[论文解读] Complete Quasi-Metrics for Hyperspaces, Continuous Valuations, and Previsions

Jean Goubault-Larrecq|arXiv (Cornell University)|Jul 12, 2017

Advanced Topology and Set Theory参考文献 30被引用 7

一句话总结

本文引入了一种广义的Kantorovich-Rubinshteïn-Hutchinson准度量，适用于超空间、连续估值和准度量空间中的预估值，证明该度量诱导出一个代数Yoneda完备的拓扑，当底空间为代数Yoneda完备时，该拓扑与弱拓扑一致。该构造将经典度量空间结果推广至非Hausdorff、非对称的设定，与计算机科学语义学密切相关。

ABSTRACT

The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in that case it metrizes the weak topology. We introduce a variant of that construction in the realm of quasi-metric spaces, and prove that it is algebraic Yoneda-complete as soon as the underlying quasi-metric space of points is algebraic Yoneda-complete, and that the associated topology is the weak topology. We do this not only for probability distributions, represented as normalized continuous valuations, but also for subprobability distributions, for various hyperspaces, and in general for different brands of functionals. Those functionals model probabilistic choice, angelic and demonic non-deterministic choice, and their combinations. The mathematics needed for those results are more demanding than in the simpler case of metric spaces. To obtain our results, we prove a few other results that have independent interest, notably: continuous Yoneda-complete spaces are consonant; on a continuous Yoneda-complete space, the Scott topology on the space of $\overline{\mathbb{R}}_+$-valued lower semicontinuous maps coincides with the compact-open and Isbell topologies, and the subspace topology on spaces of $α$-Lipschitz continuous maps also coincides with the topology of pointwise convergence, and is stably compact; we introduce and study the so-called Lipschitz-regular quasi-metric spaces, and we show that the formal ball functor induces a Kock-Zöberlein monad, of which all algebras are Lipschitz-regular; and we prove a minimax theorem where one of the spaces is not compact Hausdorff, but merely compact.

研究动机与目标

• 将已知用于波兰空间弱收敛度量化的Kantorovich-Rubinshtein度量推广至非Hausdorff、非对称的准度量空间。
• 在缺乏对称性或紧致性的情况下，为连续估值、子概率测度和预估值的空间建立完备性与拓扑一致性。
• 通过确保代数Yoneda完备性与准度量设定下的拓扑一致性，为概率性和非确定性计算语义学提供基础。

提出的方法

• 通过下半连续与Lipschitz连续映射的对偶性，引入预估值泛函上的Kantorovich-Rubinshtein-Hutchinson准度量。
• 定义并研究Lipschitz正则准度量空间，证明形式球函子诱导出一个Kock-Zöberlein单子，其代数恰好为Lipschitz正则空间。
• 利用形式球构造，在估值和预估值空间上定义准度量，借助与Lipschitz函数的对偶性。
• 证明在连续Yoneda完备空间上，扩展实值下半连续映射空间上的Scott拓扑、紧开拓扑与Isbell拓扑一致。
• 通过A/D/P重影对应用收缩与嵌入技术，证明预估值上的弱拓扑与d_KR^a-Scott拓扑一致。
• 在无需紧致Hausdorff假设的条件下，建立次线性和超线性预估值的极小化极大定理与对偶性结果。

实验结果

研究问题

• RQ1 当底空间为代数Yoneda完备时，Kantorovich-Rubinshtein-Hutchinson准度量是否在连续估值空间上诱导出代数Yoneda完备性？
• RQ2 在非Hausdorff、非对称设定下，预估值空间上的弱拓扑是否可作为d_KR^a-Scott拓扑被恢复？
• RQ3 在准透镜的Plotkin幂域上，d_KR^a准度量在何种条件下可诱导出对叉或估值空间的收缩？
• RQ4 在完备准度量空间上，所有连续估值空间是否可通过广义KR型准度量实现度量化？
• RQ5 对偶定理中的耦合是否可表示为X×X上的概率估值，其成立条件为何？

主要发现

• 当底空间为代数Yoneda完备时，连续估值上的Kantorovich-Rubinshtein-Hutchinson准度量为代数Yoneda完备。
• 在连续Yoneda完备空间上，扩展实值下半连续映射空间上的Scott拓扑、紧开拓扑与Isbell拓扑一致。
• α-Lipschitz连续映射的子空间继承了逐点收敛拓扑，并在d_KR^a准度量下保持稳定紧致性。
• 形式球函子诱导出一个Kock-Zöberlein单子，其代数恰好为Lipschitz正则准度量空间。
• 通过拓扑收缩与嵌入论证，证明预估值空间上的d_KR^a-Scott拓扑与弱拓扑一致。
• 即使其中一个空间仅为紧致而非紧致Hausdorff，仍为次线性和超线性预估值建立了极小化极大定理。