Order Λ-calculus Research Articles

CATEGORICAL MODEL CONSTRUCTION FOR PROVING SYNTACTIC PROPERTIES

It is observed that the proof of strong normalizability of typed terms of typed λ-calculus looks like the process of the model construction. The observation is clarified by the works of Hyland and Ong [8], Altenkirch [2], Stefanova and Geuvers [15], and so on. The model of Hyland and Ong is based on PL-category [14] and the model of Altenkirch or Stefanova and Geuvers is a non-categorical model that is specially designed for this purpose. In this paper, we will construct yet another categorical model for the second order λ-calculus — a non-extensional PL-category. To demonstrate the significance of the non-extensional model, we will give a model in which we can carry out the proof of the uniqueness of the β-normal form. The non-extensionality comes from the use of semi-adjoint for interpretation of abstraction and application (uncurry).

Second Order Isomorphic Types: A Proof Theoretic Study on Second Order λ-Calculus with Surjective Pairing and Terminal Object

We investigate invertible terms and isomorphic types in the second order lambda calculus extended with surjective pairs and terminal (or Unit ) type. These two topics are closely related: on one side, the study of invertibility is a necessary tool for the characterization of isomorphic types; on the other hand, we need the notion of isomorphic types to study the typed invertible terms. The result of our investigation is twofold: we give a constructive characterization of the invertible terms, extending previous work by Dezani and Bruce-Longo, and a decidable equational theory of the isomorphisms of types which hold in all models of the calculus, which is a conservative extension to the second order case of the results previously achieved for the case of first order typed calculi. Via the Curry-Howard correspondence, this work also provides a decision procedure for strong equivalence of formulae in second order intuitionistic positive propositional logic, that is suitable to search equivalent proofs in automated deduction systems.

Deciding type isomorphisms in a type-assignment framework

AbstractThis paper provides a formal treatment of isomorphic types for languages equipped with an ML style polymorphic type inference mechanism. The results obtained make less justified the commonplace feeling that (the core of) ML is a subset of second order λ-calculus: we can provide an isomorphism of types that holds in the core ML language, but not in second order λ-calculus. This new isomorphism allows to provide a complete (and decidable) axiomatization of all the types isomorphic in ML style languages, a relevant issue for thetype as specificationsparadigm in library searches. This work is a very extended version of Di Cosmo (1992): we provide both a thorough theoretical treatment of the topic and describe a practical implementation of a library search system so that the paper can be used as a reference both by those interested in the formal theory of ML style languages, and by those simply concerned with implementation issues. The new isomorphism can also be used to extend the usual ML type-inference algorithm, as suggested by Di Cosmo (1992). Building on that proposal, we introduce a better type-inference algorithm that behaves well in the presence of non-functional primitives like references and exceptions. The algorithm described here has been implemented easily as a variation to the Caml-Light 0.4 system.

Semantics of the second order lambda calculus

In the literature ther are two main notins of model for the second order λ-calculus: one by Bruce, Meyer and Mitchell (the BMM-model, for short) in set-theoretical formulation and one category-theoretical by Seely. Here we generalise Seely's notion, using semifunctors and semi-adjunctions from Hayashi, and introduce λ2-algebras, λη2-algebras, λ2-models and λη-models, similarly to the untyped λ-calculus. Non-extensional abstraction of both term and type variables is described by semi-adjunctions (essentially as in Martini's thesis). We show that also for second order sume, the β-(and commutation!) conversions correspond to semi-functoriality and that the (additional) η-conversion corresponds to ordinary fuctioriality.In the above framework various examples – well known ones and variations – are described. Also, we determine the place of the BMM-models; an earlier version of the latter has been reported by Jacobs.

Finitely stratified polymorphism

We consider predicative type-abstraction disciplines based on type quantification with finitely stratified levels. These lie in the vast middle ground between quantifier-free parametric abstraction and full impredicative abstraction. Stratified polymorphism has an unproblematic set-theoretic semantics, and may lend itself to new approaches to type inference, without sacrificing useful expressive power. Our main technical result is that the functions representable in the finitely stratified polymorphic λ-calculus are precisely the super-elementary functions, i.e., the class ε 4 in Grzegorczyk's subrecursive hierarchy. This implies that there is no super-elementary bound on the length of optimal normalization sequences, and that the equality problem for finitely stratified polymorphic λ-expressions is not super-elementary. We also observe that finitely stratified polymorphism augmented with type recursion admits functional algorithms that are not typable in the full second order λ-calculus.