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

Abstract

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.

Practical Higher-Order Pattern Unification with On-the-Fly Raising

“…We would like to understand this matter better since it is likely to shed light on the role of explicit substitutions in this setting. In a related sense, we have found it beneficial to employ explicit substitutions only implicitly in reduction procedures [4] and we would like to extend this approach also to the unification context. Finally, higher-order pattern unification offers promising possibilities for compilation.…”

“…Such a form has the structure λ(n, a(t)) where a, called the head of the term, is a universal or existential variable or a de Bruijn index. Further discussion of suitable reduction procedures may be found in [4].…”

Automation of Higher-Order Logic

“…For example, one often immediately places terms into βη-normal form after making a substitution. Since there can be an explosion in the size of terms when such normalization is made, there are compelling reasons to delay such normalization (Liang et al, 2005). Andrews (1971), for example, integrates the production of conjunctive normal and Skolem normal forms within the process of doing resolution.…”

A ρ-Calculus of Explicit Constraint Application

“…Of course, the set of substitutions can be easily extended to more sophisticated constraints like, for example, X A ∧ Y B. The main benefit of this latter approach is that several substitution applications implying several term traversals can be merged together leading thus to more efficient implementations [18].…”

“…As far as implementation issues are concerned, explicit substitution calculi are very important [18]. In all the explicit substitution calculi [1,17,23], substitutions can be delayed thanks to the Beta rule that transforms a β-redex (λx.a)b into the explicit application on a of the substitution that replaces x by b.…”