Institute for Research in Cognitive Science Typing Untyped

It was observed by Curry that when (untyped)-terms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive t ypes, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having head-normal forms, can be characterized in some systems D and D. The proofs use variants of the method of reducibility. In this paper, we p r e s e n t a uniform approach for proving several meta-theorems relating properties of-terms and their typa-bility in the systems D and D. Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we s h o w h o w t h e c haracterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easily rederived. We a l s o c haracterize the terms that have w eak head-normal forms, which appears to be new. We conclude by stating a number of challenging open problems regarding possible generalizations of the realizability method.