Comparing cubes of typed and type assignment systems

Abstract

We study the cube of type assignment systems, as introduced in Giannini et al. (Fund. Inform. 19 (1993) 87–126), and confront it with Barendregt's typed gl-cube (Barendregt, in: Handbook of Logic in Computer Science, Vol. 2, Clarenden Press, Oxford, 1992). The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property holds only for systems without polymorphism. The type assignment systems we consider satisfy the properties ‘subject reduction’ and ‘strong normalization’. Moreover, we define a new type assignment cube that is isomorphic to the typed one.