We give an axiomatic treatment of fixed-point operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixed-point operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !-continuous functions between !-complete pointed partial orders. This possesses a least-fixed-point oper...

. We give a new characterization of lambda definability in Henkin models using logical relations defined over ordered sets with varying arity. The advantage of this over earlier approaches by Plotkin and Statman is its simplicity and universality. Yet, decidability of lambda definability for hereditarily finite Henkin models remains an open problem. But if the variable set allowed in terms is also restricted to be finite then our techniques lead to a decision procedure. 1 Introduction An applicative structure consists of a family (A oe ) oe2T of sets, one for each type oe, together with a family (app oe;ø ) oe;ø 2T of application functions, where app oe;ø maps A oe!ø \Theta A oe into A ø . For an applicative structure to be a model of the simply typed lambda calculus (in which case we call it a Henkin model, following [4]), one requires two more conditions to hold. It must be extensional which means that the elements of A oe!ø are uniquely determined by their behavior under app oe;ø...

. We investigate, in a categorical setting, some completeness properties of beta-eta conversion between closed terms of the simplytyped lambda calculus. A cartesian-closed category is said to be complete if, for any two unconvertible terms, there is some interpretation of the calculus in the category that distinguishes them. It is said to have a complete interpretation if there is some interpretation that equates only interconvertible terms. We give simple necessary and sufficient conditions on the category for each of the two forms of completeness to hold. The classic completeness results of, e.g., Friedman and Plotkin are immediate consequences. As another application, we derive a syntactic theorem of Statman characterizing beta-eta conversion as a maximum consistent congruence relation satisfying a property known as typical ambiguity. 1 Introduction In 1970 Friedman proved that beta-eta conversion is complete for deriving all equalities between the (simply-typed) lambda-definable...

. We present a linear realizability technique for building Partial Equivalence Relations (PER) categories over Linear Combinatory Algebras. These PER categories turn out to be linear categories and to form an adjoint model with their co-Kleisli categories. We show that a special linear combinatory algebra of partial involutions, arising from Geometry of Interaction constructions, gives rise to a fully and faithfully complete model for ML polymorphic types of system F. Keywords: ML-polymorphic types, linear logic, PER models, Geometry of Interaction, full completeness. Introduction Recently, Game Semantics has been used to define fully-complete models for various fragments of Linear Logic ([AJ94a,AM99]), and to give fully-abstract models for many programming languages, including PCF [AJM96,HO96,Nic94], richer functional languages [McC96], and languages with non-functional features such as reference types and non-local control constructs [AM97,Lai97]. All these results are cru...

We study the operational, denotational, and axiomatic semantics of lazy and call-by-value functional languages, and use these semantics to build a new expressiveness theory for comparing functional languages. The first part of the thesis develops the theory of lazy and call-by-value languages separately, following paradigmatic studies of call-by-name functional languages. We first describe the operational semantics of two simply-typed languages, lazy PCF and call-byvalue PCF. These two languages provide enough intuition to describe general definitions of denotational models and logics for lazy and call-by-value languages. We prove, via a completeness theorem, that the definitions of models and logic coincide for both the lazy and call-by-value theories. The second part of the thesis compares the two kinds of languages via translations. Specifically, we develop the idea of a fully abstract translation and define new fully abstract translations from call-by-value PCF to lazy PCF, and vi...