Continuous semantics for strong normalisation

Abstract

We prove a general strong normalisation theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domain-theoretic semantics. The theorem applies to extensions of Gödel's system T , but also to various forms of barrecursion for which strong normalisation was hitherto unknown.

A proof of strong normalisation using domain theory

“…In particular there exists m such that if g m (S q 0) = f (S q 0) and g m (S q ⊤) = f (S q ⊤), for all q < m, then K g m = ⊤. If we define Like in [11], it is crucial for this argument that we are using a domain model. These constants make also the system proof-theoretically strong, at least the strength of secondorder arithmetic.…”

“…where l is the arity of c. Like in [11], we assume our system of constant reduction rules to be left linear, i.e. a variable occurs at most once in the left hand side of a rule, and mutually disjoint, i.e.…”

“…The paper [9] 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 [11,12], who used instead a modification of Plotkin's computational adequacy theorem [19], and could prove strong normalisation. In a way, the idea is to replace infinite terms by elements of a domain interpretation.…”

“…The main contribution of this paper is to show that, using ideas from intersection types [3,6,7,18] and Martin-Löf's domain interpretation of type theory [16], one can in turn simplify further U. Berger's argument. Contrary to [11], we build a domain model for an…”

“…The main contribution of this paper is to show that, using ideas from intersection types [3,6,7,18] and Martin-Löf's domain interpretation of type theory [16], one can in turn simplify further U. Berger's argument. Contrary to [11], we build a domain model for an untyped programming language. Compared to [12], there is no need of an extra hypothesis to deduce strong normalisation from the domain interpretation.…”