The Halting Problem

Theorem 2:

Let M₁, M₂, ... be a list of all TM using characters 1, #.

The Self-Halting Problem is the problem of finding an effective procedure to compute function s such that s(n) = 1 iff M_n doesn't halt if it begins scanning in n 1s

The Self-Halting Problem is unsolvable.

Proof:

Suppose it is solvable. Then by Church's Thesis there is a machine S that computes s.

Recall function t of Theorem 1: supposedly uncomputable.

We can compute it thus:

To compute t(n) start S and M_n in initial states

1.    If M_n does not halt, then S will eventually halt

        At that time we realise M_n doesn't halt and so f_n(n) is undefined.

        So t(n) = 1

2.    If M_n halts then the tape will tell us whether or not f_n(n) is defined.

        So we know whether t(n) = 1 or t(n) is undefined

        But this can't be.

So SHP is not solvable.