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

Abstract

Beck's distributive laws provide sufficient conditions under which two monads can be composed, and monads arising from distributive laws have many desirable theoretical properties. Unfortunately, finding and verifying distributive laws, or establishing if one even exists, can be extremely difficult and error-prone. We develop general-purpose techniques for showing when there can be no distributive law between two monads. Two approaches are presented. The first widely generalizes ideas from a counterexample attributed to Plotkin, yielding general-purpose theorems that recover the previously known situations in which no distributive law can exist. Our second approach is entirely novel, encompassing new practical situations beyond our generalization of Plotkin's approach. It negatively resolves the open question of whether the list monad distributes over itself. Our approach adopts an algebraic perspective throughout, exploiting a syntactic characterization of distributive laws. This approach is key to generalizing beyond what has been achieved by direct calculations in previous work. We show via examples many situations in which our theorems can be applied. This includes a detailed analysis of distributive laws for members of an extension of the Boom type hierarchy, well known to functional programmers.