Strong Normalization from Weak Normalization by Translation into the Lambda-I-Calculus

Abstract

This paper offers a systematic account of techniques to infer strong normalization from weak normalization that make use of syntactic translations from λ-terms to λI-terms. We present variants of such techniques due to Klop, Sorensen, Xi, Gandy, and Loader. We show that all the translations, in some cases via adjustments, are special cases of a generic scheme of translations, known as permutative inner interpretations. Having established this, which is easy, the fact that all the translations can be used to reduce strong normalization to weak normalization can be obtained from a single, general result concerning permutative inner interpretations. Furthermore, we show that each of the translations can be obtained as the composition of Klop's well-known ι-translation and some other translation of independent interest. For instance, in the case of Xi's translation, this other translation is a thunkification translation. Finally we compare the above techniques in some detail to the intimately related techniques by de Vrijer and Girard. A main contribution of the paper is to compare the techniques of Gandy, Klop, and Girard in detail. For instance, we prove the main property of Gandy's translation without reference to functionals using instead ideas from Klop's translation.