Recursive coalgebras of finitary functors

Abstract. We reformulate the bar recursion and induction principles in terms of recursive and wellfounded coalgebras. Bar induction was originally proposed by Brouwer as an axiom to recover certain classically valid theorems in a constructive setting. It is a form of induction on nonwellfounded trees satisfying certain properties. Bar recursion, introduced later by Spector, is the corresponding function definition principle. We give a generalization of these principles, by introducing the notion of barred coalgebra: a process with a branching behaviour given by a functor, such that all possible computations terminate. Coalgebraic bar recursion is the statement that every barred coalgebra is recursive; a recursive coalgebra is one that allows definition of functions by a coalgebra-to-algebra morphism. It is a framework to characterize valid forms of recursion for terminating functional programs. One application of the principle is the tabulation of continuous functions: Ghani, Hancock and Pattinson defined a type of wellfounded trees that represent continuous functions on streams. Bar recursion allows us to prove that every stably continuous function can be tabulated to such a tree where by stability we mean that the modulus of continuity is also continuous. Coalgebraic bar induction states that every barred coalgebra is wellfounded; a wellfounded coalgebra is one that admits proof by induction.

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“…(This was observed by Adámek, Lücke and Milius [2], who also called this condition the halting property. )…”

Section: Definitionmentioning

confidence: 65%

A Coalgebraic View of Bar Recursion and Bar Induction

Capretta

Uustalu

2016

Lecture Notes in Computer Science

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“…A hylomorphism encapsulates a fold after an unfold, combining a coalgebra A → F A and an algebra F B → B. The algebra and coalgebra have different carriers (A and B), but share the functor F. Their use has been explored in a wide variety of settings (Adámek et al 2007;Capretta et al 2006;Meijer et al 1991). However, we do not discuss hylomorphisms in this paper, instead using bialgebras, which combine an algebra F X → X and a coalgebra X → G X: they share the same carrier, but operate on different functors.…”

Section: Related Workmentioning

confidence: 99%

Sorting with bialgebras and distributive laws

Hinze

James

Harper

et al. 2012

Proceedings of the 8th ACM SIGPLAN Workshop on Generic Programming

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Sorting algorithms are an intrinsic part of functional programming folklore as they exemplify algorithm design using folds and unfolds. This has given rise to an informal notion of duality among sorting algorithms: insertion sorts are dual to selection sorts. Using bialgebras and distributive laws, we formalise this notion within a categorical setting. We use types as a guiding force in exposing the recursive structure of bubble, insertion, selection, quick, tree, and heap sorts. Moreover, we show how to distill the computational essence of these algorithms down to one-step operations that are expressed as natural transformations. From this vantage point, the duality is clear, and one side of the algorithmic coin will neatly lead us to the other "for free". As an optimisation, the approach is also extended to paramorphisms and apomorphisms, which allow for more efficient implementations of these algorithms than the corresponding folds and unfolds.

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“…xs = smerge (fst (α a), xs 0 , xs 1 ) ∧ fst (snd (α a)) ↓ ∞ xs 0 ∧ snd (snd (α a)) ↓ ∞ xs 1 It turns out that ∀a : A. Dom ∞ a no matter what A, α are. So in this case we do have a unique solution f for any A, α, i.e., (Str, smerge) is corecursive.…”

Section: General Case (3): Coinductive Bisimilarity Relationmentioning

confidence: 99%

“…For FX = 1 + El × X × X, A = List, α = qsplit, B = List, β = concat, the relation ↓ is defined inductively by nil ↓ nil x : El xs : List xs| ≤x ↓ ys 0 xs| >x ↓ ys 1 cons (x, xs) ↓ app (ys 0 , cons(x, ys 1 ))…”

Section: P (Cons (X Xs )) P Xsmentioning

confidence: 99%

Structured general corecursion and coinductive graphs [extended abstract]

Uustalu¹

2012

Electron. Proc. Theor. Comput. Sci.

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Bove and Capretta's popular method for justifying function definitions by general recursive equations is based on the observation that any structured general recursion equation defines an inductive subset of the intended domain (the "domain of definedness") for which the equation has a unique solution. To accept the definition, it is hence enough to prove that this subset contains the whole intended domain. This approach works very well for "terminating" definitions. But it fails to account for "productive" definitions, such as typical definitions of stream-valued functions. We argue that such definitions can be treated in a similar spirit, proceeding from a different unique solvability criterion. Any structured recursive equation defines a coinductive relation between the intended domain and intended codomain (the "coinductive graph"). This relation in turn determines a subset of the intended domain and a quotient of the intended codomain with the property that the equation is uniquely solved for the subset and quotient. The equation is therefore guaranteed to have a unique solution for the intended domain and intended codomain whenever the subset is the full set and the quotient is by equality

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Recursive coalgebras of finitary functors

Abstract: For finitary set functors preserving inverse images, recursive coalgebras A of Paul Taylor are proved to be precisely those for which the system described by A always halts in finitely many steps.

Cited by 22 publications

References 8 publications

A Coalgebraic View of Bar Recursion and Bar Induction

A Coalgebraic View of Bar Recursion and Bar Induction

Sorting with bialgebras and distributive laws

Structured general corecursion and coinductive graphs [extended abstract]

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