Reasoning about Uncertainty in Metric Spaces

TL;DR

This paper develops a formal logical framework for reasoning about uncertainty in metric spaces using belief measures, integrating probability and metric aspects, with proofs of soundness and completeness.

Contribution

It introduces a novel logical system for expected distance in metric spaces and defines a new metric for product spaces with proven properties.

Findings

Logical system is sound and complete.

New metric for product spaces has desirable properties.

Framework unifies probability and metric uncertainties.

Abstract

We set up a model for reasoning about metric spaces with belief theoretic measures. The uncertainty in these spaces stems from both probability and metric. To represent both aspect of uncertainty, we choose an expected distance function as a measure of uncertainty. A formal logical system is constructed for the reasoning about expected distance. Soundness and completeness are shown for this logic. For reasoning on product metric space with uncertainty, a new metric is defined and shown to have good properties.