. 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 types, 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 \Omega\Gamma The proofs use variants of the method of reducibility. In this paper, we present a uniform approach for proving several meta-theorems relating properties of -terms and their typability 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 show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easi...

. The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this...

The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces t together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modi ed realizability both t into this framework. We are then able to prove a meta-theorem which shows that if a property of realizers satis es some simple conditions, then it holds for the semantic interpretations of all terms. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed-terms, in particular, strong normalization and con uence. This approach clari es the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. The above approach is applied to the simply-typed-calculus (with types!,,+,and?), and to the second-order (polymorphic)-calculus (with types! and 82), for which it yields a new theorem.