Does there exist an ex-machine X that has at least one evolutionary path X --> X_1 --> X_2 --> . . . --> X_m, so at the mth stage ex-machine X_m can correctly determine for 0 <= i <= m whether M_i's execution  ...  In 1936, Alan Turing posed the halting problem for Turing machines and proved that this problem is unsolvable for Turing machines.  ...  T k is the alphabet symbol in the kth tape square. φ maps right computational step η(q, T k ) = (r, α, +1) to the affine function f (x, y) = Bx − B 2 ν(T k ), 1 B y + Bν(r) + ν(α) − ν(q) .  ...