# Quantum Random Self-Modifiable Computation. (arXiv:1807.01369v6 [cs.CC] UPDATED)

Among the fundamental questions in computer science, at least two have a deep

impact on mathematics. What can computation compute? How many steps does a

computation require to solve an instance of the 3-SAT problem? Our work

addresses the first question, by introducing a new model called the ex-machine.

The ex-machine executes Turing machine instructions and two special types of

instructions. Quantum random instructions are physically realizable with a

quantum random number generator. Meta instructions can add new states and add

new instructions to the ex-machine. A countable set of ex-machines is

constructed, each with a finite number of states and instructions; each

ex-machine can compute a Turing incomputable language, whenever the quantum

randomness measurements behave like unbiased Bernoulli trials. In 1936, Alan

Turing posed the halting problem for Turing machines and proved that this

problem is unsolvable for Turing machines. Consider an enumeration E_a(i) =

(M_i, T_i) of all Turing machines M_i and initial tapes T_i. 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 on tape T_i eventually halts? We demonstrate

an ex-machine Q(x) that has one such evolutionary path. The existence of this

evolutionary path suggests that David Hilbert was not misguided to propose in

1900 that mathematicians search for finite processes to help construct

mathematical proofs. Our refinement is that we cannot use a fixed computer

program that behaves according to a fixed set of mechanical rules. We must

pursue methods that exploit randomness and self-modification so that the

complexity of the program can increase as it computes.