Coproducts of Ideal Monads

Ghani, Neil; Uustalu, Tarmo

doi:10.1051/ita:2004016

Cited by 22 publications

(22 citation statements)

References 25 publications

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“…There are other constructions on monads that involve initial algebras, for example the coproduct of two ideal monads, as shown by Ghani and Uustalu [20]. We conjecture that a similar construction with ν in place of μ yields the coproduct in the category of ideal cims.…”

Section: Complete Iterativitymentioning

confidence: 91%

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: Complete Iterativitymentioning

confidence: 91%

The Coinductive Resumption Monad

Piróg

Gibbons

2014

Electronic Notes in Theoretical Computer Science

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“…The coproduct of two arbitrary monads is not always guaranteed to exist, but is known to exist in certain special cases. For example, monad coproducts are guaranteed to exist when the monads in question are ideal monads (Ghani & Uustalu, 2004), or when working in the category of Sets (Adámek et al, 2012), or if the monads are constructed from algebraic theories (Hyland et al, 2006). One particular special case is when one of the constituent monads is free, as we shall see in Section 8.4, below.…”

Section: Coproducts Of Monadsmentioning

confidence: 99%

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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“…1(a), with a T-layer uppermost. (3) Ghani and Uustalu [8] treat the case where both S and T are ideal monads (see Elgot [7]), corresponding to a theory whose equations are all between nontrivial terms. A nontrivial term in the sum consists of layers alternating between nontrivial terms of S and those of T, as depicted in Fig.…”

Section: Introductionmentioning

confidence: 99%

Coproducts of Monads on Set

Ad'mek

Milius

et al. 2012

2012 27th Annual IEEE Symposium on Logic in Computer Science

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Coproducts of monads on Set have arisen in both the study of computational effects and universal algebra.We describe coproducts of consistent monads on Set by an initial algebra formula, and prove also the converse: if the coproduct exists, so do the required initial algebras. That formula was, in the case of ideal monads, also used by Ghani and Uustalu. We deduce that coproduct embeddings of consistent monads are injective; and that a coproduct of injective monad morphisms is injective.Two consistent monads have a coproduct iff either they have arbitrarily large common fixpoints, or one is an exception monad, possibly modified to preserve the empty set. Hence a consistent monad has a coproduct with every monad iff it is an exception monad, possibly modified to preserve the empty set. We also show other fixpoint results, including that a functor (not constant on nonempty sets) is finitary iff every sufficiently large cardinal is a fixpoint.

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Coproducts of Ideal Monads

Cited by 22 publications

References 25 publications

The Coinductive Resumption Monad

The Coinductive Resumption Monad

Interleaving data and effects

Coproducts of Monads on Set

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