#### Title

Monotonic Bit-Invariant Permutation-Invariant Metrics on the Set of All Infinite Binary Sequences

#### Publication Date

7-1-2022

#### Abstract

In a computer, all the information about an object is described by a sequence of 0s and 1s. At any given moment of time, we only have partial information, but as we perform more measurements and observations, we get longer and longer sequence that provides a more and more accurate description of the object. In the limit, we get a perfect description by an infinite binary sequence. If the objects are similar, measurement results are similar, so the resulting binary sequences are similar. Thus, to gauge similarity of two objects, a reasonable idea is to define an appropriate metric on the set of all infinite binary sequences. Several such metrics have been proposed, but their limitation is that while the order of the bits is rather irrelevant -- if we have several simultaneous measurements, we can place them in the computer in different order -- the distance measured by current formulas change if we select a different order. It is therefore natural to look for permutation-invariant metrics, i.e., distances that do not change if we select different orders. In this paper, we provide a full description of all such metrics. We also explain thee limitation of these new metrics: that they are, in general, not computable.

## Comments

Technical Report: UTEP-CS-22-86