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39
Parameterised notions of computation
 In MSFP 2006: Workshop on mathematically structured functional programming, ed. Conor McBride and Tarmo Uustalu. Electronic Workshops in Computing, British Computer Society
, 2006
"... Moggi’s Computational Monads and Power et al’s equivalent notion of Freyd category have captured a large range of computational effects present in programming languages such as exceptions, sideeffects, input/output and continuations. We present generalisations of both constructs, which we call para ..."
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Cited by 37 (3 self)
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Moggi’s Computational Monads and Power et al’s equivalent notion of Freyd category have captured a large range of computational effects present in programming languages such as exceptions, sideeffects, input/output and continuations. We present generalisations of both constructs, which we call parameterised monads and parameterised Freyd categories, that also capture computational effects with parameters. Examples of such are composable continuations, sideeffects where the type of the state varies and input/output where the range of inputs and outputs varies. By also considering monoidal parameterisation, we extend the range of effects to cover separated sideeffects and multiple independent streams of I/O. We also present two typed λcalculi that soundly and completely model our categorical definitions — with and without monoidal parameterisation — and act as prototypical languages with parameterised effects.
Naming proofs in classical propositional logic
 IN PAWE̷L URZYCZYN, EDITOR, TYPED LAMBDA CALCULI AND APPLICATIONS, TLCA 2005, VOLUME 3461 OF LECTURE
"... We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentiali ..."
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Cited by 24 (8 self)
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We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cutelimination procedure. This gives us a “Boolean ” category, which is not a poset. With the semiring of natural numbers, we obtain a sound semantics for classical logic, in which fewer proofs are identified. Though a “real” sequentialization theorem is missing, these proof nets have a grip on complexity issues. In both cases the cut elimination procedure is closely related to its equivalent in the calculus of structures.
From proof nets to the free * autonomous category
 Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05
, 2005
"... Vol. 2 (4:3) 2006, pp. 1–44 www.lmcsonline.org ..."
L.: Constructing free Boolean categories
, 2005
"... By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *autonomous category ..."
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Cited by 21 (6 self)
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By Boolean category we mean something which is to a Boolean algebra what a category is to a poset. We propose an axiomatic system for Boolean categories, which is different in several respects from the ones proposed recently. In particular everything is done from the start in a *autonomous category and not in a weakly distributive one, which simplifies issues like the Mix rule. An important axiom, which is introduced later, is a “graphical ” condition, which is closely related to denotational semantics and the Geometry of Interaction. Then we show that a previously
Monatron: an extensible monad transformer library. Available at http://www.cs.nott.ac.uk/~mjj/pubs/monatron.pdf
"... Abstract. Monads are pervasive in functional programming. In order to reap the benefits of their abstraction power, combinator libraries for monads are necessary. Monad transformers provide the basis for such libraries, and are based on a design that has proved to be successful. In this article, we ..."
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Cited by 12 (2 self)
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Abstract. Monads are pervasive in functional programming. In order to reap the benefits of their abstraction power, combinator libraries for monads are necessary. Monad transformers provide the basis for such libraries, and are based on a design that has proved to be successful. In this article, we show that this design has a number of shortcomings and provide a new design that builds on the strengths of the traditional design, but addresses its problems. 1
Higher gauge theory I: 2Bundles
 University of California Riverside
"... Stevenson for helpful discussion about covers. And of course I thank John Baez for all of the above, as well as inspiration, guidance, and encouragement. iv Abstract of the Dissertation ..."
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Cited by 11 (0 self)
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Stevenson for helpful discussion about covers. And of course I thank John Baez for all of the above, as well as inspiration, guidance, and encouragement. iv Abstract of the Dissertation
Discrete Quantum Causal Dynamics
 International Journal of Theoretical Physics
, 2003
"... We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolut ..."
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Cited by 10 (5 self)
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We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences  such as covariance  follow directly from the functoriality of our axioms. We establish strong links between the physical picture we propose and linear logic. Specifically we show that the rened logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic. This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 1
Higher gauge theory
"... I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to ce ..."
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Cited by 9 (0 self)
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I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2groups, and bundles with a suitable notion of 2bundle. To link this with previous work, I show that certain 2categories of principal 2bundles are equivalent to certain 2categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2category of 2bundles over a given 2space under a given 2group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2space is the 2space corresponding to a given space and the 2group is the automorphism 2group of a given group, then this 2category is equivalent to the 2category of gerbes over that space under that group (being described by the same cohomological data).
When is a function a fold or an unfold
 Coalgebraic Methods in Computer Science, number 44.1 in Electronic Notes in Theoretical Computer Science
, 2001
"... We give a necessary and sufficient condition for when a settheoretic function can be written using the recursion operator fold, and a dual condition for the recursion operator unfold. The conditions are simple, practically useful, and generic in the underlying datatype. 1 ..."
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Cited by 9 (3 self)
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We give a necessary and sufficient condition for when a settheoretic function can be written using the recursion operator fold, and a dual condition for the recursion operator unfold. The conditions are simple, practically useful, and generic in the underlying datatype. 1
Modular monad transformers
 In ESOP ’09: Proceedings of the 18th European Symposium on Programming Languages and Systems
, 2009
"... Abstract. During the last two decades, monads have become an indispensable tool for structuring functional programs with computational effects. In this setting, the mathematical notion of a monad is extended with operations that allow programmers to manipulate these effects. When several effects are ..."
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Cited by 8 (2 self)
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Abstract. During the last two decades, monads have become an indispensable tool for structuring functional programs with computational effects. In this setting, the mathematical notion of a monad is extended with operations that allow programmers to manipulate these effects. When several effects are involved, monad transformers can be used to build up the required monad one effect at a time. Although this seems to be modularity nirvana, there is a catch: in addition to the construction of a monad, the effectmanipulating operations need to be lifted to the resulting monad. The traditional approach for lifting operations is nonmodular and adhoc. We solve this problem with a principled technique for lifting operations that makes monad transformers truly modular. 1