A representation theorem for second-order functionals

Jaskelioff, Mauro; O'Connor, Russell

doi:10.1017/s0956796815000088

Cited by 16 publications

(20 citation statements)

References 21 publications

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“…Moreover, they only consider lenses, not prisms and other optics. Jaskelioff and O'Connor [2015] do make thorough use of the equational consequences of Yoneda for reasoning about a class of second-order functions, including some optics under the alternative van Laarhoven representation [van Laarhoven 2009], but they do not address the more convenient profunctor representation.…”

Section: Related Workmentioning

confidence: 99%

What you needa know about Yoneda: profunctor optics and the Yoneda lemma (functional pearl)

Boisseau

Gibbons

2018

Proc. ACM Program. Lang.

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Profunctor optics are a neat and composable representation of bidirectional data accessors, including lenses, and their dual, prisms. The profunctor representation exploits higher-order functions and higher-kinded type constructor classes, but the relationship between this and the familiar representation in terms of 'getter' and 'setter' functions is not at all obvious. We derive the profunctor representation from the concrete representation, making the relationship clear. It turns out to be a fairly direct application of the Yoneda Lemma, arguably the most important result in category theory. We hope this derivation aids understanding of the profunctor representation. Conversely, it might also serve to provide some insight into the Yoneda Lemma.

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Section: Related Workmentioning

confidence: 99%

What you needa know about Yoneda: profunctor optics and the Yoneda lemma (functional pearl)

Boisseau

Gibbons

2018

Proc. ACM Program. Lang.

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“…We do not provide any additional justification for this representation as it has been extensively studied [Bird et al 2013;Jaskelioff and O'Connor 2015;O'Connor 2011]. In any case, the choice is not crucial to our work.…”

Section: Lenses and Traversalsmentioning

confidence: 99%

Generic deriving of generic traversals

Kiss

Pickering

2018

Proc. ACM Program. Lang.

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Functional programmers have an established tradition of using traversals as a design pattern to work with recursive data structures. The technique is so prolific that a whole host of libraries have been designed to help in the task of automatically providing traversals by analysing the generic structure of data types. More recently, lenses have entered the functional scene and have proved themselves to be a simple and versatile mechanism for working with product types. They make it easy to focus on the salient parts of a data structure in a composable and reusable manner.In this paper, we use the combination of lenses and traversals to give rise to an expressive and flexible library for querying and modifying complex data structures. Furthermore, since our lenses and traversals are based on the generic shape of data, we are able to use this information to produce code that is as efficient as hand-written versions. The technique leverages the structure of data to produce generic abstractions that are then eliminated by the standard workhorses of modern functional compilers: inlining and specialisation.

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“…Indeed Jaskelioff and O'Connor [18] show that there is a oneto-one correspondence between dialogue trees in Dialogue A,B and monadic functions of type ∀M, M A → M B.…”

Section: Dialogue Treesmentioning

confidence: 99%

“…Three recent articles ( [3], [18], [10]) studied the closely related topic of pure functions on a type of streams encoded by the type N → M A, its relation to dialogue trees and its use to prove the continuity of functional programs.…”

Section: Related and Future Workmentioning

confidence: 99%

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The continuity of monadic stream functions

Capretta

Fowler

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Abstract-Brouwer's continuity principle states that all functions from infinite sequences of naturals to naturals are continuous, that is, for every sequence the result depends only on a finite initial segment. It is an intuitionistic axiom that is incompatible with classical mathematics. Recently Martín Escardó proved that it is also inconsistent in type theory. We propose a reformulation of the continuity principle that may be more faithful to the original meaning by Brouwer. It applies to monadic streams, potentially unending sequences of values produced by steps triggered by a monadic action, possibly involving side effects. We consider functions on them that are uniform, in the sense that they operate in the same way independently of the particular monad that provides the specific side effects. Formally this is done by requiring a form of naturality in the monad. Functions on monadic streams have not only a foundational importance, but have also practical applications in signal processing and reactive programming. We give algorithms to determine the modulus of continuity of monadic stream functions and to generate dialogue trees for them (trees whose nodes and branches describe the interaction of the process with the environment).

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A representation theorem for second-order functionals

Cited by 16 publications

References 21 publications

What you needa know about Yoneda: profunctor optics and the Yoneda lemma (functional pearl)

What you needa know about Yoneda: profunctor optics and the Yoneda lemma (functional pearl)

Generic deriving of generic traversals

The continuity of monadic stream functions

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