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31
Relational Semantics of NonDeterministic Dataflow
, 1997
"... We recast dataflow in a modern categorical light using profunctors as a generalization of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fit ..."
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Cited by 12 (5 self)
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We recast dataflow in a modern categorical light using profunctors as a generalization of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors, (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps and (3) a treatment of higherorder dataflow as a biproduct, essentially by following the geometry of interaction programme.
Profunctors, open maps and bisimulation
 Mathematical Structures in Computer Science, To appear. Available from the Glynn Winskel’s web
, 2000
"... ..."
A Linear Metalanguage for Concurrency
 Handbook of Logic in Computer Science
, 1998
"... A metalanguage for concurrent process languages is introduced. ..."
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Cited by 11 (4 self)
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A metalanguage for concurrent process languages is introduced.
Presheaf Models for CCSlike Languages
 THEORETICAL COMPUTER SCIENCE
, 1999
"... The aim of this paper is to harness the mathematical machinery around presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel proposed a general definition of bisimulation from open maps. Here we show that openmap bisimulations within a range of presheaf models are congruences ..."
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Cited by 8 (2 self)
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The aim of this paper is to harness the mathematical machinery around presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel proposed a general definition of bisimulation from open maps. Here we show that openmap bisimulations within a range of presheaf models are congruences for a general process language, in which CCS and related languages are easily encoded. The results are then transferred to traditional models for processes. By first establishing the congruence results for presheaf models, abstract, general proofs of congruence properties can be provided and the awkwardness caused through traditional models not always possessing the cartesian liftings, used in the breakdown of process operations, are sidestepped. The abstract results are applied to show that hereditary historypreserving bisimulation is a congruence for CCSlike languages to which is added a refinement operator on event structures as proposed by van Glabbeek and Goltz.
Unique Factorisation Lifting Functors and Categories of LinearlyControlled Processes
 Mathematical Structures in Computer Science
, 1999
"... We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearlycontrolled ones. To this end, we introduce ..."
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Cited by 7 (2 self)
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We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearlycontrolled ones. To this end, we introduce and study a notion of pathlinearisable category in which any two paths of morphisms with equal composites can be linearised (or interleaved) in a canonical fashion. Our main result is that categories of linearlycontrolled processes (viz., processes controlled by pathlinearisable categories) are sheaf models. Introduction This work is an investigation into the mathematical structure of processes. The processes to be considered embody a notion of state space varying according to a control. This we formalise as a category of states (and their interrelations) Xequipped with a control functor X C f . There are different ways in which the control category C may be required to control t...
Categorical Models for Concurrency: Independence, Fairness and Dataflow
 BRICS DISSERTATION SERIES DS001
, 2000
"... 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 t ..."
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Cited by 6 (4 self)
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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 different 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. The work
Categories in Concurrency
 Semantics and Logics of Computation, Publications of the Newton Institute
, 1997
"... These notes survey a range of models for parallel computation, including interleaving models like transition systems, synchronisation trees and languages (often called Hoare traces in this context), and models like asynchronous transition systems, event structures, pomsets and Mazurkiewicz traces wh ..."
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Cited by 5 (0 self)
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These notes survey a range of models for parallel computation, including interleaving models like transition systems, synchronisation trees and languages (often called Hoare traces in this context), and models like asynchronous transition systems, event structures, pomsets and Mazurkiewicz traces where concurrency is represented more explicitly by a form of causal independence. The presentation is unified by casting the models in a categorytheoretic framework. One aim is to use category theory to provide abstract characterisations of constructions like parallel composition valid throughout a range of different models and to provide formal means for translating between different models. The framework helps in extending the useful concept of bisimulation equivalence from its familiar situation on transition systems, to independence models where without a careful analysis even the appropriate definition of bisimulation is not clearcut. 1 Contents 1 Introduction 3 2 Transition systems 5...
Weak Bisimulation and Open Maps (Extended Abstract)
, 1999
"... A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles , we are able to account for weak morphisms (rough ..."
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Cited by 4 (4 self)
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A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles , we are able to account for weak morphisms (roughly only required to preserve observable paths) and to derive a saturation monad (on the category of presheaves over the category of paths). Weak morphisms may be encoded as strong ones via the Kleisli construction associated to the saturation monad. A general
Event Structures as Presheaves  Two Representation Theorems
 In Proc. CONCUR'99, LNCS
, 1999
"... The category of event structures is known to embed fully and faithfully in the category of presheaves over pomsets. Here a characterisation of the presheaves represented by event structures is presented. The proof goes via a characterisation of the presheaves represented by event structures when the ..."
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Cited by 4 (2 self)
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The category of event structures is known to embed fully and faithfully in the category of presheaves over pomsets. Here a characterisation of the presheaves represented by event structures is presented. The proof goes via a characterisation of the presheaves represented by event structures when the morphisms on event structures are "strict" in that they preserve the partial order of causal dependency. 1
The SheafTheoretic Structure Of NonLocality and Contextuality
, 2011
"... Locality and noncontextuality are intuitively appealing features of classical physics, which are contradicted by quantum mechanics. The goal of the classic nogo theorems by Bell, KochenSpecker, et al. is to show that nonlocality and contextuality are necessary features of any theory whose predic ..."
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Cited by 4 (3 self)
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Locality and noncontextuality are intuitively appealing features of classical physics, which are contradicted by quantum mechanics. The goal of the classic nogo theorems by Bell, KochenSpecker, et al. is to show that nonlocality and contextuality are necessary features of any theory whose predictions agree with those of quantum mechanics. We use the mathematics of sheaf theory to analyze the structure of nonlocality and contextuality in a very general setting. Starting from a simple experimental scenario, and the kind of probabilistic models familiar from discussions of Bell’s theorem, we show that there is a very direct, compelling formalization of these notions in sheaftheoretic terms. Moreover, on the basis of this formulation, we show that the phenomena of nonlocality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. We give linear algebraic methods for computing these obstructions, and use these methods to obtain a number of new insights into nonlocality and contextuality. For example, we distinguish a proper hierarchy of strengths of nogo theorems, and show that three leading examples — due to Bell, Hardy, and Greenberger, Horne and Zeilinger, respectively — occupy successively higher levels of this hierarchy. We show how our abstract setting can be represented in quantum mechanics. In doing so, we uncover a strengthening of the usual nosignalling theorem, which shows that quantum mechanics obeys nosignalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.