## Thursday, August 21, 2014

### Turing's Oracle

My daughter had a summer project to read and summarize some popular science articles. Having heard me talk about Alan Turing more than a few times, she picked a cover story from a recent New Scientist. The cover copy says "Turing's Oracle: Will the universe let us build the ultimate thinking machine" sounds like an AI story but in fact more of an attack on the Church-Turing thesis. The story is behind a paywall but here is an excerpt:
He called it the "oracle". But in his PhD thesis of 1938, Alan Turing specified no further what shape it might take...Turing has shown with his universal machine that any regular computer would have inescapable limitations. With the oracle, he showed how you might smash through them.
This is a fundamental misinterpretation of Turing's oracle model. Here is what Turing said in his paper Systems of Logic Based on Ordinals, Section 4.
Let us suppose we are supplied with some unspecified means of solving number-theoretic problems; a kind of oracle as it were. We shall not go any further into the nature of the oracle apart from saying it cannot be a machine. (emphasis mine)
The rest of the section defines the oracle model and basically argues that for any oracle O, the halting problem relative to O is not computable relative to O. Turing is arguing here that there is no single hardest problem, there is always something harder.

If you take O to be the usual halting problem then a Turing machine equipped with O can solve the halting problem, just by querying the oracle. But that doesn't mean that you have some machine that solves the halting problem for, as Turing has so eloquently argued in Section 9 of his On Computable Numbers, no machine can compute such an O. Turing created the oracle model, not because he thought it would lead to a process that would solve the halting problem, but because it allowed him to show there are problems even more difficult.

Turing's oracle model, like so much of his work, has played a major role in both computability and computational complexity theory. But one shouldn't twist this model to think the oracle could lead to machines that solve non-computable problems and it is sacrilege to suggest that Turing himself would think that.

1. new scientist tends to hype sometimes & be quite breathless in its tone esp in the cover story. it presumably sells more magazines. sci am has adopted that style somewhat in the last few years also. as for the church-turing thesis, there is no reason to regard it religiously. arguably models of computation now exist that have now demonstrably "gone beyond" it, in particular QM computing, based on QM nondeterminism. even probabilistic algorithms now widely used are not so much envisioned by Turing.

2. Sorry, I see that vzn already referred to Sci Am. My comment and link just adds that they explicitly made the same error.