Defunctionalized interpreters for programming languages

Abstract

This document illustrates how functional implementations of formal semantics (structural operational semantics, reduction semantics, small-step and big-step abstract machines, natural semantics, and denotational semantics) can be transformed into each other. These transformations were foreshadowed by Reynolds in "Definitional Interpreters for Higher-Order Programming Languages" for functional implementations of denotational semantics, natural semantics, and big-step abstract machines using closure conversion, CPS transformation, and defunctionalization. Over the last few years, the author and his students have further observed that functional implementations of small-step and of big-step abstract machines are related using fusion by fixed-point promotion and that functional implementations of reduction semantics and of smallstep abstract machines are related using refocusing and transition compression. It furthermore appears that functional implementations of structural operational semantics and of reduction semantics are related as well, also using CPS transformation and defunctionalization. This further relation provides an element of answer to Felleisen's conjecture that any structural operational semantics can be expressed as a reduction semantics: for deterministic languages, a reduction semantics is a structural operational semantics in continuation style, where the reduction context is a defunctionalized continuation. As the defunctionalized counterpart of the continuation of a one-step reduction function, a reduction context represents the rest of the reduction, just as an evaluation context represents the rest of the evaluation since it is the defunctionalized counterpart of the continuation of an evaluation function.

A Context-based Approach to Proving Termination of Evaluation

“…Not surprisingly, the computational content of our proofs are instances of normalization by evaluation; the extracted programs are evaluators in continuation-passing style, whose continuations arise by extraction from a context reducibility predicate. Thus the present article provides a logical confirmation of the connection between continuations and contexts, previously observed and investigated by Danvy [11,12].…”

“…Moreover, they are in continuation-passing style and the continuations arise as the computational content of the reducibility predicates for reduction contexts. This latter fact shows a logical connection between continuations and contexts; the correspondence between them has previously been observed and investigated by Danvy in the setting of program transformations [11,12]. The method of proof developed in this paper relies on the formalism of reduction semantics and is therefore fitted for languages with context-sensitive notion of reduction.…”

“…Felleisen's contexts represent "the surrounding term" of the current subterm, or "the rest of the computation", and they directly correspond to continuations: the latter can be seen as functional representations of contexts. More precisely, Danvy observed that reduction contexts arise as defunctionalized continuations of a one-step reduction function whereas evaluation contexts arise as defunctionalized continuations of an evaluation function (i.e., big-step) [11,12]. Since these defunctionalized representations of continuations are in both cases the same, the terms "evaluation context" and "reduction context" are usually used interchangeably, and we will adhere to this practice in the remainder of this article.…”

Inter-deriving semantic artifacts for object-oriented programming

“…It is usually left unspecified in the literature [29]. We define it here in a form that time and again we have found convenient [19], namely as a big-step abstract machine. Here, this abstract machine has (a) two intermediate states: a term and a context, and a context and a value; and (b) three state-transition functions decompose_term, decompose_term_neg, and decompose_context.…”

“…Our goal here is to apply Danvy et al's 'syntactic correspondence' and 'functional correspondence' [3,8,11,18,19,[43][44][45] to Abadi and Cardelli's untyped non-imperative calculus of objects [1,Chapter 6]. In the rest of this introduction, we describe in turn the syntactic correspondence, the functional correspondence, and Abadi and Cardelli's calculus of objects, before detailing our contribution.…”