Recursive Coalgebras from Comonads

Folds over inductive datatypes are well understood and widely used. In their plain form, they are quite restricted; but many dis-parate generalisations have been proposed that enjoy similar cal-culational benefits. There have also been attempts to unify the various generalisations: two prominent such unifications are the 're-cursion schemes from comonads' of Uustalu, Vene and Pardo, and our own 'adjoint folds'. Until now, these two unified schemes have appeared incompatible. We show that this appearance is illusory: in fact, adjoint folds subsume recursion schemes from comonads. The proof of this claim involves standard constructions in category theory that are nevertheless not well known in functional programming: Eilenberg-Moore categories and bialgebras.

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“…Another interesting direction is to explore the use of recursive coalgebras [5] (or corecursive algebras) in λ -bialgebras.…”

Section: Resultsmentioning

confidence: 99%

Unifying structured recursion schemes

Hinze

Gibbons

2013

Proceedings of the 18th ACM SIGPLAN International Conference on Functional Programming

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“…(An F λ -algebra is also known as a λ -bialgebra; F λ -coalgebras have been called λ -dicoalgebras [6].) If (C, c) is a recursive F-coalgebra, we know that (3.11) has a unique solution in C .…”

Section: Proof We Have To Show That the Equation Inmentioning

confidence: 99%

“…With these definitions in place, we proceed by describing two fundamental techniques for establishing uniqueness: Eppendahl's Basic Lemma [7], which we call the rolling rule, and a symmetric version of Proposition 12 of [6], which we call the conjugate rule. Indeed, we will be careful to emphasize symmetry, even though our main focus is on recursive coalgebras.…”

Section: Example 32mentioning

confidence: 99%

“…In a sense the ensuing recursion scheme, conjugate hylomorphisms, can be considered the mother of all recursion schemes, as almost every recursion scheme can be framed as an instance: folds and unfolds [10,21,23], folds with parameters [26], mutu-and zygomorphisms [8,20], histo-and futumorphisms [29], generalized folds [5], recursion schemes from comonads [31], and adjoint folds [11]. This is not the first time this topic has been visited, and indeed most of the results in Section 3 appear in Capretta et al [6], albeit using different machinery-comonads and distributive laws-and with substantially longer proofs. We demonstrate that each of their comonadic schemes also arises by instantiating the conjugate rule to an adjunction that generates the comonad.…”

Section: Introductionmentioning

confidence: 95%

“…In particular, we can restrict ourselves to so-called recursive coalgebras [6]-those that by definition yield unique solutions to the hylo equation, whatever the algebra. Dually, we can work with corecursive algebrasthose that yield unique solutions whatever the coalgebra.…”

Section: Introductionmentioning

confidence: 99%

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Conjugate Hylomorphisms -- Or

Hinze

Gibbons

2015

Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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The past decades have witnessed an extensive study of structured recursion schemes. A general scheme is the hylomorphism, which captures the essence of divide-and-conquer: a problem is broken into sub-problems by a coalgebra; sub-problems are solved recursively; the sub-solutions are combined by an algebra to form a solution. In this paper we develop a simple toolbox for assembling recursive coalgebras, which by definition ensure that their hylo equations have unique solutions, whatever the algebra. Our main tool is the conjugate rule, a generic rule parametrized by an adjunction and a conjugate pair of natural transformations. We show that many basic adjunctions induce useful recursion schemes. In fact, almost every structured recursion scheme seems to arise as an instance of the conjugate rule. Further, we adapt our toolbox to the more expressive setting of parametrically recursive coalgebras, where the original input is also passed to the algebra. The formal development is complemented by a series of worked-out examples in Haskell.

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On Well-Founded and Recursive Coalgebras

Adámek

Milius

Moss

2020

Lecture Notes in Computer Science

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This paper studies fundamental questions concerning categorytheoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving endofunctors on complete and well-powered categories every coalgebra has a well-founded part, and we provide a new, shorter proof that this is the coreflection in the category of all well-founded coalgebras. We present a new more general proof of Taylor's General Recursion Theorem that every well-founded coalgebra is recursive, and we study under which hypothesis the converse holds. In addition, we present a new equivalent characterization of well-foundedness: a coalgebra is well-founded iff it admits a coalgebra-to-algebra morphism to the initial algebra.

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Recursive Coalgebras from Comonads

Cited by 9 publications

References 29 publications

Unifying structured recursion schemes

Unifying structured recursion schemes

Conjugate Hylomorphisms -- Or

On Well-Founded and Recursive Coalgebras

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