The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logical-relations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesian-closed-category models of the simply typed lambda calculus and PL-category models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.

We describe a sequent calculus, based on work of Herbelin, of which the cut-free derivations are in 1-1 correspondence with the normal natural deduction proofs of intuitionistic logic. We present a simple proof of Herbelin's strong cutelimination theorem for the calculus, using the recursive path ordering theorem of Dershowitz.

This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a category-theoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined. 1 Introduction Many modern programming languages feature rather sophisticated typing mechanisms. In particular, languages such as ML include polymorphic data types, which allow considerable programming flexibility. Several notions of polymorphism were introduced into computer science by Strachey [Str67], among them the important notion of parametric polymorphism. Strachey's intuitive definition is that a polymorphic function is parametric if it has a uniformly given algorithm in all types, that is, if the function's behavior is independent of the type at which the function is instantiated. Reynolds [Rey83] proposed a mathematical definition of parametric polymorphic functions by means of invariance with respect to certain relations induced by typ...

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Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types and between types. Reynolds's approach set the basis for most of the current work on parametricity, as we will review below (.3). Some twelve years earlier, Girard had given just a simple hint towards another understanding of the properties of "computing with types". In [Gir71], it is shown, as a side remark, that, given a type A, if one defines a term J A such that, for any type B, J A B reduces to 1, if A = B, and reduces to 0, if A ¹ B, then F + J A does not normalize. In particular, then, J A is not definable in F. This remark on how terms may depend on types is inspired by a view of types which is quite different from Reynolds's. System F was born as the theory of proofs of second order intuitionis...

We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL+MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces. This paper

Abstract. We give a semantic footing to the fold/build syntax of programming with inductive types, covering shortcut deforestation, based on a universal property. Specifically, we give a semantics for inductive types based on limits of algebra structure forgetting functors and show that it is equivalent to the usual initial algebra semantics. We also give a similar semantic account of the augment generalization of build and of the unfold/destroy syntax of coinductive types. 1

Dedicated to the memory of Moez Alimohamed

. We investigate a simple form of parametricity, based on adding "abstract" copies of pre-existing types. Connections are made with the Reynolds-Ma theory of parametricity by logical relations, with the theory of parametricity via dinaturality, and with the categorical notion of equivalence. Introduction In his fundamental paper on the notion of parametricity in connection with type theories [Rey83], John Reynolds links the notion of parametricity firmly to the notion of data abstraction. This, unlike Strachey's earlier characterization via algorithm re-use, is a need-driven analysis. We need things to be parametric because otherwise our data abstractions will no longer be abstract. In his subsequent paper with Ma [MR91], two further points are made. One is that the problems reside more at the level of parametrized types than at the level of the quantified polymorphic types, and the other is that the notion of parametricity is not absolute, but relative. The Ma-Reynolds work produces ...

) R.F. Blute 1 P.J. Scott 1 Dept. of Mathematics University of Ottawa Ottawa, Ontario K1N 6N5 CANADA E. N. T. C. S. Elsevier Science B. V. Abstract We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic known as cyclic linear logic (CyLL), first defined by Yetter. The semantics is obtained by considering dinatural transformations on a category of topological vector spaces which are invariant under certain actions of a noncocommutative Hopf algebra, called the shuffle algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that the space has the denotations of cut-free proofs in CyLL+MIX as a basis. This work is a natural extension of the authors' previous work, "Linear Lauchli Semantics", where a similar theorem is obtained for the commutative logic. In that paper, we consider dinaturals which are invariant under certain actions of the additive group of integers. The passage from group...