# No-Go Theorems for Distributive Laws

@article{Zwart2019NoGoTF, title={No-Go Theorems for Distributive Laws}, author={Maaike Zwart and Dan Marsden}, journal={2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)}, year={2019}, pages={1-13} }

Monads are commonplace in computer science, and can be composed using Beck's distributive laws. Unfortunately, finding distributive laws can be extremely difficult and error-prone. The literature contains some principles for constructing distributive laws. However, until now there have been no general techniques for establishing when no such law exists. We present two families of theorems for showing when there can be no distributive law for two monads. The first widely generalizes a…

## 22 Citations

Weakening and Iterating Laws using String Diagrams

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Distributive laws are a standard way of combining two monads, providing a compositional approach for reasoning about computational eﬀects in semantics. Situations where no such law exists can…

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One of the main reasons for the correspondence of regular languages and monadic secondorder logic is that the class of regular languages is closed under images of surjective letter-toletter…

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This paper proves the existence of a weak distributive law of the powerset monad over the finite distribution monad, and retrieves the well-known convex powersetmonad as a weak lifting of the power monad to the category of convex algebras.

Convexity via Weak Distributive Laws

- MathematicsArXiv
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The canonical weak distributive law δ of the powerset monad over the semimodule monad for a certain class of semirings containing, in particular, positive semifields is studied, characterising δ as a convex closure in the free semimmodule of a set.

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- Mathematics, Computer ScienceArXiv
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This work provides abstract compositionality results, a generalized determinization procedure, and systematic soundness of up-to techniques of alternating automata as a motivating example and extends this framework to the case when one of the monads is only a functor.

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It is proved that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg–Moore algebrAs for a suitable monad structure on the functor id +M, which is called the semifree monadM.

Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces

- MathematicsICALP
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This work derives the canonical Egli-Milner extension of the powerset to the category of relations in three different frameworks and proves that it corresponds to a monotone weak distributive law in each case by showing that the multiplication extends to relations but the unit does not.

Algebraic Presentation of Semifree Monads

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. Monads and their composition via distributive laws have many applications in program semantics and functional programming. For many interesting monads, distributive laws fail to exist, and this has…

Degrading Lists

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This work asks if a graded monad can be extended to a monad, and when such a degrading is in some sense canonical, and shows that, in both cases, there exist infinitely many monad structures.

Lattices do not distribute over powerset

- Mathematics
- 2020

. We show that there is no distributive law of the free lattice monad over the powerset monad. The proof presented here also works for other classes of lattices such as (bounded) distributive/modular…

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