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The relation of inclusion between types has been suggested by the practice of programming as it enriches the polymorphism of functional languages. We propose a simple (and linear) sequent calculus for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subtyping from a few, logically meaningful sequents. In particular, transitivity and anti-symmetry will be derived from elementary logical principles. 1 Introduction 1.1 Motivations, theories and models In recent years, several extensions of core functional languages have been proposed to deal with the notion of subtyping; see, for example, [CW85, Mit88, BL90, BCGS91, CMMS91, CG92, PS94, Tiu96, TU96]. These extensions were suggested by the practice of programming in computer science. In particular, they were inspired by the notion of inheritance as used in object-oriented programming languages, or by other concrete implementations of the following form of polymorphism: data living in a type oe, ...

Genericity is the idea that the same program can work at many dierent data types. Longo, Milstead and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type 8X:A[X ], are equal at any given instance A[T ], then they are equal at all instances. They proved that this rule is admissible in a certain extension of System F, but nding a semantically motivated model satisfying this principle remained an open problem.

We show that an enriched version of Freyd's principle of versality holds in the Kleisli category of a commutative strong monad with fixed-point object. This gives a general categorical setting in which it is possible to model recursive types involving the usual datatype constructors.

Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a category-theoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...

In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositions-as-types and proofs-as-constructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logical-operations-as-adjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9g-fragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and single-valued relations. However, when enriched with proofs-as-arrows, this familiar concept must be supplied with an additional conversion rule, conn...

Introduction to the Polymorphic Lambda Calculus John C. Reynolds Carnegie Mellon University December 23, 1994 The polymorphic (or second-order) typed lambda calculus was invented by Jean-Yves Girard in 1971 [11, 10], and independently reinvented by myself in 1974 [24]. It is extraordinary that essentially the same programming language was formulated independently by the two of us, especially since we were led to the language by entirely different motivations. In my own case, I was seeking to extend conventional typed programming languages to permit the definition of "polymorphic" procedures that could accept arguments of a variety of types. I started with the ordinary typed lambda calculus and added the ability to pass types as parameters (an idea that was "in the air" at the time, e.g. [4]). For example, as in the ordinary typed lambda calculus one can write f int!int : x int : f(f (x)) to denote the "doubling" function for the type int, which accepts a function from integers

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Abstract not found