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Towards a quantum programming language
 Mathematical Structures in Computer Science
, 2004
"... The field of quantum computation suffers from a lack of syntax. In the absence of a convenient programming language, algorithms are frequently expressed in terms of hardware circuits or Turing machines. Neither approach particularly encourages structured programming or abstractions such as data type ..."
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The field of quantum computation suffers from a lack of syntax. In the absence of a convenient programming language, algorithms are frequently expressed in terms of hardware circuits or Turing machines. Neither approach particularly encourages structured programming or abstractions such as data types. In this paper, we describe the syntax and semantics of a simple quantum programming language. This language provides highlevel features such as loops, recursive procedures, and structured data types. It is statically typed, and it has an interesting denotational semantics in terms of complete partial orders of superoperators. 1
A Relational Model of NonDeterministic Dataflow
 In CONCUR'98, volume 1466 of LNCS
, 1998
"... . We recast dataflow in a modern categorical light using profunctors as a generalisation 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 ..."
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Cited by 27 (13 self)
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. We recast dataflow in a modern categorical light using profunctors as a generalisation 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. 1 Introduction A fundament...
Physical traces: Quantum vs. classical information processing
 In Proceedings of Category Theory and Computer Science 2002 (CTCS’02), volume 69 of Electronic Notes in Theoretical Computer Science. Elsevier Science
, 2003
"... a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation/realization of the traced monoidal categories of finitedimensional vector spaces with tensor product as monoidal structure and of finite sets an ..."
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Cited by 19 (5 self)
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a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpretation/realization of the traced monoidal categories of finitedimensional vector spaces with tensor product as monoidal structure and of finite sets and relations with Cartesian product as monoidal structure, both of them providing a socalled wavestyle GoI. The developments in this paper reveal that envisioning state update due to quantum measurement as a process provides a powerful tool for developing highlevel approaches to quantum information processing.
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
Event structures with persistence
, 2008
"... 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 computation ..."
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Cited by 1 (0 self)
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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 nondeterministic 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
Categories of Circuits
, 1999
"... Traced monoidal categories of circuits are introduced in order to give a compositional account of circuits. Behaviour functors of circuits are dened, and compositionality results are obtained in terms of the structurepreserving properties of these functors. The theory includes models of the asynchr ..."
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Traced monoidal categories of circuits are introduced in order to give a compositional account of circuits. Behaviour functors of circuits are dened, and compositionality results are obtained in terms of the structurepreserving properties of these functors. The theory includes models of the asynchronous binary circuits constructed from instantaneous functional circuits and upbounded inertial binary delays, as well as asynchronous pulse circuits such as pipelines. Via the notion of a circuit network, an explicit connection is made with the General MultipleWinner circuit model.
*autonomous categories, Unique decomposition categories.
"... We analyze the categorical foundations of Girard’s Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of ..."
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We analyze the categorical foundations of Girard’s Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of Abramsky’s GoI situations–ones based on Unique Decomposition Categories (UDC’s)–exactly captures Girard’s functional analytic models in his first GoI paper, including Girard’s original Execution formula in Hilbert spaces, his notions of orthogonality, types, datum, algorithm, etc. Here we associate to a UDCbased GoI Situation a denotational model (a ∗autonomous category (without units) with additional exponential structure). We then relate this model to some of the standard GoI models via a fullyfaithful embedding into a doublegluing category, thus connecting up GoI with earlier Full Completeness
A Categorical Model for the Geometry of Interaction Abstract
"... We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard’s original approach to GoI 1 via ..."
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We consider the multiplicative and exponential fragment of linear logic (MELL) and give a Geometry of Interaction (GoI) semantics for it based on unique decomposition categories. We prove a Soundness and Finiteness Theorem for this interpretation. We show that Girard’s original approach to GoI 1 via operator algebras is exactly captured in this categorical framework.
Some Algebraic Properties of (Co)Spans
, 2000
"... The paper investigates the algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions. We analyze the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets, show ..."
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The paper investigates the algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions. We analyze the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets, showing that they characterize dierent selfdual algebraic structures. Furthermore, our results shed some light on their relationship with the categories of (multi)relations and of equivalence relations. Introduction Recent years have seen a large diusion of equational presentations for dierent graphlike structures. The motivations underlying these works are quite varied, and dicult to recast in a unitary thread, even if they could be roughly divided into two halves, albeit tightly intertwined. On the one hand, there have been the studies regarding the syntax for diagrammatic presentations of, e.g., nets and circuits. The initial works can be considered those introducing the ownomial c...