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}
}
  • M. Zwart, Dan Marsden
  • Published 1 June 2019
  • Mathematics
  • 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
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… 

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References

SHOWING 1-10 OF 47 REFERENCES

Don't Try This at Home: No-Go Theorems for Distributive Laws

TLDR
This work develops general-purpose techniques for showing when there can be no distributive law between two monads, and adopts an algebraic perspective throughout, exploiting a syntactic characterization of distributive laws.

Iterated distributive laws

  • Eugenia Cheng
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 2011
Abstract We give a framework for combining n monads on the same category via distributive laws satisfying Yang–Baxter equations, extending the classical result of Beck which combines two monads via

Distributive laws for Lawvere theories

Distributive laws give a way of combining two algebraic structures expressed as monads; in this paper we propose a theory of distributive laws for combining algebraic structures expressed as Lawvere

Presenting Distributive Laws

TLDR
This paper describes how to obtain a distributive law for a monad with an equational presentation from a distributives law for the underlying free monad and applies this result to show the equivalence between two different representations of context-free languages.

MONAD COMPOSITIONS I: GENERAL CONSTRUCTIONS AND RECURSIVE DISTRIBUTIVE LAWS

New techniques for constructing a distributive law of a monad over another are studied using submonads, quotient monads, product mon- ads, recursively-defined distributive laws, and linear equations.

Monad compositions II: Kleisli strength

TLDR
The concept of Kleisli strength for monads in an arbitrary symmetric monoidal category is introduced and generalises the notion of commutative monad and gives new examples, even in the cartesian-closed category of sets.

Distributing probability over non-determinism

TLDR
The notion of indexed valuations is used to define a new monad that can be combined with the usual non-deterministic monad via a categorical distributive law and an equational characterisation of the construction is given.

Distributing probability over nondeterminism

We study the combination of probability and nondeterminism from a categorical point of view. In category theory, nondeterminism and probability are represented by suitable monads. Those two monads do

A divertimento on MonadPlus and nondeterminism

  • T. Uustalu
  • Mathematics
    J. Log. Algebraic Methods Program.
  • 2016