) to model the alternative and sequencing operations of BNF. This paper describes a method for constructing parser combinators that can be used to build efficient predictive parsers which accurately report the cause of parsing errors. The method uses parsers that return values (parse trees or error indications) decorated with one of four tags.

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37

Krishnamurthy, E. V., and B. P. Vickers. "Compact numeral representation with combinators." Journal of Symbolic Logic 52, no. 2 (1987): 519–25. http://dx.doi.org/10.2307/2274398.

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AbstractThis paper is concerned with the combinator representation of numeral systems with logarithmic space complexity of symbols. The principle used is based on the lexicographic ordering of words over a finite alphabet.

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38

Andrews, James H. "An untyped higher order logic with Y combinator." Journal of Symbolic Logic 72, no. 4 (2007): 1385–404. http://dx.doi.org/10.2178/jsl/1203350794.

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AbstractWe define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.

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39

Bessai, Jan, and Anna Vasileva. "User Support for the Combinator Logic Synthesizer Framework." Electronic Proceedings in Theoretical Computer Science 284 (November 27, 2018): 16–25. http://dx.doi.org/10.4204/eptcs.284.2.

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40

Farmer, William M., John D. Ramsdell, and Ronald J. Watro. "A correctness proof for combinator reduction with cycles." ACM Transactions on Programming Languages and Systems 12, no. 1 (1990): 123–34. http://dx.doi.org/10.1145/77606.77612.

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41

Goldberg, Mayer. "A Variadic Extension of Curry's Fixed-Point Combinator." Higher-Order and Symbolic Computation 18, no. 3-4 (2005): 371–88. http://dx.doi.org/10.1007/s10990-005-4881-8.

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42

Montenyohl, Margaret, and Mitchell Wand. "Incorporating static analysis in a combinator-based compiler." Information and Computation 82, no. 2 (1989): 151–84. http://dx.doi.org/10.1016/0890-5401(89)90052-7.

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43

Båge, Göran, and Gary Lindstrom. "Combinator evaluation of functional programs with logical variables." Lisp and Symbolic Computation 3, no. 3 (1990): 289–320. http://dx.doi.org/10.1007/bf01806101.

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44

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 look like some sort of assembly language for mathematical logic. A Neural Algebra, modeling the way we think, constitutes an interesting model of Combinatory Logic. There are other models, also based on the Graph Model (Engeler 1981), such as software testing. This paper investigates what Combinatory Logic contributes to modern software testing. Keywords: combinatory logic, combinatory algebra, autonomous real-time testing, recursion, software testing, artificial intelligence

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45

Hutton, Graham. "Higher-order functions for parsing." Journal of Functional Programming 2, no. 3 (1992): 323–43. http://dx.doi.org/10.1017/s0956796800000411.

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AbstractIn combinator parsing, the text of parsers resembles BNF notation. We present the basic method, and a number of extensions. We address the special problems presented by white-space, and parsers with separate lexical and syntactic phases. In particular, a combining form for handling the ‘offside rule’ is given. Other extensions to the basic method include an ‘into’ combining form with many useful applications, and a simple means by which combinator parsers can produce more informative error messages.

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46

Manzonetto, Giulio, Andrew Polonsky, Alexis Saurin, and Jakob Grue Simonsen. "The fixed point property and a technique to harness double fixed point combinators." Journal of Logic and Computation 29, no. 5 (2019): 831–80. http://dx.doi.org/10.1093/logcom/exz013.

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Abstract The ${\lambda }$-calculus enjoys the property that each ${\lambda }$-term has at least one fixed point, which is due to the existence of a fixed point combinator. It is unknown whether it enjoys the ‘fixed point property’ stating that each ${\lambda }$-term has either one or infinitely many pairwise distinct fixed points. We show that the fixed point property holds when considering possibly open fixed points. The problem of counting fixed points in the closed setting remains open, but we provide sufficient conditions for a ${\lambda }$-term to have either one or infinitely many fixed points. In the main result of this paper we prove that in every sensible ${\lambda }$-theory there exists a ${\lambda }$-term that violates the fixed point property. We then study the open problem concerning the existence of a double fixed point combinator and propose a proof technique that could lead towards a negative solution. We consider interpretations of the ${\lambda } {\mathtt{Y}}$-calculus into the ${\lambda }$-calculus together with two reduction extension properties, whose validity would entail the non-existence of any double fixed point combinators. We conjecture that both properties hold when typed ${\lambda } {\mathtt{Y}}$-terms are interpreted by arbitrary fixed point combinators. We prove reduction extension property I for a large class of fixed point combinators. Finally, we prove that the ${\lambda }{\mathtt{Y}}$-theory generated by the equation characterizing double fixed point combinators is a conservative extension of the ${\lambda }$-calculus.

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47

Scholz, Enno. "Imperative streams—a monadic combinator library for synchronous programming." ACM SIGPLAN Notices 34, no. 1 (1999): 261–72. http://dx.doi.org/10.1145/291251.289454.

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48

Sarwar, S. M., S. J. Hahn, and J. A. Davis. "Implementing functional languages on a combinator-based reduction machine." ACM SIGPLAN Notices 23, no. 4 (1988): 65–70. http://dx.doi.org/10.1145/44326.44333.

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49

Statman, Rick. "The word problem for Smullyan's lark combinator is decidable." Journal of Symbolic Computation 7, no. 2 (1989): 103–12. http://dx.doi.org/10.1016/s0747-7171(89)80044-6.

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50

Bimbó, Katalin. "Admissibility of Cut in LC with Fixed Point Combinator." Studia Logica 81, no. 3 (2005): 399–423. http://dx.doi.org/10.1007/s11225-005-4651-y.

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