## A COMBINATORY ACCOUNT OF INTERNAL STRUCTURE

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@MISC{Jay_acombinatory,

author = {Barry Jay and Thomas Given-wilson},

title = {A COMBINATORY ACCOUNT OF INTERNAL STRUCTURE},

year = {}

}

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### Abstract

Abstract. Traditional combinatory logic is able to represent all Turing computable functions on natural numbers, but there are effectively calculable functions on the combinators themselves that cannot be so represented, because they have direct access to the internal structure of their arguments. Some of this expressive power is captured by adding a factorisation combinator. It supports structural equality, and more generally, a large class of generic queries for updating of, and selecting from, arbitrary structures. The resulting combinatory logic is structure complete in the sense of being able to represent pattern-matching functions, as well as simple abstractions. §1. Introduction. Traditional combinatory logic [21, 4, 10] is computationally equivalent to pure λ-calculus [3] and able to represent all of the Turing computable functions on natural numbers [23], but there are effectively calculable functions on the combinators themselves that cannot be so represented, as they examine the internal structure of their arguments.

### Citations

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(Show Context)
Citation Context ...hen P is already a head normal form. For example, in SK-calculus, the head normal forms are those combinators of the form S,SM,SMN,K and KM. The SK-calculus can be translated to λ-calculus as follows =-=[4, 5, 1, 10]-=- For example for any combinator X. [[S]] = λg.λf.λx.g x (f x) [[K]] = λx.λy.x [[MN]] = [[M]] [[N]] . [[SKX]] = (λg.λf.λx.g x (f x)) (λx.λy.x) [[X]] −→ (λf.λx.(λx.λy.x) x (f x)) [[X]] −→ λx.(λx.λy.x) x... |

365 |
Confluent reductions: Abstract properties and applications to term rewriting systems
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(Show Context)
Citation Context ...ied to the components. The combinator K is defined to be K ≡ FF and then I ≡ SKK as before. Theorem 4.1. Reduction is confluent. Proof. It is enough to observe that the reduction rules are orthogonal =-=[20, 11]-=-, since matchable forms are stable under reduction. ⊣ Theorem 4.2. Every normal form is a matchable form. Proof. Trivial. Despite this result, note that there are (equivalence classes of) combinators ... |

351 | Recursive functions of symbolic expressions and their computation by machine
- McCarthy
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(Show Context)
Citation Context ...ing structures that generalise the usual database queries from rows and tables to arbitrary structures. These queries are slightly more general than those of pattern calculus [15, 13, 14, 12] or Lisp =-=[19]-=- since they interact with arbitrary normal forms, while the latter are limited to data structures or S-expressions. Indeed, the present work has been motivated by the observation that the factorisatio... |

267 |
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(Show Context)
Citation Context ...nse of being able to represent pattern-matching functions, as well as simple abstractions. §1. Introduction. Traditional combinatory logic [21, 4, 10] is computationally equivalent to pure λ-calculus =-=[3]-=- and able to represent all of the Turing computable functions on natural numbers [23], but there are effectively calculable functions on the combinators themselves that cannot be so represented, as th... |

227 |
Une extension de l’interprétation de Gödel à l’analyse, et son application à l’elimination des coupures dans l’analyse et la théorie des types
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(Show Context)
Citation Context ...mponents of an application are not determined by the type of application itself. However, the missing information can be acknowledged by using existential quantification in System F of variable types =-=[7, 6]-=-. Although SF-combinatory logic is structure complete, there remain symbolic computations that it does not represent, which in turn suggest other novel combinators. Examples considered include a gener... |

208 |
Feys R. \Combinatory Logic
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(Show Context)
Citation Context ... functions whose domain is limited to normal forms. The combinator F can be typed using an existential type to represent internal type information. §1. Introduction. The traditional combinatory logic =-=[22, 4, 10]-=- built from combinators S and K is able to represent all the extensional functions described by λ-calculus [3, 1], and all the Turing-computable functions [24] on natural numbers [18]. However, there ... |

139 |
Introduction to Higher-order Categorical Logic
- Lambek, Scott
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(Show Context)
Citation Context ... .20 BARRY JAY AND THOMAS GIVEN-WILSON However, it is not clear how to handle the examination of internal structure, i.e. what it means to factor elements of a partial order, or arrows in a category =-=[19]-=-. In mathematical logic, structural induction [2] is the analogue of factorisation, but the relationship has not been formalised. Pattern calculi also support factorisation, albeit only for data struc... |

113 |
Introduction to combinators and λcalculus
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(Show Context)
Citation Context ...e resulting combinatory logic is structure complete in the sense of being able to represent pattern-matching functions, as well as simple abstractions. §1. Introduction. Traditional combinatory logic =-=[21, 4, 10]-=- is computationally equivalent to pure λ-calculus [3] and able to represent all of the Turing computable functions on natural numbers [23], but there are effectively calculable functions on the combin... |

89 | Proving properties of programs by structural induction
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- 1969
(Show Context)
Citation Context ...partial orders [9]. However, it is not clear how to handle the examination of internal structure, i.e. what it means to factor elements of a partial order. In mathematical logic, structural induction =-=[2]-=- is the analogue of factorisation, but the relationship has not been formalised. Pattern calculus also supports factorisation, albeit only for data structures. Future work will show how to translate t... |

79 |
Über die Bausteine der mathematischen Logik. Mathematische Annalen, 92:305–316
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(Show Context)
Citation Context ...e resulting combinatory logic is structure complete in the sense of being able to represent pattern-matching functions, as well as simple abstractions. §1. Introduction. Traditional combinatory logic =-=[21, 4, 10]-=- is computationally equivalent to pure λ-calculus [3] and able to represent all of the Turing computable functions on natural numbers [23], but there are effectively calculable functions on the combin... |

72 |
Tree-manipulating systems and Church-Rosser theorems
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(Show Context)
Citation Context ...ied to the components. The combinator K is defined to be K ≡ FF and then I ≡ SKK as before. Theorem 4.1. Reduction is confluent. Proof. It is enough to observe that the reduction rules are orthogonal =-=[20, 11]-=-, since matchable forms are stable under reduction. ⊣ Theorem 4.2. Every normal form is a matchable form. Proof. Trivial. Despite this result, note that there are (equivalence classes of) combinators ... |

66 | Pure pattern calculus
- Jay, Kesner
- 2006
(Show Context)
Citation Context ...es for selecting or updating structures that generalise the usual database queries from rows and tables to arbitrary structures. These queries are slightly more general than those of pattern calculus =-=[15, 13, 14, 12]-=- or Lisp [19] since they interact with arbitrary normal forms, while the latter are limited to data structures or S-expressions. Indeed, the present work has been motivated by the observation that the... |

39 |
Introduction to Metamathematics. North-Holland, seventh edition
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- 1952
(Show Context)
Citation Context ... traditional theorems about computable functions of numbers, but imposes severe constraints upon the interpretation of Church’s thesis, that all effectively calculable functions are general recursive =-=[18]-=-. The theoretical implications will be considered in Section 3, but the practical consequence is that the expressive power of traditional combinatory logic can be increased by adding new combinators. ... |

18 | First-class patterns
- Jay, Kesner
(Show Context)
Citation Context ...es for selecting or updating structures that generalise the usual database queries from rows and tables to arbitrary structures. These queries are slightly more general than those of pattern calculus =-=[15, 13, 14, 12]-=- or Lisp [19] since they interact with arbitrary normal forms, while the latter are limited to data structures or S-expressions. Indeed, the present work has been motivated by the observation that the... |

18 |
Computability and λ-definability
- Turing
- 1937
(Show Context)
Citation Context ...The traditional combinatory logic [22, 4, 10] built from combinators S and K is able to represent all the extensional functions described by λ-calculus [3, 1], and all the Turing-computable functions =-=[24]-=- on natural numbers [18]. However, there is a Turing-computable function that distinguishes the combinators SKK and SKS, that cannot be represented by application of an SK-combinator, since SKK and SK... |

13 |
Pattern Calculus: Computing with Functions and Structures
- Jay
- 2009
(Show Context)
Citation Context ...es for selecting or updating structures that generalise the usual database queries from rows and tables to arbitrary structures. These queries are slightly more general than those of pattern calculus =-=[15, 13, 14, 12]-=- or Lisp [19] since they interact with arbitrary normal forms, while the latter are limited to data structures or S-expressions. Indeed, the present work has been motivated by the observation that the... |

12 |
Computability and lambda-definability
- Turing
- 1937
(Show Context)
Citation Context ...tions. §1. Introduction. Traditional combinatory logic [21, 4, 10] is computationally equivalent to pure λ-calculus [3] and able to represent all of the Turing computable functions on natural numbers =-=[23]-=-, but there are effectively calculable functions on the combinators themselves that cannot be so represented, as they examine the internal structure of their arguments. This is consistent with the tra... |

8 |
Uniformly reflexive structures: on the nature of Gödelizations and relative computability
- Wagner
- 1969
(Show Context)
Citation Context ...uctures. Combinator equality has been considered indirectly by appealing to: meta-level operations [4, p. 245]; or partial combinatory algebras (not logics) such as the uniformly reflexive structures =-=[24]-=-; or discriminators [16, 17]. However, this is the first account we know of that is strictly within combinatory logic. Further, one may define generic queries for selecting or updating structures that... |

3 |
Semantic domains, Handbook of theoretical computer science, vol
- Gunter, Scott
- 1990
(Show Context)
Citation Context ...ting system, it is natural to ask about its denotational semantics. Dana Scott showed how to model pure λ-calculus (and hence SK-calculus) using continuous lattices and then ω-complete partial orders =-=[9]-=-. However, it is not clear how to handle the examination of internal structure, i.e. what it means to factor elements of a partial order. In mathematical logic, structural induction [2] is the analogu... |

2 |
Interpreting the untyped pattern calculus in bondi, Honours Thesis
- Given-Wilson
- 2007
(Show Context)
Citation Context ... PQ is a data structure DXMN −→ M otherwise, if X is matchable is(C) C −→ K is(C) X −→ KI otherwise, if X is matchable. Although, it is routine to show that SDC-calculus is to static pattern calculus =-=[8, 12]-=- as SK-calculus is to the λ-calculus, this would require a detailed account of static pattern calculus. We intend to cover all of this within a full treatment of the relationship between pattern calcu... |

1 |
Pure pattern calculus, Programming languages and systems
- Jay, Kesner
- 2006
(Show Context)
Citation Context ...erns supported. In mainstream function programming, patterns for data structures are constrained by the type system so that cases can be translated into λabstractions. Recent work in pattern calculus =-=[15, 13, 8, 14, 12]-=- drops these limitations on patterns for data structures, but does not allow matching of cases. Here, the class of patterns is expanded to include all terms in normal form, even those representing cas... |

1 |
Combinatory logic with discriminators, The
- Kearns
- 1969
(Show Context)
Citation Context ...lity has been considered indirectly by appealing to: meta-level operations [4, p. 245]; or partial combinatory algebras (not logics) such as the uniformly reflexive structures [24]; or discriminators =-=[16, 17]-=-. However, this is the first account we know of that is strictly within combinatory logic. Further, one may define generic queries for selecting or updating structures that generalise the usual databa... |

1 |
semantics, metamathematics, Intentions in communication
- Tarski, Logic
- 1956
(Show Context)
Citation Context ...antics. The existing translation from SK-logic to λ-calculus serves well enough for numerical computations, but does not provide a representation of F. As F is a meta-function, in the sense of Tarksi =-=[22]-=-, it is tempting to translate combinators to binary trees and then use tree operations to factorise. However, it is not clear what semantics is being preserved. Further, one may consider effective cal... |