44 - The Church–Turing Thesis

Summary

We now bring together Church's Thesis and Turing's Thesis into the Church- Turing Thesis, explain what the Thesis claims when properly interpreted, and say something about its history and about its status.

Putting things together

Right back in Chapter 3 we stated Turing's Thesis: a numerical (total) function is effectively computable by some algorithmic routine if and only if it is computable by a Turing machine. Of course, we initially gave almost no explanation of the Thesis. It was only very much later, in Chapter 41, that we eventually developed the idea of a Turing machine and saw the roots of Turing's Thesis in his general analysis of the fundamental constituents of any computation.

Meanwhile, in Chapter 38, we introduced the idea of a μ-recursive function and noted the initial plausibility of Church's Thesis: a numerical (total) function is effectively computable by an algorithmic routine if and only if it is μ-recursive. Then finally, in Chapter 42, we outlined the proof that a total function is Turing-computable if and only if it is μ-recursive. Our two Theses are therefore equivalent. And given that equivalence, we can now talk of

The Church–Turing Thesis The effectively computable total numerical functions are the μ-recursive/Turing-computable functions.

Crucially, this Thesis links what would otherwise be merely technical results about μ-recursiveness/Turing-computability with intuitive claims about effective computability; and similarly it links claims about recursive decidability with intuitive claims about effective decidability.