Fibrational Induction Meets Effects

Atkey, Robert; Ghani, Neil; Jacobs, Bart; Johann, Patricia

doi:10.1007/978-3-642-28729-9_3

Cited by 9 publications

(17 citation statements)

References 13 publications

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“…Different programming patterns that involve resumptions were discussed by Claessen [14], Kiselyov [28], Harrison [23], and the present authors [38]. Interleaving of data and effects in the algebraic context was also studied by Filinski and Støvring [17], and Atkey et al [10].…”

Section: Related Workmentioning

confidence: 87%

The Coinductive Resumption Monad

Piróg

Gibbons

2014

Electronic Notes in Theoretical Computer Science

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Resumptions appear in many forms as a convenient abstraction, such as in semantics of concurrency and as a programming pattern. In this paper we introduce generalised resumptions in a category-theoretic, coalgebraic context and show their basic properties: they form a monad, they come equipped with a corecursion scheme in the sense of Adámek et al.'s notion of completely iterative monads (cims), and they enjoy a certain universal property, which specialises to the coproduct with a free cim in the category of cims.

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Section: Related Workmentioning

confidence: 87%

The Coinductive Resumption Monad

Piróg

Gibbons

2014

Electronic Notes in Theoretical Computer Science

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“…A further line of future work lies in deeper investigation of the categorical properties of the category of f -and-m-algebras. In the present work, we constructed the category of f -and-m-algebras and showed that it had initial objects from first principles, while in Atkey et al's previous work (Atkey et al, 2012), this was demonstrated by constructing an adjunction between the category of f -and-m-algebras and the category of ( f • m)-algebras. An anonymous reviewer has pointed out the interesting property that the category of f -and-m-algebras is isomorphic to the category of ((FreeM f ) + m)-EilenbergMoore algebras, showing that the monad coproduct construction in Section 8 has a deeper significance.…”

Section: Future Workmentioning

confidence: 58%

“…The concept of initial f -and-m-algebras is originally due to Filinski and Støvring in the specific setting of Cpo (the category of complete partial orders and continuous functions) (Filinski & Støvring, 2007), and was subsequently extended to a general category-theoretic setting for arbitrary functors f by Atkey, Ghani, Jacobs, and Johann (Atkey et al, 2012). In this article, we aim to introduce the concept of initial f -and-m-algebras to a general functional programming audience and show how they can be used to structure and reason about functional programs in practice, without the heavy category-theoretic prerequisites of Atkey et al's work.…”

mentioning

confidence: 99%

“…To solve the problems we have identified with the direct use of f -algebras, we use the concept of f -and-m-algebras, originally introduced by Filinski and Støvring (Filinski & Støvring, 2007), and generalised to arbitrary functors by Atkey, Ghani, Jacobs and Johann (Atkey et al, 2012). As the name may imply, f -and-m-algebras are simultaneously falgebras and m-algebras.…”

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confidence: 99%

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Interleaving data and effects

Atkey

Johann

2015

J. Funct. Prog.

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The study of programming with and reasoning about inductive datatypes such as lists and trees has benefited from the simple categorical principle of initial algebras. In initial algebra semantics, each inductive datatype is represented by an initial f -algebra for an appropriate functor f . The initial algebra principle then supports the straightforward derivation of definitional principles and proof principles for these datatypes. This technique has been expanded to a whole methodology of structured functional programming, often called origami programming.In this article we show how to extend initial algebra semantics from pure inductive datatypes to inductive datatypes interleaved with computational effects. Inductive datatypes interleaved with effects arise naturally in many computational settings. For example, incrementally reading characters from a file generates a list of characters interleaved with input/output actions, and lazily constructed infinite values can be represented by pure data interleaved with the possibility of non-terminating computation. Straightforward application of initial algebra techniques to effectful datatypes leads either to unsound conclusions if we ignore the possibility of effects, or to unnecessarily complicated reasoning because the pure and effectful concerns must be considered simultaneously. We show how pure and effectful concerns can be separated using the abstraction of initial f -and-m-algebras, where the functor f describes the pure part of a datatype and the monad m describes the interleaved effects. Because initial f -and-m-algebras are the analogue for the effectful setting of initial f -algebras, they support the extension of the standard definitional and proof principles to the effectful setting.Initial f -and-m-algebras are originally due to Filinski and Støvring, who studied them in the category Cpo. They were subsequently generalised to arbitrary categories by Atkey, Ghani, Jacobs, and Johann in a FoSSaCS 2012 paper. In this article we aim to introduce the general concept of initial f -and-m-algebras to a general functional programming audience.

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“…(Fibration) 1 For any I ∈ C, Y ∈ E and f : I → pY, there exists X ∈ E above I such that f : X→Y and the following property holds: for any Z ∈ E and g : pZ…”

Section: Definitionmentioning

confidence: 99%

Relating Computational Effects by ⊤ ⊤-Lifting

Katsumata

2011

Automata, Languages and Programming

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We consider the problem of establishing a relationship between two interpretations of base type terms of a λ c -calculus extended with algebraic operations. We show that the given relationship holds if it satisfies a set of natural conditions. We apply this result to 1) comparing two monadic semantics related by a strong monad morphism, and 2) comparing two monadic semantics of fresh name creation: Stark's new name creation monad [32], and the global counter monad. We also consider the same problem, relating semantics of computational effects, in the presence of recursive functions. We apply this additional by extending the previous monad morphism comparison result to the recursive case.

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Fibrational Induction Meets Effects

Cited by 9 publications

References 13 publications

The Coinductive Resumption Monad

The Coinductive Resumption Monad

Interleaving data and effects

Relating Computational Effects by ⊤ ⊤-Lifting

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