We introduce CoCasl as a simple coalgebraic extension of the algebraic specification language Casl. CoCasl allows the nested combination of algebraic datatypes and coalgebraic process types. We show that the well-known coalgebraic modal logic can be expressed in CoCasl. We present sufficient criteria for the existence of cofree models, also for several variants of nested cofree and free specifications. Moreover, we describe an extension of the existing proof support for Casl (in the shape of an encoding into higher-order logic) to CoCasl.

We investigate the difference between two well-known notions of independence bisimilarity, history-preserving bisimulation and hereditary history-preserving bisimulation. We characterise the difference between the two bisimulations in trace-theoretical terms, advocating the view that the first is (just) a bisimulation for causality, while the second is a bisimulation for concurrency. We explore the frontier zone between the two notions by defining a hierarchy of bounded backtracking bisimulations.

Spans of open maps have been proposed by Joyal, Nielsen, and Winskel as a way of adjoining an abstract equivalence, P-bisimilarity, to a category of models of computation M, where P is an arbitrary subcategory of observations. Part of the motivation was to recast and generalise Milner's well-known strong bisimulation in this categorical setting. An issue

We present a coalgebraic approach to trace equivalence semantics based on lifting behaviour endofunctors for deterministic action to Kleisli categories of monads for non-deterministic choice. In Set , this gives a category with ordinary transition systems as objects and with morphisms characterised in terms of a linear notion of bisimulation. The final object in this category is the canonical abstract model for trace equivalence and can be obtained by extending the final coalgebra of the deterministic action behaviour to the Kleisli category of the non-empty powerset monad. The corresponding final coalgebra semantics is fully abstract with respect to trace equivalence.

This thesis is concerned with formal semantics and models for concurrent computational systems, that is, systems consisting of a number of parallel computing sequential systems, interacting with each other and the environment. A formal semantics gives meaning to computational systems by describing their behaviour in a mathematical model. For concurrent systems the interesting aspect of their computation is often how they interact with the environment during a computation and not in which state they terminate, indeed they may not be intended to terminate at all. For this reason they are often referred to as reactive systems, to distinguish them from traditional calculational systems, as e.g. a program calculating your income tax, for which the interesting behaviour is the answer it gives when (or if) it terminates, in other words the (possibly partial) function it computes between input and output. Church's thesis tells us that regardless of whether we choose the lambda calculus, Turing machines, or almost any modern programming language such as C or Java to describe calculational systems, we are able to describe exactly the same class of functions. However, there is no agreement on observable behaviour for concurrent reactive systems, and consequently there is no correspondent to Church's thesis. A result of this fact is that an overwhelming number of di#erent and often competing notions of observable behaviours, primitive operations, languages and mathematical models for describing their semantics, have been proposed in the litterature on concurrency.

We present a presheaf model for the observation of infinite as well as finite computations. We apply it to give a denotational semantics of SCCS with finite delay, in which the meanings of recursion are given by final coalgebras and meanings of finite delay by initial algebras of the process equations for delay. This can be viewed as a first step in representing fairness in presheaf semantics. We give a concrete representation of the presheaf model as a category of generalised synchronisation trees and show that it is coreflective in a category of generalised transition systems, which are a special case of the general transition systems of Hennessy and Stirling. The open map bisimulation is shown to coincide with extended bisimulation of Hennessy and Stirling, which is essentially fair CTL*-bisimulation. Finally we formulate Milners operation semantics of SCCS with finite delay in terms of generalised transition systems and prove that the presheaf semantics is fully abstract with respect to extended bisimulation.

Formal models for real-time systems have been studied intensively over the past decade. Much of the theory of untimed systems have been lifted to real-time settings. One example is the notion of bisimulation applied to timed transition systems, which is studied here within the general categorical framework of open maps. We define a category of timed transition systems, and show how to characterize standard timed bisimulation in terms of spans of open maps with a natural choice of a path category. This allows us to apply general results from the theory of open maps, e.g. the existence of canonical models and characteristic logics. Also, we obtain here an alternative proof of decidability of bisimulation for finite transition systems, and illustrate the use of open maps in finite presentations of bisimulations