Normalization and Partial Evaluation

Abstract

Abstract. We give an introduction to normalization by evaluation and type-directed partial evaluation. We first present normalization by evaluation for a combinatory version of Gödel System T. Then we show normalization by evaluation for typed lambda calculus with β and η conversion. Finally, we introduce the notion of binding time, and explain the method of type-directed partial evaluation for a small PCF-style functional programming language. We give algorithms for both call-by-name and call-by-value versions of this language.

Canonicity for 2-dimensional type theory

“…We do not attempt a proof-theoretic proof of consistency of definitional equality, e.g. using reduction or normalization-by-evaluation [7]. Proving consistency using these approaches typically involves giving a decision procedure for definitional equality, and equality for 2-dimensional type theory is not decidable, due to the equality reflection rules for identity types discussed in in Section 4.…”

Formal Aspects of Polyvariant Specialization

“…We work in a call-by-value denotational setting [43], where our approach -inspired by type-directed partial evaluation [14,19,21] -is to phrase polyvariant specialization as a non-standard, code-generating semantics (also called a residualizing semantics). This approach immediately suggests certain proof principles, namely structural induction (by compositionality) and fixed-point induction.…”

“…Our closest related work is Filinski's formalizations of type-directed (monovariant) partial evaluation [14,15,19,21,22]. In fact, the technical aspects of the present work are inspired by Filinski's work: for call-by-value type-directed partial evaluation with computational effects for the simply-typed lambda calculus with constants, he uses a denotational semantics parameterized by a monad.…”

“…This would allow more radical program rearrangements, e.g., via the continuation monad [19]. In particular, an "unlift" operation, @, (namely bounded static variation) can be defined:…”

On Normalization by Evaluation for Object Calculi

“…The method is robust and widely applicable; it has been adapted to various type theories, going well beyond simply typed lambda calculus (for instance, see [4,3,8,7]). An introductory survey, including applications of NBE to partial evaluation of functional programs, can be found in [10].…”