A Universal Embedding for the Higher Order Structure of Computational Effects

Power, John

doi:10.1007/3-540-44904-3_21

Cited by 4 publications

(12 citation statements)

References 17 publications

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“…Finally, combining Sections 3 with 5, we can now say exactly how the various results relate to last year's paper [20]. In studying computational effects, one invariably has a signature of operations one wishes to model.…”

Section: Introductionmentioning

confidence: 95%

“…In this section, we briefly recall the notions of F reyd-category and closed F reydcategory as used in [20] to model the first-order fragment of the λ c -calculus and the whole calculus respectively. The λ c -calculus is a fragment of a call-by-value programming language such as ML or the idealised language F P C (see for instance [3]).…”

Section: Models For the λ C -Calculusmentioning

confidence: 99%

“…Last year, at the Typed Lambda Calculus and Applications conference, I proved that every category theoretic model of what I defined to be the first-order fragment of Moggi's computational λ-calculus can be canonically embedded into a model of the whole calculus, the embedding satisfying an elegant and natural universal property [20]. After my talk, Carolyn Talcott asked what I regarded, and continue to regard, as an excellent question.…”

Section: Introductionmentioning

confidence: 98%

“…Our starting point has typically been Eugenio Moggi's computational λ-calculus or λ c -calculus, which was introduced in [10,11], with four distinct sound and complete classes of category theoretic models explained in [18], and with further abstract semantic development in [8,19,20,21]. The ideas surrounding the calculus have been applied extensively by the functional programming community, albeit typically using the computational metalanguage rather than the λ c -calculus: a recent overview appears as [1].…”

Section: Introductionmentioning

confidence: 99%

“…That model satisfies a natural universal property detailed in Section 5. The main result of [20] shows how to extend that model to a model of the whole λ c -calculus (and it can be adapted to preserve the semantics of the sum types and the type of natural numbers), giving a universal property that exhibits its definitiveness. A mild systematic variant of the construction of [20] allows one to recover the standard models on Set listed by Moggi [10,11].…”

Section: Introductionmentioning

confidence: 99%

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Canonical Models for Computational Effects

Power

2004

Lecture Notes in Computer Science

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Abstract. Given a signature of basic operations for a computational effect such as side-effects, interactive input/output, or exceptions, we give a unified construction that determines equations that should hold between derived operations of the same arity. We then show how to construct a canonical model for the signature, together with the firstorder fragment of the computational λ-calculus, subject to the equations, done at the level of generality of an arbitrary computational effect. We prove a universality theorem that characterises the canonical model, and we recall, from a previous paper, how to extend such models to the full computational λ-calculus. Our leading example is that of side-effects, with occasional reference to interactive input/output, exceptions, and nondeterminism.

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Section: Introductionmentioning

confidence: 95%

Section: Models For the λ C -Calculusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Canonical Models for Computational Effects

Power

2004

Lecture Notes in Computer Science

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Generic models for computational effects

Power

2006

Theoretical Computer Science

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A Freyd-category is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in call-by-value programming languages, such as the computational -calculus, with computational effects. We develop the theory of Freyd-categories with that in mind. We first show that any countable Lawvere theory, hence any signature of operations with countable arity subject to equations, directly generates a Freyd-category. We then give canonical, universal embeddings of Freyd-categories into closed Freyd-categories, characterised by being free cocompletions. The combination of the two constructions sends a signature of operations and equations to the Kleisli category for the monad on the category Set generated by it, thus refining the analysis of computational effects given by monads. That in turn allows a more structural analysis of the c -calculus. Our leading examples of signatures arise from side-effects, interactive input/output and exceptions. We extend our analysis to an enriched setting in order to account for recursion and for computational effects and signatures that inherently involve it, such as partiality, nondeterminism and probabilistic nondeterminism.

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