Midwest Computability Seminar

XXV
Part i



The Midwest Computability Seminar is meeting remotely in the fall of 2020. The recurring Zoom link is:

https://notredame.zoom.us/j/99754332165?pwd=RytjK1RFZU5KWnZxZ3VFK0g4YTMyQT09

Meeting ID: 997 5433 2165

Passcode: midwest



YouTube video    Panopto video


This session will be held jointly with the Computability Theory and Applications Online Seminar.


DATE: Tuesday, August 18th, 2020

TIME: 3:00 - 4:00 PM CDT

SPEAKER: Joe Miller - University of Wisconsin–Madison

TITLE:
Redundancy of information: lowering effective dimension

ABSTRACT:
A natural way to measure the similarity between two infinite binary sequences X and Y is to take the (upper) density of their symmetric difference. This is the Besicovitch distance on Cantor space. If d(X,Y) = 0, then we say that X and Y are coarsely equivalent. Greenberg, M., Shen, and Westrick (2018) proved that a binary sequence has effective (Hausdorff) dimension 1 if and only if it is coarsely equivalent to a Martin-Löf random sequence. They went on to determine the best and worst cases for the distance from a dimension t sequence to the nearest dimension s>t sequence. Thus, the difficulty of increasing dimension is understood.

Greenberg, et al. also determined the distance from a dimension 1 sequence to the nearest dimension t sequence. But they left open the general problem of reducing dimension, which is made difficult by the fact that the information in a dimension s sequence can be coded (at least somewhat) redundantly. Goh, M., Soskova, and Westrick recently gave a complete solution.

I will talk about both the results in the 2018 paper and the more recent work. In particular, I will discuss some of the coding theory behind these results. No previous knowledge of coding theory is assumed.



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