Results 1  10
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19
Algebraic foundations for effectdependent optimisations
 In POPL
, 2012
"... We present a general theory of Giffordstyle type and effect annotations, where effect annotations are sets of effects. Generality is achieved by recourse to the theory of algebraic effects, a development of Moggi’s monadic theory of computational effects that emphasises the operations causing the e ..."
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We present a general theory of Giffordstyle type and effect annotations, where effect annotations are sets of effects. Generality is achieved by recourse to the theory of algebraic effects, a development of Moggi’s monadic theory of computational effects that emphasises the operations causing the effects at hand and their equational theory. The key observation is that annotation effects can be identified with operation symbols. We develop an annotated version of Levy’s CallbyPushValue language with a kind of computations for every effect set; it can be thought of as a sequential, annotated intermediate language. We develop a range of validated optimisations (i.e., equivalences), generalising many existing ones and adding new ones. We classify these optimisations as structural, algebraic, or abstract: structural optimisations always hold; algebraic ones depend on the effect theory at hand; and abstract ones depend on the global nature of that theory (we give modularlycheckable sufficient conditions for their validity).
Coalgebraic Components in a ManySorted Microcosm
"... Abstract. The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a ..."
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Abstract. The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a manysorted setting. Then we can show that the coalgebraic component calculi of Barbosa are examples, with compositionality of behavior following from microcosm structure. The algebraic structure on these coalgebraic components corresponds to variants of Hughes’ notion of arrow, introduced to organize computations in functional programming. 1
Just do it: Simple monadic equational reasoning
 In Proceedings of the 16th International Conference on Functional Programming (ICFP’11
, 2011
"... One of the appeals of pure functional programming is that it is so amenable to equational reasoning. One of the problems of pure functional programming is that it rules out computational effects. Moggi and Wadler showed how to get round this problem by using monads to encapsulate the effects, leadin ..."
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One of the appeals of pure functional programming is that it is so amenable to equational reasoning. One of the problems of pure functional programming is that it rules out computational effects. Moggi and Wadler showed how to get round this problem by using monads to encapsulate the effects, leading in essence to a phase distinction—a pure functional evaluation yielding an impure imperative computation. Still, it has not been clear how to reconcile that phase distinction with the continuing appeal of functional programming; does the impure imperative part become inaccessible to equational reasoning? We think not; and to back that up, we present a simple axiomatic approach to reasoning about programs with computational effects.
Monads with arities and their associated theories
 J. of Pure and Applied Algebra
"... Abstract. After a review of the concept of “monad with arities ” we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere’s algebraic theories to a general correspondence between mona ..."
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Abstract. After a review of the concept of “monad with arities ” we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere’s algebraic theories to a general correspondence between monads and theories for a given category with arities. As application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids. Introduction. In his seminal work [20] Lawvere constructed for every variety of algebras, defined by finitary operations and relations on sets, an algebraic theory whose nary operations are the elements of the free algebra on n elements. He showed that the variety of algebras is equivalent to the category of models of the associated algebraic
Secondorder algebraic theories
, 2010
"... Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet: ..."
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Cited by 3 (2 self)
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Technical reports published by the University of Cambridge Computer Laboratory are freely available via the Internet:
Semantics for computational effects: from global to local
, 2006
"... We give a general construct that extends denotational semantics for a global computational effect to yield denotational semantics for a corresponding local computational effect. Our leading example yields a construction of the usual denotational semantics for local state from that for global state ..."
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We give a general construct that extends denotational semantics for a global computational effect to yield denotational semantics for a corresponding local computational effect. Our leading example yields a construction of the usual denotational semantics for local state from that for global state. Given any Lawvere theory L, possibly countable and possibly enriched, modelling a specific computational effect, we first give a universal construction that extends L, hence the global operations and equations of a given effect, to incorporate worlds of arbitrary finite size. Then, making delicate use of the final comodel of L, we give a construct that uniformly allows us to model block, the universality of the final comodel yielding a universal property of the construct. We illustrate both the universal extension of L and the canonical construction of block by seeing how they work primarily for state but also for nondeterminism, timing, exceptions, and interactive I/O. Key words: computational effects, Lawvere theory, indexed Lawvere theory, model, monad, global state, local state.
Monoidal indeterminates and categories of possible worlds
 In Proc. of MFPS XXV
, 2009
"... Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffin ..."
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Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffine category (symmetric monoidal category with initial unit) on a given small symmetric monoidal category. We then exhibit all the known categories of “possible worlds ” used to treat languages that allow for dynamic creation of “new ” variables, locations, or names as instances of this construction and explicate their associated universality properties. As an application of the resulting characterisation of O(W), Oles’s category of possible worlds, we present an O(W)indexed Lawvere theory of manysorted storage, generalizing the singlesorted one introduced by J. Power, and we describe explicitly an associated
BarQL: Collaborating Through Change
"... Applications such as Google Docs, Office 365, and Dropbox show a growing trend towards incorporating multiuser collaboration functionality into web applications. These collaborative applications share a need to efficiently express shared state, typically through a shared log abstraction. Extensive ..."
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Applications such as Google Docs, Office 365, and Dropbox show a growing trend towards incorporating multiuser collaboration functionality into web applications. These collaborative applications share a need to efficiently express shared state, typically through a shared log abstraction. Extensive research efforts on log abstractions by the database, programming languages, and distributed systems communities have identified a variety of analysis techniques based on the algebraic properties of updates (i.e., pairwise commutativity, subsumption, and idempotence). Although these techniques have been applied to specific application domains, to the best of our knowledge, no attempt has been made to create a general framework for such analyses in the context of a nontrivial update language. In this paper, we introduce monadic logs, a semantically rich state abstraction that provides a powerful, expressive framework for reasoning about a variety of application state properties. We also define BarQL, a general purpose stateupdate language, and show how the monadic log abstraction allows us to reason about the properties of updates expressed in BarQL. Finally, we show how such analyses can be expressed declaratively using the SPARQL graph query language. 1.
Investigations into Algebra and Topology over Nominal Sets
, 2011
"... The last decade has seen a surge of interest in nominal sets and their applications to formal methods for programming languages. This thesis studies two subjects: algebra and duality in the nominal setting. In the first part, we study universal algebra over nominal sets. At the heart of our approach ..."
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The last decade has seen a surge of interest in nominal sets and their applications to formal methods for programming languages. This thesis studies two subjects: algebra and duality in the nominal setting. In the first part, we study universal algebra over nominal sets. At the heart of our approach lies the existence of an adjunction of descent type between nominal sets and a category of manysorted sets. Hence nominal sets are a full reflective subcategory of a manysorted variety. This is presented in Chapter 2. Chapter 3 introduces functors over manysorted varieties that can be presented by operations and equations. These are precisely the functors that preserve sifted colimits. They play a central role in Chapter 4, which shows how one can systematically transfer results of universal algebra from a manysorted variety to nominal sets. However, the equational logic obtained is more expressive than the nominal equational logic of Clouston and Pitts, respectively, the nominal algebra of Gabbay and Mathijssen. A uniform fragment of our logic with the same expressivity
Adjunctions for exceptions
, 2012
"... Abstract. The exceptions form a computational effect, in the sense that there is an apparent mismatch between the syntax of exceptions and their intended semantics. We solve this apparent contradiction by defining a logic for exceptions with a proof system which is close to their syntax and where th ..."
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Abstract. The exceptions form a computational effect, in the sense that there is an apparent mismatch between the syntax of exceptions and their intended semantics. We solve this apparent contradiction by defining a logic for exceptions with a proof system which is close to their syntax and where their intended semantics can be seen as a model. This requires a robust framework for logics and their morphisms, which is provided by diagrammatic logics.