A notation for lambda terms a generalization of environments

Nadathur, Gopalan; Wilson, Debra Sue

doi:10.1016/s0304-3975(97)00184-9

Cited by 30 publications

(22 citation statements)

References 17 publications

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“…In order to understand (and implement) what entailment involving λProlog programs should be, one simply needs to understand intuitionistic reasoning in STT. A large literature also exists that describes that logic and various ways to implement it, e.g., goal-directed search [19], higher-order unification [12,16], backtracking search [20], term representation to support efficient λ-reduction and unification [21], etc. While a particular implementation of logic programming is a particular piece of technology, that logic programming language and the checker implemented in it are not tied to that technology.…”

Section: Logic Foundations For These Featuresmentioning

confidence: 99%

Foundational Proof Certificates in First-Order Logic

Chihani

Miller

Renaud

2013

Automated Deduction – CADE-24

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We present the design philosophy of a proof checker based on a notion of foundational proof certificates. At the heart of this design is a semantics of proof evidence that arises from recent advances in the theory of proofs for classical and intuitionistic logic. That semantics is then performed by a (higher-order) logic program: successful performance means that a formal proof of a theorem has been found. We describe how the λProlog programming language provides several features that help guarantee such a soundness claim. Some of these features (such as strong typing, abstract datatypes, and higher-order programming) were features of the ML programming language when it was first proposed as a proof checker for LCF. Other features of λProlog (such as support for bindings, substitution, and backtracking search) turn out to be equally important for describing and checking the proof evidence encoded in proof certificates. Since trusting our proof checker requires trusting a programming language implementation, we discuss various avenues for enhancing one's trust of such a checker.

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Section: Logic Foundations For These Featuresmentioning

confidence: 99%

Foundational Proof Certificates in First-Order Logic

Chihani

Miller

Renaud

2013

Automated Deduction – CADE-24

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“…Moreover, these calculi reflect suspended substitutions directly into term structure, thereby offering a flexibility in ordering computations. We study the benefits of this flexibility in this paper based on a particular calculus known as the suspension notation [NW98]. We outline this notation in this section to facilitate this discussion.…”

Section: The Explicit Treatment Of Substitutionmentioning

confidence: 99%

“…The suspension notation that we have described here is actually a restricted version of the calculus presented in [NW98]. 6 In particular, the full calculus allows for the transformation .…”

Section: Variations On the Suspension Notationmentioning

confidence: 99%

“…A well-known solution to the problem of identifying two lambda terms that differ only in the names chosen for bound variables is, for instance, to transform them into a nameless form using a scheme due to de Bruijn [Bru72]. Similarly, several new notations for the lambda calculus have been described in recent years that have the purpose of making substitutions explicit (e.g., see [ACCL91,BBLRD96,KR97,NW98]). However, the actual manner in which all these devices should be deployed in a practical context is far from clear.…”

Section: Introductionmentioning

confidence: 99%

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Choices in Representation and Reduction Strategies for Lambda Terms in Intensional Contexts

Liang

Nadathur

2004

J Autom Reasoning

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Higher-order representations of objects such as programs, proofs, formulas and types have become important to many symbolic computation tasks. Systems that support such representations usually depend on the implementation of an intensional view of the terms of some variant of the typed lambda calculus. New notations have been proposed for the lambda calculus that provide an excellent basis for realizing such implementations. There are, however, several choices in the actual deployment of these notations the practical consequences of which are not currently well understood. We attempt to develop such an understanding here by examining the impact on performance of different combinations of the features afforded by such notations. Amongst the facets examined are the treatment of bound variables, eagerness and laziness in substitution and reduction, the ability to merge different structure traversals into one and the virtues of annotations on terms that indicate their dependence on variables bound by external abstractions. We complement qualitative assessments with experiments conducted by executing programs in a language that supports an intensional view of lambda terms while varying relevant aspects of the implementation of the language. Our study provides insights into the preferred approaches to representing and reducing lambda terms and also exposes characteristics of computations that have a somewhat unanticipated effect on performance.

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“…A remarkable exception is λProlog, for which its explicit substitutions calculus, the suspension calculus, has been extracted and formally studied [NaWi98].…”

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confidence: 99%

Principal Typings for Explicit Substitutions Calculi

Ventura

Ayala-Rincón

Kamareddine

Logic and Theory of Algorithms

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Abstract. Having principal typings (for short PT) is an important property of type systems. In simply typed systems, this property guarantees the possibility of a complete and terminating type inference mechanism. It is well-known that the simply typed λ-calculus has this property but recently J.B. Wells has introduced a system-independent definition of PT, which allows to prove that some type systems, e.g. the Hindley/Milner type system, do not satisfy PT. Explicit substitutions address a major computational drawback of the λ-calculus and allow the explicit treatment of the substitution operation to formally correspond to its implementation. Several extensions of the λ-calculus with explicit substitution have been given but some of which do not preserve basic properties such as the preservation of strong normalization. We consider two systems of explicit substitutions (λse and λσ) and show that they can be accommodated with an adequate notion of PT. Specifically, our results are as follows:• We introduce PT notions for the simply typed versions of the λse-and the λσ-calculi and prove that they agree with Wells' notion of PT.• We show that these versions satisfy PT by revisiting previously introduced type inference algorithms.

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A notation for lambda terms a generalization of environments

Cited by 30 publications

References 17 publications

Foundational Proof Certificates in First-Order Logic

Foundational Proof Certificates in First-Order Logic

Choices in Representation and Reduction Strategies for Lambda Terms in Intensional Contexts

Principal Typings for Explicit Substitutions Calculi

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