Generic models for computational effects

Abstract

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.

Axiomatics for Data Refinement in Call by Value Programming Languages

“…More specifically, one can take C = Set or the category of ω-cpo's, both of which are cartesian closed; and one can take a strong monad on them, such as a lifting monad or ones for modelling side-effects, exceptions, continuations, etcetera. More specifically again, the paper [27] shows how every countable Lawvere theory gives rise to a canonical premonoidal category, including all the examples just cited.…”

“…It is common to see a let constructor in descriptions of the λ c -calculus, with let x = e in e being syntactic sugar for (λx.e )e. It only plays a substantial role when one wants to consider a first-order fragment of the calculus [27], so, for simplicity, we omit it here.…”

Semantics for probabilistic programming

“…Second, the standard construction for building a Cartesian closed category out of a distributive one is based on functor categories (e.g. [30]). …”

Algebraic Effects, Linearity, and Quantum Programming Languages

“…By understanding quantum computation as an algebraic effect, we are able to begin applying other techniques developed for algebraic effects in general, such as compiler optimizations and static analyses [24] and normalization by evaluation [4]. Another general method is building models of higher order computation with effects [38] by using monads. Indeed, the equational theory of quantum computation is not a theory in the sense of classical universal algebra, but rather a theory enriched in the functor category [Bij, Set], which is why we used actions of Bij to discuss models.…”