Completely iterative algebras and completely iterative monads

Milius, Stefan

doi:10.1016/j.ic.2004.05.003

Cited by 63 publications

(96 citation statements)

References 17 publications

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“…Obviously, not every monad is coinductive in this sense, because the notion of multiplication of infinitely many layers is not always well-defined. Thus, to capture the notion of coinduction in the monadic context, we adopt a property called complete iterativity, introduced by Elgot et al [16] and later studied by Adámek et al [4,30]. A monad M is completely iterative (is a 'cim') if it is equipped with an additional coinductive structure: certain ('guarded') morphisms e : X → M (X + A), where X represents variables (seeds of the corecursion) and A is an object of parameters (final values), have unique solutions e † : X → MA coherent with the monadic structure of M .…”

Section: Coinductionmentioning

confidence: 99%

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: Coinductionmentioning

confidence: 99%

The Coinductive Resumption Monad

Piróg

Gibbons

2014

Electronic Notes in Theoretical Computer Science

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“…In category-theoretic terms, the existence and uniqueness of a solution means that we maintain on J a completely iterative algebra [51] (whose signature is gradually incremented with each additional friend).…”

Section: Reasoning Principlesmentioning

confidence: 99%

Friends with Benefits

Blanchette

Bouzy

Lochbihler

et al. 2017

Programming Languages and Systems

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Abstract. We introduce AmiCo, a tool that extends a proof assistant, Isabelle/ HOL, with flexible function definitions well beyond primitive corecursion. All definitions are certified by the assistant's inference kernel to guard against inconsistencies. A central notion is that of friends: functions that preserve the productivity of their arguments and that are allowed in corecursive call contexts. As new friends are registered, corecursion benefits by becoming more expressive. We describe this process and its implementation, from the user's specification to the synthesis of a higher-order definition to the registration of a friend. We show some substantial case studies where our approach makes a difference.

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“…The algebra A is called completely iterative (or, shortly cia), see [13], if every flat equation morphism has a unique solution. Example: let T Z be the terminal coalgebra of H(−) + Z.…”

Section: Extensive Categories Cia's and Iterative Algebrasmentioning

confidence: 99%