No-Go Theorems for Distributive Laws

  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)},
  • 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… 

Tables from this paper

Weakening and Iterating Laws using String Diagrams

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

From Multisets over Distributions to Distributions over Multisets

  • B. Jacobs
  • Mathematics
    2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2021
This article shows that there is a rich underlying theory relating multisets and probability distributions and it is shown that the new distributive law, called parallel multinomial law, can be defined in (at least) four equivalent ways.

Monadic Monadic Second Order Logic

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

Combining probabilistic and non-deterministic choice via weak distributive laws

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

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.

Combining Weak Distributive Laws: Application to Up-To Techniques

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.

Semialgebras and Weak Distributive Laws

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

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

. 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

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.



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

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

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.


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

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

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