We propose Interaction Categories as a new paradigm for the semantics of functional and concurrent computation. Interaction categories have specifications as objects, processes as morphisms, and interaction as composition. We introduce two key examples of interaction categories for concurrent computation and indicate how a general axiomatisation can be developed. The upshot of our approach is that traditional process calculus is reconstituted in functorial form, and integrated with type theory and functional programming.

Introduction 2 The semantic universe: transducers Similar ideas appeared independently in the work of Hans Bekic [Bek71]. Samson Abramsky Laboratory for the Foundations of Computer Science University of Edinburgh The very existence of the conference bears witness to the fact that "concurrency theory" has developed into a subject unto itself, with substantially di#erent emphases and techniques to those prominent elsewhere in the semantics of computation. Whatever the past merits of this separate development, it seems timely to look for some convergence and unification. In addressing these issues, I have found it instructive to trace some of the received ideas in concurrency back to their origins in the early 1970's. In particular, I want to focus on a seminal paper by Robin Milner [Mil75] , which led in a fairly direct line to his enormously influential work on [Mil80, Mil89]. I will take (to the extreme) the liberty of of applying hindsight, and show how some di

We propose a typed calculus of synchronous processes based on the structure of interaction categories. Our aim has been to develop a calculus for concurrency that is canonical in the sense that the typed -calculus is canonical for functional computation. We show strong connections between syntax, logic and semantics, analogous to the familiar correspondence between the typed -calculus, intuitionistic logic and cartesian closed categories. 1 Introduction T ypes are fundamental to the study of functional computation, for both theoretical and practical reasons. On the foundational side there are elegant connections between the typed -calculus, intuitionistic logic and cartesian closed categories, leading to the Propositions as Types paradigm [14] and the development of categorical logic [9,17]. From a practical point of view, compile-time type reconstruction is a boon to the programmer in languages such as Standard ML and Haskell. Turning to concurrency, the situation is much less sati...

Many different notions of "property of interest" and methods of verifying such properties arise naturally in programming. A general framework of "Specification Structures" is presented for combining different notions and methods in a coherent fashion. This is then applied to concurrency in the setting of Interaction Categories.

Synchronous languages have been designed to ease the development of reactive systems, by providing a methodological framework for assisting system designers from the early stages of requirement specifications to the final stages of code generation or circuit production. Synchronous languages enable a very high-level specification and an extremely modular design of complex reactive systems by structural decomposition of them into elementary processes. We define an order-theoretical model that gives a unified mathematical formalisation of all the above aspects of the synchronous methodology and characterises the essentials of the synchronous paradigm.

. We provide a systematic reconstruction of Abramsky's category SProc of synchronous processes [Abr93]: SProc is isomorphic to a span category on a category of traces. The significance of the work is twofold: It shows that the original presentation of SProc in mixed formulations is unnecessary --- a simple categorical description exists. Furthermore, the techniques employed in the reconstruction suggest a general method of obtaining process categories with structure similar to SProc. In particular, the method of obtaining bisimulation equivalence in our setting, which represents an extension of the work of Joyal, Nielsen and Winskel [JNW93], has natural application in many settings. 1 Introduction In [Abr93], Abramsky proposed a new paradigm for the semantics of computation, interaction categories, where the following substitutions are made: Denotational semantics Categories Interaction categories Domains objects Interface specifications Continuous functions maps Commun...

The purpose of this paper is to show how one may construct from a synchronous interaction category, such as SProc, a corresponding asynchronous version. Significantly, it is not a simple Kleisli construction, but rather arises due to particular properties of a monad combined with the existence of a certain type of distributive law. Following earlier work we consider those synchronous interaction categories which arise from model categories through a quotiented span construction: SProc arises in this way from labelled transition systems. The quotienting is determined by a cover system which expresses bisimulation. Asynchrony is introduced into a model category by a monad which, in the case of transition systems, adds the ability to idle. To form a process category atop this two further ingredients are required: pullbacks in the Kleisli category, and a cover system to express (weak) bisimulation. The technical results of the paper provide necessary and sufficient conditions for a Kleisli...

. The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications. In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing particular specification logics. We make use of `classifying categories', to present categories of algebras as functor categories and to obtain necessary basic results for particular specification logics in a uniform manner. The specification logics considered are: equational logic for total algebras, conditional equational logic for partial algebras, and rewrite logic for concurrent systems. 1 Category Theory and Applications in Computer Science Category theory has been developed as a mathematical theory over 50 years and has influenced not only almost all branches of structural mathematics but also the development of several areas of computer science. It is the aim of this paper to review t...

1 Introduction It has been known for some years that formal systems of various kinds correspond to certain flavours of categorical structure. The first person to observe this phenomenon seems to have been Lawvere, who formulated a connection between certain kinds of algebraic theories and categories with finite products. Since Lawvere's original insight, there has been much progress in understanding

this paper. We will, however, consider specific bicategories-with-feedback and define the notion of a category-withfeedback, which includes traced symmetric monoidal categories ([21]). In this context, we will consider functors between categories-with-feedback -- functors which model and facilitate the interplay between systems and their behaviours which occurs when specifying, building and analysing machines or programs.