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Media type:
E-Article
Title:
Flexible presentations of graded monads
Contributor:
Katsumata, Shin-ya;
McDermott, Dylan;
Uustalu, Tarmo;
Wu, Nicolas
Published:
Association for Computing Machinery (ACM), 2022
Published in:
Proceedings of the ACM on Programming Languages, 6 (2022) ICFP, Seite 902-930
Language:
English
DOI:
10.1145/3547654
ISSN:
2475-1421
Origination:
Footnote:
Description:
A large class of monads used to model computational effects have natural presentations by operations and equations, for example, the list monad can be presented by a constant and a binary operation subject to unitality and associativity. Graded monads are a generalization of monads that enable us to track quantitative information about the effects being modelled. Correspondingly, a large class of graded monads can be presented using an existing notion of graded presentation. However, the existing notion has some deficiencies, in particular many effects do not have natural graded presentations. We introduce a notion of flexibly graded presentation that does not suffer from these issues, and develop the associated theory. We show that every flexibly graded presentation induces a graded monad equipped with interpretations of the operations of the presentation, and that all graded monads satisfying a particular condition on colimits have a flexibly graded presentation. As part of this, we show that the usual algebra-preserving correspondence between presentations and a class of monads transfers to an algebra-preserving correspondence between flexibly graded presentations and a class of flexibly graded monads.