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Typed Applicative Structures and Normalization by Evaluation for System F

Summary: Typed Applicative Structures and
Normalization by Evaluation for System F
Andreas Abel
Department of Computer Science
Ludwig-Maximilians-University Munich
Abstract. We present a normalization-by-evaluation (NbE) algorithm for Sys-
tem F
with -equality, the simplest impredicative type theory with computa-
tion on the type level. Values are kept abstract and requirements on values are kept
to a minimum, allowing many different implementations of the algorithm. The al-
gorithm is verified through a general model construction using typed applicative
structures, called type and object structures. Both soundness and completeness of
NbE are conceived as an instance of a single fundamental theorem.
1 Introduction and Related Work
The Curry-Howard isomorphism, which identifies proofs with programs and proposi-
tions with types, is the basis of several type-theoretic theorem provers, such as Coq
[INR07], Agda [Nor07], Epigram [CAM07] and LEGO [Pol94a]. In these systems,
checking the validity of proofs or typings relies on deciding equality of types. Types
are recognized as equal if they have the same normal form, this is why normalization
plays a key role in the study of the theories underlying these theorem provers, such as


Source: Abel, Andreas - Theoretische Informatik, Ludwig-Maximilians-Universit√§t M√ľnchen


Collections: Computer Technologies and Information Sciences