• Corpus ID: 15972097

Recursion Schemes from Comonads

@article{Uustalu2001RecursionSF,
  title={Recursion Schemes from Comonads},
  author={Tarmo Uustalu and Varmo Vene and Alberto Pardo},
  journal={Nord. J. Comput.},
  year={2001},
  volume={8},
  pages={366-390}
}
Within the setting of the categorical approach to total functional programming, we introduce a "many-in-one" recursion scheme that neatly unifies a variety of seemingly diverging strengthenings of the basic recursion scheme of iteration. The new scheme is doubly generic: in addition to being parametric in a functor capturing the signature of an inductive type, it is also parametric in a comonad and a distributive law (of the functor over the comonad) that together encode the recursive call… 
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References

SHOWING 1-10 OF 26 REFERENCES
Primitive (Co)Recursion and Course-of-Value (Co)Iteration, Categorically
TLDR
It is shown on ex-amples that primitive corecursion is a useful function definition scheme and two novel constructions, viz., histomorphisms and futumorphisms, that capture the powerful schemes of course-of-value iteration and its dual are argued.
Towards Merging Recursion and Comonads
TLDR
Two applications are shown that naturally lead to versions of a comonadic fold operator on the product comonad that capture functions that require extra arguments for their computation and are related with the notion of strong datatype.
Notions of Computation and Monads
  • E. Moggi
  • Computer Science
    Inf. Comput.
  • 1991
Fusion of recursive programs with computational effects
Monadic Maps and Folds for Arbitrary Datatypes
Each datatype constructor comes equiped not only with a so-called map and fold ( catamorphism ), as is widely known, but, under some condition, also with a kind of map and fold that are related to an
Universal Algebra for Computer Scientists
  • W. Wechler
  • Computer Science, Mathematics
    EATCS Monographs on Theoretical Computer Science
  • 1992
TLDR
This volume offers a new model-theoretic approach to universal algebra and presents a systematic development of the methods of results of universal algebra that are useful in a variety of applications in computer science.
Merging Monads and Folds for Functional Programming
TLDR
The simultaneous use of generalised fold operators and monads to structure functional programs and how generalised monadic folds aid in calculating an efficient graph reduction engine from an inefficient specification is discussed.
The essence of functional programming
TLDR
This paper explores the use monads to structure functional programs and describes the relation between monads and the continuation-passing style in a compiler for Haskell that is written in Haskell.
...
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