CPS Translations and Applications: The Cube and Beyond

Continuation passing style (CPS) translations of typed -calculi numerous applications. However, the range of these applications is con ned by the fact that CPS translations are known for nondependent type systems only, thus excluding well-known systems like the calculus of constructions (CC) and the logical frameworks (LF). This paper presents techniques for CPS translating systems with dependent types. In the rst part of the paper we review several lines of work in which the need for CPS translations of dependent type systems has arisen, and discuss the di culties involved with CPS translating such systems. One way of overcoming these di culties is to work with so-called domain-free type systems. Thus, instead of Barendregt's -cube we shall consider the domain-free -cube, and instead of traditional pure type systems, we shall consider domain-free pure type systems. We therefore begin the second part by reviewing the domain-free -cube, which includes domainfree versions of CC and LF, and then present CPS translations of the systems of the domain-free -cube. Direct style (DS) (i.e., inverse CPS) translations have been used in a number of applications, but have been formulated for untyped and simply-typed languages only. We introduce DS translations for all the systems of the domain-free -cube. The CPS translations are then generalized further to a certain class of domain-free pure type systems which includes all the systems of the domain-free -cube as well as domain-free versions of many the non-dependent logical systems of Coquand and Herbelin. Many translations that appear in the literature arise as special cases of our translation. In the third part of the paper we present two approaches to CPS translations of traditional pure type systems. The rst, indirect, technique uses some results that relate derivations in domain-free and traditional pure type systems to modify the above mentioned translation of domain-free pure type systems to the analogous class of traditional pure type systems, thus including Barendregt's -cube and many of the non-dependent logical pure type systems of Coquand and Herbelin. The second, direct, approach translates traditional pure type systems, requiring a certain order to be well-founded. The technique works for Barendregt's -cube, but requires rather strong means.