# Reasoning about Measures of Unmeasurable Sets

Marco Console, Matthias Hofer, Leonid Libkin

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract

In a variety of reasoning tasks, one estimates the likelihood of events by means of volumes of sets they define. Such sets need to be measurable, which is usually achieved by putting bounds, sometimes ad hoc, on them. We address the question how unbounded or unmeasurable sets can be measured nonetheless. Intuitively, we want to know how likely a randomly chosen point is to be in a given set, even in the absence of a uniform distribution over the entire space.

To address this, we follow a recently proposed approach of taking intersection of a set with balls of increasing radius, and defining the measure by means of the asymptotic behavior of the proportion of such balls taken by the set. We show that this approach works for every set definable in first-order logic with the usual arithmetic over the reals (addition, multiplication, exponentiation, etc.), and every uniform measure over the space, of which the usual Lebesgue measure (area, volume, etc.) is an example. In fact we establish a correspondence between the good asymptotic behavior and the finiteness of the VC dimension of definable families of sets. Towards computing the measure thus defined, we show how to avoid the asymptotics and characterize it via a specific subset of the unit sphere. Using definability of this set, and known techniques for sampling from the unit sphere, we give two algorithms for estimating our measure of unbounded unmeasurable sets, with deterministic and probabilistic guarantees, the latter being more efficient. Finally we show that a discrete analog of this measure exists and is similarly well-behaved.
Original language English Proceedings of 17th International Conference on Principles of Knowledge Representation and Reasoning International Joint Conferences on Artificial Intelligence Organization 264-273 10 978-0-9992411-7-2 https://doi.org/10.24963/kr.2020/27 Published - 12 Sep 2020 17th International Conference on Principles of Knowledge Representation and Reasoning - Rhodes, GreeceDuration: 12 Sep 2020 → 18 Sep 2020https://kr2020.inf.unibz.it/

### Publication series

Name IJCAI Organization 2334-1033

### Conference

Conference 17th International Conference on Principles of Knowledge Representation and Reasoning KR 2020 Greece Rhodes 12/09/20 → 18/09/20 https://kr2020.inf.unibz.it/