A Proof of Strong Normalisation Using Domain Theory

Thierry Coquand 1 Arnaud Spiwack 2, 3
2 TYPICAL - Types, Logic and computing
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : In 1961, Spector presented an extension of Gödel's system T by a new schema of definition called bar recursion. With this new schema, he was able to give an interpretation of Analysis, extending Gödel's Dialectica interpretation of Arithmetic, and completing preliminary results of Kreisel. Tait proved a normalisation theorem for Spector's bar recursion, by embedding it in a system with infinite terms. In a paper by Berardi, Bezem and Coquand, an alternative form of bar recursion was introduced. This allowed to give an interpretation of Analysis by modified realisability, instead of Dialectica interpretation. It presented also a normalisation proof for this new schema, but this proof, which used Tait's method of introducing infinite terms, was quite complex. It was simplified significantly by U. Berger, who used instead a modification of Plotkin's computational adequacy theorem, and could prove strong normalisation. In a way, the idea is to replace infinite terms by elements of a domain interpretation. This domain has the property that a term is strongly normalisable if its semantics is not ⊥. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Löf's domain interpretation of type theory, one can in turn simplify further U. Berger's argument. Contrary to him, we build a domain model for an untyped programming language. Compared to other works, there is no need of an extra hypothesis to deduce strong normalisation from the domain interpretation. A noteworthy feature of this domain model is that it is in a natural way a complete lattice, and in particular it has a top element which can be seen as the interpretation of a top-level exception in programming language. We think that this model can be the basis of modular proofs of strong normalisation for various type systems. As a main application, we show that Martin-Löf dependent type theory extended with a program for Spector double negation shift, similar to bar recursion, has the strong normalisation property.
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Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2007, 16 p. 〈10.2168/LMCS-3(4:12)2007〉
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Thierry Coquand, Arnaud Spiwack. A Proof of Strong Normalisation Using Domain Theory. Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2007, 16 p. 〈10.2168/LMCS-3(4:12)2007〉. 〈inria-00432448〉

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