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A Notation for Lambda Terms I: A Generalization of Environments
 THEORETICAL COMPUTER SCIENCE
, 1994
"... A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms ..."
Abstract

Cited by 33 (12 self)
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A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a ficontraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...
A FineGrained Notation for Lambda Terms and Its Use in Intensional Operations
 Journal of Functional and Logic Programming
, 1996
"... We discuss issues relevant to the practical use of a previously proposed notation for lambda terms in contexts where the intensions of such terms have to be manipulated. This notation uses the `nameless' scheme of de Bruijn, includes expressions for encoding terms together with substitutions to be p ..."
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Cited by 25 (9 self)
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We discuss issues relevant to the practical use of a previously proposed notation for lambda terms in contexts where the intensions of such terms have to be manipulated. This notation uses the `nameless' scheme of de Bruijn, includes expressions for encoding terms together with substitutions to be performed on them and contains a mechanism for combining such substitutions so that they can be effected in a common structure traversal. The combination mechanism is a general one and consequently difficult to implement. We propose a simplification to it that retains its functionality in situations that occur commonly in fireduction. We then describe a system for annotating terms to determine if they can be affected by substitutions generated by external ficontractions. These annotations can lead to a conservation of space and time in implementations of reduction by permitting substitutions to be performed trivially in certain situations. The use of the resulting notation in the reduction...
A Notation for Lambda Terms II: Refinements and Applications
, 1994
"... Issues that are relevant to the representation of lambda terms in contexts where their intensions have to be manipulated are examined. The basis for such a representation is provided by the suspension notation for lambda terms that is described in a companion paper. This notation obviates ffconver ..."
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Cited by 12 (2 self)
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Issues that are relevant to the representation of lambda terms in contexts where their intensions have to be manipulated are examined. The basis for such a representation is provided by the suspension notation for lambda terms that is described in a companion paper. This notation obviates ffconversion in the comparison of terms by using the `nameless' scheme of de Bruijn and also permits a delaying of substitutions by including a class of terms that encode other terms together with substitutions to be performed on them. The suspension notation contains a mechanism for `merging' substitutions so that they can be effected in a common structure traversal. The mechanism is cumbersome to implement in its full generality and a simplification to it is considered. In particular, the old merging operations are eliminated in favor of new ones that capture some of their functionality and that permit a simplified syntax for terms. The resulting notation is refined by the addition of annotations ...
Tradeoffs in the Intensional Representation of Lambda Terms
 Rewriting Techniques and Applications (RTA 2002), volume 2378 of LNCS
, 2002
"... Higherorder representations of objects such as programs, specifications and proofs are important to many metaprogramming and symbolic computation tasks. Systems that support such representations often depend on the implementation of an intensional view of the terms of suitable typed lambda calculi. ..."
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Cited by 10 (3 self)
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Higherorder representations of objects such as programs, specifications and proofs are important to many metaprogramming and symbolic computation tasks. Systems that support such representations often depend on the implementation of an intensional view of the terms of suitable typed lambda calculi. Refined lambda calculus notations have been proposed that can be used in realizing such implementations. There are, however, choices in the actual deployment of such notations whose practical consequences are not well understood. Towards addressing this lacuna, the impact of three specific ideas is examined: the de Bruijn representation of bound variables, the explicit encoding of substitutions in terms and the annotation of terms to indicate their independence on external abstractions. Qualitative assessments are complemented by experiments over actual computations using the lambdaProlog language.
Choices in representation and reduction strategies for lambda terms in intensional contexts
 J. Autom. Reasoning
, 2004
"... ..."
A Treatment of HigherOrder Features in Logic Programming
, 2003
"... The logic programming paradigm provides the basis for a new intensional view of higherorder notions. This view is realized primarily by employing the terms of a typed lambda calculus as representational devices and by using a richer form of unification for probing their structures. These additions ..."
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Cited by 4 (0 self)
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The logic programming paradigm provides the basis for a new intensional view of higherorder notions. This view is realized primarily by employing the terms of a typed lambda calculus as representational devices and by using a richer form of unification for probing their structures. These additions have important metaprogramming applications but they also pose nontrivial implementation problems. One issue concerns the machine representation of lambda terms suitable to their intended use: an adequate encoding must facilitate comparison operations over terms in addition to supporting the usual reduction computation. Another aspect relates to the treatment of a unification operation that has a branching character and that sometimes calls for the delaying of the solution of unification problems. A final issue concerns the execution of goals whose structures becomes apparent only in the course of computation. These various problems are exposed in this paper and solutions to them are described. A satisfactory representation for lambda terms is developed by exploiting the nameless notation of de Bruijn as well as explicit encodings of substitutions. Special mechanisms are molded into the structure of traditional Prolog implementations to support branching in unification and carrying of unication problems over other computation steps; a premium is placed in this context on exploiting determinism and on emulating usual firstorder behaviour. An extended compilation model is presented that treats higherorder unification and also handles dynamically emergent goals. The ideas described here have been employed in the Teyjus implementation of the Prolog language, a fact that is used to obtain a preliminary assessment of their efficacy.
Under consideration for publication in Theory and Practice of Logic Programming 1 A Treatment of HigherOrder Features in Logic Programming
, 2003
"... The logic programming paradigm provides the basis for a new intensional view of higherorder notions. This view is realized primarily by employing the terms of a typed lambda calculus as representational devices and by using a richer form of unification for probing their structures. These additions h ..."
Abstract
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The logic programming paradigm provides the basis for a new intensional view of higherorder notions. This view is realized primarily by employing the terms of a typed lambda calculus as representational devices and by using a richer form of unification for probing their structures. These additions have important metaprogramming applications but they also pose nontrivial implementation problems. One issue concerns the machine representation of lambda terms suitable to their intended use: an adequate encoding must facilitate comparison operations over terms in addition to supporting the usual reduction computation. Another aspect relates to the treatment of a unification operation that has a branching character and that sometimes calls for the delaying of the solution of unification problems. A final issue concerns the execution of goals whose structures become apparent only in the course of computation. These various problems are exposed in this paper and solutions to them are described. A satisfactory representation for lambda terms is developed by exploiting the nameless notation of de Bruijn as well as explicit encodings of substitutions. Special mechanisms are molded into the structure of traditional Prolog