Combinators, λ-Terms and Proof Theory

Stenlund, Sören

doi:10.1007/978-94-010-2913-1

Cited by 68 publications

(18 citation statements)

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“…Examples of type definitions used in the paper are bool for booleans, nat for natural numbers, list nat for lists of natural numbers (we do not consider polymorphic types here), tree and list tree for the mutually inductive types of trees and lists of trees, proc for process expressions [44] (δ denotes the deadlock, ";" the sequencing, + the choice operator and Σ the dependent choice), ord for well-founded trees, i.e. Brouwer's ordinals [45], form for formulas of the predicate calculus and R for expressions built upon real numbers [42]:…”

Section: Assumptionmentioning

confidence: 99%

Inductive-data-type systems

Blanqui

Jouannaud

Okada

2002

Theoretical Computer Science

102

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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λ-calculus enriched by pattern-matching definitions following a certain format, called the "General Schema", which generalizes the usual recursor definitions for natural numbers and similar "basic inductive types". This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called "strictly positive", and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.

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Section: Assumptionmentioning

confidence: 99%

Inductive-data-type systems

Blanqui

Jouannaud

Okada

2002

Theoretical Computer Science

102

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“…We briefly review a few known results essential to our exposition. Those seeking a fuller treatment should consult Hindley and Seldin [7] or Stenlund [20].…”

Section: Introductionmentioning

confidence: 99%

Visualizing the turing tarpit

Hemann

Holk

2013

Proceedings of the First ACM SIGPLAN Workshop on Functional Art, Music, Modeling &Amp; Design

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Minimal programming languages like Jot generate limited interest outside of the community of languages enthusiasts. This is unfortunate, because the simplicity of these languages endows them with an inherent beauty and provides deep insight into the nature of computation. We present a way of visualizing the behavior of many Jot programs at once, providing interesting images and also hinting at somewhat non-obvious relationships between programs.In the same way that fractals research has yielded new mathematical insights, research into visualization such as that presented here could produce new perspectives on the structure and nature of computation. A gallery containing the visualizations presented herein can be found at http://tarpit.github.io/ TarpitGazer.

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“…This problem cannot have a unique solution because one knows that the technical content of the formulae-as-types principle, namely the Curry-Howard isomorphism [3,11,14,21], is strongly related to the constructive aspects of intuitionistic logic. Therefore, when dealing with classical logic, one has to drop some of the properties that exist in the intuitionistic case.…”

Section: Introductionmentioning

confidence: 99%

Strong normalization in a non-deterministic typed lambda-calculus

Groote

1994

Logical Foundations of Computer Science

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Abstract. In a previous paper [4], we introduced a non-deterministic λ-calculus (λ-LK) whose type system corresponds exactly to Gentzen's cut-free LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, it is not possible to define an effective notion of reduction. In the present paper, we address this problem. We consider a weak version of the implicative fragment of λ-LK, and we define for it a relation of reduction that models, at the level of the terms, the appropriate proof-theoretic notion of proof reduction. This reduction relation satisfies several properties of interest, among others, the property of strong normalization. We prove this last result by using a reducibility argumentà la Tait.

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Combinators, λ-Terms and Proof Theory

Cited by 68 publications

References 0 publications

Inductive-data-type systems

Inductive-data-type systems

Visualizing the turing tarpit

Strong normalization in a non-deterministic typed lambda-calculus

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