Usually types of PCF are interpreted as cpos and terms as continuous functions. It is then the case that non-termination of a closed term of ground type corresponds to the interpretation being bottom; we say that the semantics is adequate. We shall here present an axiomatic approach to adequacy for PCF in the sense that we will introduce categorical axioms enabling an adequate semantics to be given. We assume the presence of certain "bottom" maps with the role of being the interpretation of non-terminating terms, but the order-structure is left out. This is different from previous approaches where some kind of order-theoretic structure has been considered as part of an adequate categorical model for PCF. We take the point of view that partiality is the fundamental notion from which order-structure should be derived, which is corroborated by the observation that our categorical model induces an order-theoretic model for PCF in a canonical way.

The relationship between Hyland-Ong-style games and Berry-Curien sequential algorithms is investigated, with the object of describing semantic solutions to two problems | to characterise eectively the \minimal models" of the simply-typed -calculus and the fully abstract model of PCF with control operators | which are shown to be equivalent.

We give a categorical description of a class of sound and adequate models of a functional language with assignable variables. This is based on a notion of \sequoidal category", a symmetric monoidal category with an additional non-commutative and non-associative tensor product. We describe a category G of games and strategies, and show that it satis es our axioms. We give an axiomatic characterization of those categories (including G) which give rise to fully abstract models.

. 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 present an axiomatic characterization of models fully-complete for ML-polymorphic types of system F. This axiomatization is given for hyperdoctrine models, which arise as adjoint models, i.e. co-Kleisli categories of suitable linear categories. Examples of adjoint models can be obtained from categories of Partial Equivalence Relations over Linear Combinatory Algebras. We show that a special linear combinatory algebra of partial involutions induces an hyperdoctrine which satisfies our axiomatization, and hence it provides a fully-complete model for ML-types.