一句话：为概率选择操作与 Wasserstein 距离建立定量代数公理，刻画其自由完备代数。
本文研究编程语言中概率效应的定量代数理论，目标是为完备度量空间上的 Wasserstein 距离 W_p（p≥1）给出代数化描述。作者提出一族以 p 为参数的公理，刻画由概率选择操作构成的代数：这些操作形成重心代数，且度量满足 W_p 的典型性质。主要结果为：任意完备度量空间上，满足该公理的自由完备代数正是具有 p 阶有限矩的 Radon 概率测度空间，其度量为 Wasserstein 距离，操作为通常的二元凸和。这为概率效应的定量代数语义提供了自由构造与表示定理。
We present an algebraic account of the Wasserstein distances $W_p$ on complete metric spaces, for $p \geq 1$. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in $p$, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance $W_p$. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order $p$, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations.
@article{paperbot376,
title = {Free complete Wasserstein algebras},
author = {Radu Mardare and Prakash Panangaden and Gordon D. Plotkin},
journal = {Logical Methods in Computer Science},
volume = {Volume 14, Issue 3},
year = {2018},
doi = {10.23638/lmcs-14(3:19)2018}
}