Sketches

Theoretical Computer Science

2006

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156

228

We seek a unified account of modularity for computational effects. We begin by reformulating Moggi's monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggi's exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggi's side-effects monad transformer. Finally, we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.

Section: A = (T H * ) and : H (T H * )X → T H * X Is The Compositionmentioning

confidence: 99%

Combining effects: Sum and tensor

Hyland

Plotkin

Theoretical Computer Science

2006

Self Cite

156

228

“…The tensor product exists because it is determined by the free theory on an enriched sketch [18]. But it may equally, indeed more elegantly, be proved to exist by appeal to an enrichment of the delicate 2-categorical analysis of the form appearing in [9], from which the following result also follows:…”

Section: Definition 6 [11] Given Discrete Countable Lawvere V -Theormentioning

confidence: 99%

“…So, in particular, if L is freely generated by some sort of signature and equations, equivalently by some generalised notion of sketch (see [18] for a general definition and treatment of the sketch idea), it simply amounts to giving a model of each constructor of the generalised signature subject to the generalised equations, exactly as for Lawvere V -theories as in Section 2 but with arities restricted to be objects of ℵ 1 rather than arbitrary countably presentable objects of V . One can construct a forgetful functor from the category Law V of countable Lawvere V -theories to DLaw V as follows.…”

Section: Is a (Necessarily Strict Countable-product Preserving) V -Fumentioning

confidence: 99%

Discrete Lawvere Theories

Algebra and Coalgebra in Computer Science

2005

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Abstract. We introduce the notion of discrete countable Lawvere Vtheory and study constructions that may be made on it. The notion of discrete countable Lawvere V -theory extends that of ordinary countable Lawvere theory by allowing the homsets of an ordinary countable Lawvere theory to become homobjects of a well-behaved axiomatically defined category such as that of ω-cpo's. Every discrete countable Lawvere V -theory induces a V -enriched monad, equivalently a strong monad, on V . We show that discrete countable Lawvere V -theories allow us to model all the leading examples of computational effects other than continuations, and that they are closed under constructions of sum, tensor and distributive tensor, which are the fundamental ways in which one combines such effects. We also show that discrete countable Lawvere V -theories are closed under taking an image, allowing one to treat observation as a mathematical primitive in modelling effects.

“…Then T h(S) would be the free closed F reyd-category determined by the base types and terms, hence the free model for the λ c -calculus with those types and terms. See [14] for examples of sketches for monads on Cat and their use for modelling signatures in call by name programming languages, and see [18] for more detail of this idea.…”

Section: Definition 53 a T -Sketch S Consists Of A Small [→ Set]-camentioning

confidence: 99%

“…We do not address representation independence, the topic of [14], in this paper, but the techniques of [14], based on the T -sketches in [18], extend to the setting of this paper. We plan to make that extension, but it is not entirely clear how to do so yet.…”

Section: Introductionmentioning

confidence: 99%

Axiomatics for Data Refinement in Call by Value Programming Languages

Electronic Notes in Theoretical Computer Science

Tanaka

2009

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We give a systematic category theoretic axiomatics for modelling data refinement in call by value programming languages. Our leading examples of call by value languages are extensions of the computational λ-calculus, such as F P C and languages for modelling nondeterminism, and extensions of the first order fragment of the computational λ-calculus, such as a CP S language. We give a category theoretic account of the basic setting, then show how to model contexts, then arbitrary type and term constructors, then signatures, and finally data refinement. This extends and clarifies Kinoshita and Power's work on lax logical relations for call by value languages.