We define a new class of Kripke structures for the second-order λ-calculus, and investigate the soundness and completeness of some proof systems for proving inequalities as well as equations. The Kripke structures under consideration are equipped with preorders that correspond to an abstract form of reduction, and they are not necessarily extensional. A novelty of our approach is that we define these structures directly as functors A: → Preor equipped with certain natural transformations corresponding to application and abstraction . We (...) make use of an explicit construction of the exponential of functors in the Cartesian-closed category Preor, and we also define a kind of exponential ΠΦs ε T to take care of type abstraction. However, we strive for simplicity, and we only use very elementary categorical concepts. Consequently, we believe that the models described in this paper are more palatable than abstract categorical models which require much more sophisticated machinery . We obtain soundness and completeness theorems that generalize some results of Mitchell and Moggi to the second-order λ-calculus, and to sets of inequalities. (shrink)

It was observed by Curry that when λ-terms can be assigned types, for example, simple types, these terms have nice properties . Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of 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 and Ω. 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 λ-terms and their typability in the systems and Ω. 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 easily rederived. We also characterize the terms that have weak 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. (shrink)