Studies of the mathematical properties of impredicative polymorphic types have for the most part focused on the polymorphic lambda calculus of Girard–Reynolds, which is a calculus of total polymorphic functions. This paper considers polymorphic types from a functional programming perspective, where the partialness arising from the presence of fixpoint recursion complicates the nature of potentially infinite (‘lazy’) data types. An approach to Reynolds' notion of relational parametricity is developed that works directly on the syntax of a programming language, using a novel closure operator to relate operational behaviour to parametricity properties of types. Working with an extension of Plotkin's PCF with ∀-types, lazy lists and existential types, we show by example how the resulting logical relation can be used to prove properties of polymorphic types up to operational equivalence.

This thesis contains a study of the proof theory of classical logic and addresses the prob-lem of giving a computational interpretation to classical proofs. This interpretation aims to capture features of computation that go beyond what can be expressed in intuitionisticlogic. We introduce several strongly normalising cut-elimination procedures for classicallogic. Our procedures are less restrictive than previous strongly normalising procedures, while at the same time retaining the strong normalisation property, which various standardcut-elimination procedures lack. In order to apply proof techniques from term rewriting, including symmetric reducibility candidates and recursive path ordering, we develop termannotations for sequent proofs of classical logic. We then present a sequence-conclusion natural deduction calculus for classical logicand study the correspondence between cut-elimination and normalisation. In contrast to earlier work, which analysed this correspondence in various fragments of intuitionisticlogic, we establish the correspondence in classical logic. Finally, we study applications of cut-elimination. In particular, we analyse severalclassical proofs with respect to their behaviour under cut-elimination. Because our cutelimination procedures impose fewer constraints than previous procedures, we are ableto show how a fragment of classical logic can be seen as a typing system for the simplytyped lambda calculus extended with an erratic choice operator. As a pleasing conse-quence, we can give a simple computational interpretation to Lafont's example.

for enforcing strong end-to-end confidentiality and integrity policies. Such policies, however, are usually specified in term of static information---data is labeled high or low security at compile time. In practice, the confidentiality of data may depend on information available only while the system is running This paper studies language support for run-time principals, a mechanism for specifying information-flow security policies that depend on which principals interact with the system. We establish the basic property of noninterference for programs written in such language, and use run-time principals for specifying run-time authority in downgrading mechanisms such as declassification.

Submitted for publication to Information and Computation. A summary of this paper appeared in TACS '97.

. Existential types have proved useful for classifying various kinds of information hiding in programming languages, such as occurs in abstract datatypes and objects. In this paper we address the question of when two elements of an existential type are semantically equivalent. Of course, it depends what one means by `semantic equivalence'. Here we take a syntactic approach---so semantic equivalence will mean some kind of operational equivalence. The paper begins by surveying some of the literature on this topic involving `logical relations'. Matters become quite complicated if the programming language mixes existential types with function types and features involving non-termination (such as recursive definitions). We give an example (suggested by Ian Stark) to show that in this case the existence of suitable relations is sufficient, but not necessary for proving operational equivalences at existential types. Properties of this and other examples are proved using a new form of operatio...

The virtual class [15] construct was first introduced in the language Beta to provide added expressiveness when used with inheritance. Unfortunately, the virtual class construct in Beta is not statically type-safe. In this paper we show how a generalization of the semantics of object-oriented languages with a MyType construct leads to a variant of virtual classes which needs no run-time checks. This results in an object-oriented language in which both parametric types and virtual classes (or types) are well-integrated, and which is statically type-safe.

Abstract. The OCL 1.4 specification introduces let-declarations for adding auxiliary class features in static structures of the UML. We provide a type inference system and a big-step operational semantics for the OCL 1.4 that treat UML static structures and UML object models abstractly and accommodate for additional declarations; the operational semantics satisfies a subject reduction property with respect to the type inference system. We also discuss an alternative, non-operational interpretation of let-declarations as constraints. 1

We develop a theory of program specification using the notion of refinement type. This provides a notion of structured specification, useful for verification and program development. We axiomatise the satisfaction of specifications by programs as a generalised typing relation and give rules for refining specifications. A per semantics based on Henkin models is given, for which the system is proven to be sound and complete. Keywords Specification, refinement, verification, type theory, Henkin models 1

We propose a framework which extends Antitonic Logic Programs [13] to an arbitrary complete bilattice of truth-values, where belief and doubt are explicitly represented. Inspired by Ginsberg and Fitting 's bilattice approaches, this framework allows a precise de nition of important operators found in logic programming, such as explicit and default negation. In particular, it leads to a natural semantical integration of explicit and default negation through the Coherence Principle [38], according to which explicit negation entails default negation. We then de ne Coherent Answer Sets, and the Paraconsistent Well-founded Model semantics, generalising many paraconsistent semantics for logic programs. In particular, Paraconsistent Well-Founded Semantics with eXplicit negation (WFSXp ) [3, 11]. The framework is an extension of Antitonic Logic Programs for most cases, and is general enough to capture Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs, and Fuzzy Logic Programming. Thus, we have a powerful mathematical formalism for dealing simultaneously with default, paraconsistency, and uncertainty reasoning. Results are provided about how our semantical framework deals with inconsistent information and with its propagation by the rules of the program.

. Essential concepts of algebraic specification refinement are translated into a type-theoretic setting involving System F and Reynolds' relational parametricity assertion as expressed in Plotkin and Abadi's logic for parametric polymorphism. At first order, the type-theoretic setting provides a canonical picture of algebraic specification refinement. At higher order, the type-theoretic setting allows future generalisation of the principles of algebraic specification refinement to higher order and polymorphism. We show the equivalence of the acquired type-theoretic notion of specification refinement with that from algebraic specification. To do this, a generic algebraic-specification strategy for behavioural refinement proofs is mirrored in the type-theoretic setting. 1 Introduction This paper aims to express in type theory certain essential concepts of algebraic specification refinement. The benefit to algebraic specification is that inherently first-order concepts are tra...