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Introenumerable sets and the cototal enumeration degrees

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SASW09 - International conference on computability, complexity and randomness

In 2015, Emmanuel Jeandel pointed out two interesting properties of the language L of a minimal subshift. First, it is enumeration reducible to its complement; we say that L is cototal. Second, there is an enumeration operator that recovers L from any infinite subset; we say that L is uniformly introenumerable. The first observation motivated the study of cototal sets and their enumation degrees, which has been fruitful. Less attention has been paid to the second observation. In 2018, McCarthy showed that every cototal enumeration degree contains the language of a minimal subshift, and hence contains a uniformly introenumerable set. This leaves open the question of whether all uniformly introenumerable sets have cototal degree. Goh, Jacobsen-Grocott, Soskova, and I have answered this question in the negative. In this talk, I will give the background on cototality and its connection to symbolic dynamics and computable structure theory. I will put the result above in a larger conext, describing other related subclasses of the enumeration degrees. I will also discuss the forcing partial order that we use to prove the separation between uniform introenumerability and cototality.

This talk is part of the Isaac Newton Institute Seminar Series series.

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