Multisets and Distributions

TL;DR

The paper addresses the challenge of combining nondeterminism and probability by constructing a distributive law between the finite multiset monad and the finite distribution monad without recourse to combinatorics. The authors first develop a distributive law for lists over distributions and then transfer it to multisets via the Parikh map, supported by a general 2-categorical theorem: equations between 2-cells are preserved by epic 2-natural transformations. This yields a distributive law $\otimes^M: MF \to FM$ whose specialization to the finite multiset and distribution case gives the Beck distributive law $MD \to DM$; the construction relies on the parity between order (lists) and multisets. The approach provides a generally applicable tool for deriving monad and distributive-law results via 2-categorical transfer and suggests potential extensions to the Giry monad and other contexts where order-forgetting maps like the Parikh map are available.

Abstract

We give a lightweight alternative construction of Jacobs's distributive law for multisets and distributions that does not involve any combinatorics. We first give a distributive law for lists and distributions, then apply a general theorem on 2-categories that allows properties of lists to be transferred automatically to multisets. The theorem states that equations between 2-cells are preserved by epic 2-natural transformations. In our application, the appropriate epic 2-natural transformation is defined in terms of the Parikh map, familiar from formal language theory, that takes a list to its multiset of elements.

Figures (3)

• Figure 1: Axioms for the distributive law $\otimes^L:LF\to FL$
• Figure 2: Naturality of $\otimes^M$ from $\otimes^L$
• Figure 3: An axiom of distributive laws

Theorems & Definitions (17)

• theorem 1
• proof
• lemma 1
• proof
• lemma 2
• proof
• theorem 2
• proof
• theorem 3
• proof