2006 Scheme and Functional Programming Papers, University of

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118 Scheme and Functional Programming, 2006
A Self-Hosting Evaluator using HOAS A Scheme Pearl Eli Barzilay Northeastern University eli@barzilay.org Abstract We demonstrate a tiny, yet non-trivial evaluator that is powerful enough to run practical code, including itself. This is made possible using a Higher-Order Abstract Syntax (HOAS) representation — a technique that has become popular in syntax-related research during the past decade. With a HOAS encoding, we use functions to encode binders in syntax values, leading to an advantages of reflecting binders rather than re-implementing them. In Scheme, hygienic macros cover problems that are associated with binders in an elegant way, but only when extending the language, i.e., when we work at the meta-level. In contrast, HOAS is a useful object-level technique, used when we need to represent syntax values that contain bindings — and this is achieved in a way that is simple, robust, and efficient. We gradually develop the code, explaining the technique and its benefits, while playing with the evaluator. 1. Introduction Higher-Order Abstract Syntax (HOAS) is a technique for representing syntax with bindings using functions. This is a form of reflection in the sense that binders in the object level are represented using binders in the meta level. The result is simple (no need for sophisticated substitution mechanisms), robust (the meta-level is our language, which better implement scope correctly), and efficient (as it is part of the core implementation). HOAS has been in use for a while now [13], a good overview is given by Hofmann [9], and in [1, Section 4.7]. However, it is more popular in the strictly typed world than it is in Scheme. In part, this may be due to Scheme’s hygienic macro facility, which allows hooking new kinds of syntactic constructs into the language in a way that respects lexical scope. Using a high level macro system means that Schemers rarely need to represent syntax directly — instead, they work at the meta-level, extending the language itself. This is unfortunate, as HOAS can be a useful tool for syntax-related work. Furthermore, it is possible to formalize HOAS in a way that is more natural in a dynamically-typed language than it is in statically-typed ones, which corresponds to a HOAS formalization in Nuprl [3] that builds on the predicative nature of its type theory [2, 1, 12]. The purpose of this Scheme pearl is to demonstrate the use of HOAS in Scheme, with the goal of making this technique more accessible to Schemers 1 . As demonstrated below, using macros in Scheme facilitates the use of HOAS, as there is no need for an external tool for translating concrete syntax into HOAS representations. In itself, HOAS is a representation tool for object-level values, not for meta-level work where bindings are directly accessible in some way. It is, however, used in some meta-linguistic systems for implementing syntactic tools 2 . For example, it could be used as the underlying term representation in a language-manipulating tool like PLT’s Reduction Semantics [11]. 2. A Toy Evaluator The presentation begins with a simple evaluator. Our goal is to evaluate a Lambda-Calculus-like language using reductions, so we need a representation for lambda abstractions and applications. To make this example more practical, we also throw in a conditional if special form, make it handle multiple arguments, and use callby-value. A common evaluator sketch for such a language is 3 : (define (ev expr) (cond [(not (pair? expr)) expr] [(eq? ’if (car expr)) (ev (if (ev (cadr expr)) (caddr expr) (cadddr expr)))] [else (ev (let ([f (ev (car expr))]) (substitute (body-of f) (args-of f) (map ev (cdr expr)))))])) where an application is always assumed to have an abstraction in its head position, and the args-of and body-of functions pull out the corresponding parts. As expected, the main issue here is implementing a proper substitution function. Common approaches include using symbols and renaming when needed, or using symbols ‘enriched’ with lexical binding information (‘colors’). Proceedings of the 2006 Scheme and Functional Programming Workshop University of Chicago Technical Report TR-2006-06 1 The presentation is loosely based on a comp.lang.scheme post from October 2002. 2 It might be possible that HOAS can be used for implementing a lowlevel macro facility that the high-level hygienic macro system builds on. HOAS should not be confused with current low-level syntactic systems like syntactic closures or syntax-case. 3 Note that details like error checking are omitted, and that we use atomic Scheme values such as booleans and numbers to represent themselves. 119
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2006 Scheme and Functional Programming Papers, University of

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