*This was certainly true in Turing's case -- indeed he spent the years after his Ph D, from 1939 to 1943, studying certain abstruse symmetry transformations on a 26-letter alphabet.*Anyway, we say that problem A is Turing-reducible to problem B, if A is solvable by a Turing machine given an oracle for B.

of this claim, not as a mere hypothesis in need of continual reexamination and justification.

In 1988 Robin Gandy said that Turing’s analysis “proves a theorem.” However, Stephen C.

One concept we'll need again and again in this class is that of an oracle.

The idea is a pretty obvious one: we assume we have a "black box" or "oracle" that immediately solves some hard computational problem, and then see what the consequences are!

So for example, the problem of whether a statement can be proved from the axioms of set theory is Turing-equivalent to the halting problem: if you can solve one, you can solve the other.

Now, a Turing-degree is the set of all problems that are Turing-equivalent to a given problem. Well, we've already seen two examples: (1) the set of computable problems, and (2) the set of problems that are Turing-equivalent to the halting problem.Already there's an obvious question: what sort of claim is this?Is it an empirical claim, about which functions can be computed in physical reality?) Oracles were apparently first studied by Turing, in his 1938 Ph D thesis.Obviously, anyone who could write a whole thesis about these fictitious entities would have to be an extremely pure theorist, someone who wouldn't be caught dead doing anything relevant.But in my view, so far there hasn't been any serious challenge to the original Church-Turing Thesis -- neither as a claim about physical reality, nor as a definition of computable.' There have been plenty of non-serious challenges to the Church-Turing Thesis.In fact there are whole conferences and journals devoted to these challenges -- google "hypercomputation." I've read some of this stuff, and it's mostly along the lines of, well, suppose you could do the first step of a computation in one second, the next step in a half second, the next step in a quarter second, the next step in an eighth second, and so on.Basically, the thesis is that any function "naturally to be regarded as computable" is computable by a Turing machine.Or in other words, any "reasonable" model of computation will give you either the same set of computable functions as the Turing machine model, or else a proper subset.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...

## Comments Turing Oracle Thesis