From Self-Interpreters to Normalization by Evaluation

We characterize normalization by evaluation as the composition of a self-interpreter with a self-reducer using a special representation scheme, in the sense of Mogensen (1992). We do so by deriving in a systematic way an untyped normalization by evaluation algorithm from a standard interpreter for the λ-calculus. The derived algorithm is not novel and indeed other published algorithms may be obtained in the same manner through appropriate adaptations to the representation scheme. 1. Self-interpreters and self-reducers What is a self-interpreter? For the untyped λ-calculus, Mogensen (1992) offers the following definition, given an injective mapping x y (the representation scheme) that yields representations of arbitrary terms: E xMy β M. That is, a self-interpreter is a term E of the λ-calculus such that when applied to the representation xMy of any term M, the result is a convertible term (modulo renaming). The x y mapping cannot of course be defined within the λ-calculus itself, but we posit its existence as a primitive operation of the calculus. The representation of a term is a piece of data, something that can be manipulated, transformed and inspected within the calculus itself. It is natural to represent data as terms in normal form, so that datamay be regarded as constant with regard to term reduction. Consider the following grammar for terms and normal terms: Var Q x, y, z Term Q t F x | λx.t | t t Term  TermNF Q tn F ta | λx.tn Term  TermA Q ta F x | ta tn The representation scheme can be typed as x y : Term Ñ TermNF. All manner of representation schema are possible, but Mogensen commits to a particularly simple representation scheme, one that enables him to implement a trivially simple selfinterpreter that not only yields convertible terms from their representations, but in fact whose weak head normal form when applied to a normal term M is identical to M, up to renaming of variables. Let us call this particular self-interpreter Eα. We have that Eα xMy ÝÑwhnf M. Mogensen goes on to define a self-reducer as a transformation on representations: R xMy β xNFMy, whereNFM stands for the normal formofM, if one exists. Equippedwith such a contraption, we can define a special kind of self-interpreter with the additional property that all representations 1This is an analogue of the quote special form of Scheme. 35 36 MATHIEU BOESPFLUG of terms evaluate to normal forms. For all M, ENF xMŷEαpR xMyq ÝÑwhnf NFM. After a small detour towards concrete implementations of the above, we will show how through successive transformations we may obtain an untyped normalization by evaluation algorithm, in fact precisely the algorithm presented in (Boespflug, 2009). This algorithm is but one of several variants of untyped normalization by evaluation algorithms, and indeed the transformations presented here can readily be adapted to derive the algorithms of Aehlig et al. (2008) and Filinski and Rohde (2004). Because the starting points of these transformations are in fact standard definitions of evaluators and normalizers, it is possible to obtain the correctness of the normalization by evaluation algorithm as a corollary of the correctness of the meaning preserving transformations we use.