Director strings as combinators

Abstract

A simple calculus (the Director String Calculus-DSC) for expressing abstractions is introduced, which captures the essence of the "long reach" combinators introduced by Turner. We present abstraction rules that preserve the applicative structure of the original lambda term, and that cannot increase the number of subterms in the translation.A translated lambda term can be reduced according to the evaluation rules of DSC. If this terminates with a DSC normal form, this can be translated into a lambda term using rules presetid below. We call this process of abstracting a lambda term, reducing to normal form in the space of DSC terms, and translating back to a lambda term an impZ.ementation.We show that our implementation of the lambda calculus is correct: For lambda terms with a normal form that contains no lambdas (ground term), the implementation is shown to yield a lambda calculus normal form. For lambda terms whose normal forms represent functions, it is shown that the implementation yields lambda terms that are beta-convertible in zero or more steps to the normal form of the original lambda term. In this sense, our implementation involves weak reduction according to Hindley et al. [9].

Translating Higher-Order Clauses to First-Order Clauses

“…Upon closer investigation, we discovered that Turner's optimizations were not yielding dramatic reductions in the size of the output. Despite Turner's claim [24, p. 269] that "the sizes of the successive terms now [form] only a linear progression," the translation as a whole is still quadratic [9]. We observed reductions of approximately 30%, in the number of combinators produced.…”

Capturing Strong Reduction in Director String Calculus

“…In addition, our approach intuitively supports the nonatomic nature of substitutiion as explained in [11,13] and lends itself to parallel and optimal reduction [4,7,9,10,14]. We assume that the readers are familiar with [8] throughout the rest of this paper.…”

“…Kennaway and Sleep [8] introduced director strings as combinators to formally study the approach taken by Turner. Director strings also provide an intuitive interpretation of the reduction rules of the combinators to explain what combinators achieve.…”

“…Director strings also provide an intuitive interpretation of the reduction rules of the combinators to explain what combinators achieve. Director string implementation of A-calculus as shown in [8] involves weak reduction according to Hindley et al [5]. In this paper, we introduce a modified version of director string calculus and obtain the strong reduction.…”

Higher-Order Code Splicing

“…This idea corresponds exactly to director strings [16], where S , B , and C correspond to Λ, \, and /. The make-xxx functions compute the free variables on the fly at compile time.…”