provide a formal but intuitive way of presenting and reasoning about programs, which is widely used in practice, although in an informal or semi-formal fashion. In this thesis, we investigate categorical models of programming languages based on a graphical presentation. In the first part, we use a graphical presentation of processes to motivate a categorical model of processes which provides process types and constructors similar to those available in categories of graphs. The model is parametrised on a base category of processes, and may therefore be used to model a variety of process calculi or languages. We present a concrete instance of this model, based on the process calculus CCS, and show that it arises as a syntactic category of an extension of the base calculus. In the second part of the thesis, we use a graphical semantics due to Jeffrey to model and prove correct a step in the compilation of higher-order functional programming languages: closure conversion -- a program tra

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.

One emerging approach to reducing the labour and costs of software development favours the specialisation of techniques to particular application domains. The rationale is that programs within a given domain often share enough common features and assumptions to enable the incorporation of substantial support mechanisms into domain-specific programming languages and associated tools. Instead of being machine-oriented, algorithmic implementations, programs in many domain-specific languages (DSLs) are rather user-level, problem-oriented specifications of solutions. Taken further, this view suggests that the most appropriate representation of programs in many domains is diagrammatic, in a way which derives from existing design notations in the domain. This thesis conducts an investigation, using mathematical techniques and supported by case studies, of issues arising from the use of diagrammatic representations in DSLs. Its structure is conceptually divided into two parts: the first is co...

. Beginning with the category Dom of Scott domains and continuous maps, we introduce a syntax for dataflow networks as "systems of inequalities," and provide an associated operational semantics. We observe that, under this semantics, a system of inequalities determines a two-sided fibration in Dom. This leads to the introduction of a certain class of cartesian arrows of spans as a notion of morphism for systems. The resulting structure Sys, consisting of domains, systems, and morphisms, forms a bicategory that embeds Dom up to equivalence and is suitable as a semantic model for nondeterministic networks. Isomorphism in Sys amounts to a notion of system equivalence "up to deterministic internal computations." 1 Introduction Since the seminal paper of Kahn [Kah74], it has been known that networks of concurrently and asynchronously executing deterministic processes, communicating with each other by sending data values over unbounded FIFO communication channels, admit a simple ...

Increasingly, the style of computation is changing. Instead of one machine running a program sequentially, we have systems with many individual agents running in parallel. The need for mathematical models of such computations is therefore ever greater. There are many models of concurrent computations. Such models can, for example, provide a semantics to process calculi and thereby suggest behavioural equivalences between processes. They are also key to the development of automated tools for reasoning about concurrent systems. In this thesis we explore some applications and generalisations of one particular model – event structures. We describe a variety of kinds of morphism between event structures. Each kind expresses a different sort of behavioural relationship. We demonstrate the way in which event structures can model both processes and types of processes by recalling a semantics for Affine HOPLA, a higher order process language. This is given in terms of asymmetric spans of event structures. We show that such spans support a trace construction. This allows the modelling of feedback and suggests a semantics for non-deterministic dataflow processes in terms of spans. The semantics given is shown to be consistent with Kahn’s fixed point construction when we consider spans modelling deterministic processes. A generalisation of event structures to include persistent events is proposed. Based on previously described morphisms between classical event structures, we define several categories of event structures with persistence. We show that, unlike for the corresponding categories of classical event structures, all are isomorphic to Kleisli categories of monads

Programming embedded and cyber-physical systems requires attention not only to functional behavior and correctness, but also to non-functional aspects and specifically timing and performance constraints. A structured, compositional, model-based approach based on stepwise refinement and abstraction techniques can support the development process, increase its quality and reduce development time through automation of synthesis, analysis or verification. For this purpose, we introduce in this paper a general theory of timed actor interfaces. Our theory supports a notion of refinement that is based on the principle of worst-case design that permeates the world of performance-critical systems. This is in contrast with the classical behavioral and functional refinements based on restricting or enlarging sets of behaviors. An important feature of our refinement is that it allows time-deterministic abstractions to be made of time-non-deterministic systems, improving efficiency and reducing complexity of formal analysis. We also show how our theory relates to, and can be used to reconcile a number of existing time and performance models and how their established theories can be exploited to represent and analyze interface specifications and refinement steps. 1

In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginning, process networks have been drawn as particular graphs, but this syntax is never formalized. We take the opportunity to clarify it by giving a precise definition of these graphs,

We show how a simple intuitive conception of nondeterministic computation leads to a construction, given a locally ordered bicategory D having finite bicategorical products (for example, a bicategory of domains and continuous functions with extensionally ordered homs), of a bicategory N D of “nondeterministic arrows” that embeds D as a locally full sub-bicategory. The cartesian product on D extends to a pseudofunctorial tensor product on N D. We show that, in case the homs of D are bounded-complete directed complete partial orders with composition respecting local directed colimits, then a nondeterministic arrow in N D is a left adjoint (i.e. a “map”) if and only if it is isomorphic to a strict, sup-preserving arrow of D. Under the additional assumption that D has local terminal objects (i.e each hom has a “top”), then N D is a cartesian bicategory in the sense of Carboni, et al. In addition, we note that the “trace ” that exists on D via the fixed point theorem extends in a natural way to N D, thus pointing the way to the use of N D for defining the semantics of nondeterministic programming constructs. Keywords: cartesian bicategory, denotational semantics, domain, nondeterminism, trace

We study bialgebras in the compact closed category Rel of sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of

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