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Generic Models for Computational Effects
"... A Freydcategory is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in callbyvalue programming languages, such as the computational λcalculus, with computational effects. We develop the theory of Freydcategories with that in min ..."
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A Freydcategory is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in callbyvalue programming languages, such as the computational λcalculus, with computational effects. We develop the theory of Freydcategories 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 Freydcategory. We then give canonical, universal embeddings of Freydcategories into closed Freydcategories, 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 λccalculus. Our leading examples of signatures arise from sideeffects, 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. Key words: Freydcategory, enriched Yoneda embedding, conical colimit completion, canonical model
Logic for Computational Effects: work in progress
"... Abstract We outline a possible logic that will allow us to give a unified approach to reasoning about computational effects. The logic is given by extending Moggi's computational *calculus by basic types and a signature, the latter given by constant symbols, function symbols, and operation sym ..."
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Abstract We outline a possible logic that will allow us to give a unified approach to reasoning about computational effects. The logic is given by extending Moggi's computational *calculus by basic types and a signature, the latter given by constant symbols, function symbols, and operation symbols, and by including a _ operator. We give both syntax and semantics for the logic except for _. We consider a number of sound and complete classes of models, all given in categorytheoretic terms. We illustrate the ideas with some of our leading examples of computational effects, and we observe that operations give rise to natural modalities.