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# Turing Oracle Thesis

In other words, "A is no harder than B": if we had a hypothetical device to solve B, then we could also solve A.

Two problems are Turing-equivalent if each is Turing-reducible to the other.

Since Friedberg and Muchnik's breakthrough, the structure of the Turing degrees has been studied in more detail than you can possibly imagine.

Here's one of the simplest questions: if two problems A and B are both reducible to the halting problem, then must there be a problem C that's reducible to A and B, such that any problem that's reducible to both A and B is also reducible to C? But this is the point where some of us say, maybe we should move on to the next topic...

This question was first asked by Emil Post in 1944, and was finally answered in 1956, by Richard Friedberg in the US and (independently) A. Unfortunately, the resulting problems are extremely contrived; they don't look like anything that might arise in practice.

And even today, we don't have a single example of a "natural" problem with intermediate Turing degree.

This is important to determine what Turing actually accomplished.