 
Summary: Under consideration for publication in Math. Struct. in Comp. Science
Operational Aspects of Untyped
Normalization by Evaluation
KLAUS AEHL I G y and FEL I X JOACH I MSKI y
Mathematisches Institut der LudwigMaximiliansUniversitat Munchen
Theresienstrasse 39, 80333 Munchen, Germany
Received 15 April 2003
A purely syntactic and untyped variant of Normalization by Evaluation for the
calculus is presented in the framework of a twolevel calculus with rewrite rules to
model the inverse of the evaluation functional. Among its operational properties gures
a standardization theorem that formally establishes adequacy of implementation in
functional programming languages. An example implementation in Haskell is provided.
The relation to usual typedirected Normalization by Evaluation is highlighted, using a
short analysis of expansion that leads to a perspicuous strong normalization and
con
uence proof for "reduction as a byproduct.
Introduction
Normalization by Evaluation uses the evaluation mechanism of a metalanguage to nor
malize terms, typically of the calculus. By means of an interpretation function [[]] ,
terms are embedded into this metalanguage; an \inverse of the evaluation functional"
(Berger and Schwichtenberg, 1991) # serves to recover terms from the semantics. The
