Random Self-modifiable Computation

Abstract

A new computational model, called the ex-machine, executes standard instructions, meta instructions, and random instructions. Standard instructions behave similarly to machine instructions in a digital computer. Meta instructions self-modify the ex-machine’s program during its execution. We construct a countable set of ex-machines; each can compute a Turing incomputable language, whenever the quantum random measurements in the random instructions behave like unbiased Bernoulli trials. In 1936, Turing posed the halting problem and proved that this problem is unsolvable for Turing machines. Let $\{({M}_i, T_i): i \in \mathbb {N} \}$ be a Turing computable enumeration of all Turing machines M _i and finite tapes T _i. Does there exist an ex-machine $\mathcal {X}$ that has at least one evolutionary path $\mathcal {X} \rightarrow \mathcal {X}_1 \rightarrow \mathcal {X}_2 \rightarrow \dots \rightarrow \mathcal {X}_m$, so at the mth stage, the ex-machine $\mathcal {X}_m$ can correctly determine for 0 ≤ i ≤ m whether M _i’s execution on tape T _i eventually halts? We construct an ex-machine Z(x) that has one such evolutionary path; at stage m, Z(x) has used a finite amount of computational resources. The existence of this path suggests that David Hilbert (Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematische-Physikalische Klasse. 3:253–297, 1900) may not have been misguided to propose that mathematicians search for finite methods to help construct proofs. Our refinement is to not use a program that behaves according to a fixed set of mechanical rules. We must pursue computation that exploits randomness and self-modification so that the complexity of the program can increase during computation.

Appendix: A Turing Machine Is an Autonomous Dynamical System

Fix a Turing machine M. Transformation ϕ maps a machine configuration to a point in the complex plane $\mathbb {C}$; ϕ also maps each of M’s instructions to a finite number |A| of unique affine functions each with a distinct domain. These affine functions can be extended to a function F on a bounded region W in $\mathbb {C}$, containing these domains and a disjoint bounded set, called the halting attractor. Via ϕ, one computational step of M corresponds to one iteration of the discrete autonomous dynamical system (F, W).

Let machine states Q = {q ₁, …, q _m}. Let alphabet A = {a ₁, …, a _n}, where a ₁ is the blank symbol. Halt state h is a special state that is not in Q. Function η : Q × A →Q ∪{h}× A ×{−1, +1} is the machine M’s program. A single instruction is η(q, a) = (r, b, x), where q ∈Q, r ∈Q ∪{h}, a, b ∈ A, and x ∈{−1, +1}. Set B = |A| + |Q| + 1. Define symbol value function $\nu : \{h\} \cup \mathsf {Q} \cup A \rightarrow \mathbb {N}$ as ν(a ₁) = 0, …, ν(a _i) = i − 1, …, ν(a _n) = |A|− 1, ν(h) = |A|, ν(q ₁) = |A| + 1, …, ν(q _i) = |A| + i, …, ν(q _m) = |A| + |Q|.

Fig. 2

Machine configuration (q, k, T) before executing a standard instruction

$T: \mathbb {Z} \rightarrow A$ is the tape and is finite. T _k is the alphabet symbol in tape square k. Machine configuration (q, k, T) lies in $\mathsf {Q} \times \mathbb {Z} \times A^{\mathbb {Z}}$ and maps to the complex number:

(1)

In Eq. 1, the infinite series in both the real and imaginary parts sums to rational numbers because the initial tape squares contain a finite number of non-blank symbols.

Next, we define how ϕ maps each instruction in program η to a finite set of affine functions. When instruction η(q, T _k) = (r, b, +1) executes, state q moves to state r, symbol b replaces T _k on tape square k, and the head moves to tape square k + 1.

The right affine functions corresponding to instruction η(q, T _k) = (r, b, +1) are of the form f(x + yi) =  f ₁(x) + f ₂(y) i, where f ₁(x) = |A|x + m and $f_2(y) = \frac {1}{|A|} y + n$. Using Eq. 1 and Fig. 3 to solve for m and n, ϕ maps instruction η(q, T _k) = (r, α, +1) to the affine function f(x + yi) =  f ₁(x) + f ₂(y) i, where

$$\displaystyle \begin{aligned} f_1(x) = |A|x + (|A|-1) \nu(T_{k+1}) - |A|{}^2 \nu(T_k) \end{aligned} $$

(2)

$$\displaystyle \begin{aligned} f_2(y) = \frac{1}{|A|} y + B \nu(r) + \nu(b) - \frac{B}{|A|} \nu(q). \end{aligned} $$

(3)

Fig. 3

Machine configuration after executing instruction η(q, T _k) = (r, b, +1)

For each of the |A| distinct values v(T _k+1) in f ₁, f is a different affine function. Thus, there are |A| distinct affine functions that correspond to instruction η(q, T _k) = (r, b, +1). The domain of each right affine function is $U_{j,k} = \big {\{} x + yi \in \mathbb {C}: j \le x < j+1$ and k ≤ y < k + |A|}, where j = |A|ν(T _k) + ν(T _k+1) and k = Bν(q).

When η(q, T _k) = (r, b, −1) executes, state q moves to state r, symbol b replaces T _k on square k, and the head moves to tape square k − 1.

From Eq. 1 and Fig. 4, ϕ maps instruction η(q, T _k) = (r, b, −1) to affine function g(x + yi) =  g ₁(x) + g ₂(y) i, where

$$\displaystyle \begin{aligned} g_1(x) = \frac{1}{|A|}x + |A| \nu(T_{k-1}) + \nu(b) - \nu(T_k) \end{aligned} $$

(4)

$$\displaystyle \begin{aligned} g_2(y) = |A|y + B \nu(r) - |A| B \nu(q) - |A| \nu(T_{k-1}). \end{aligned} $$

(5)

Fig. 4

Machine configuration after executing instruction η(q, T _k) = (r, b, −1)

For each of the |A| distinct values v(T _k−1) in g ₁ and g ₂, g is a different affine function. Thus, there are |A| distinct left affine functions that correspond to instruction η(q, T _k) = (r, b, −1). The domain of each left affine function is $V_{j,k} = \big {\{} x + yi \in \mathbb {C}: j \le x < j + |A|$ and k ≤ y < k + 1}, where j = |A|ν(T _k) and k = Bν(q) + ν(T _k−1).

Define halting attractor $H = \big {\{} x + yi \in \mathbb {C}:$ 0 ≤ x < |A|² and B|A|≤ y ≤ (B + 1)|A|}. The points in $\mathbb {C}$ that correspond to halting configurations (h, k, T) are called halting points. Using elementary algebra and simple geometric series calculations, one can verify that the halting points are a subset of H. Define halting map $\mathfrak {h}: H \rightarrow H$, where $\mathfrak {h}(x + yi) = x + yi$ on H. Every point in the halting attractor is a fixed point of $\mathfrak {h}$. Moreover, the intersection of each affine function’s domain and H is empty. This implies that $\mathfrak {h}$ and all left and right affine functions corresponding to η’s instructions can be extended to a function F on domain W that contains H and all domains U _j,k and V _j,k.

Overall, the ϕ correspondence transforms Turing’s halting problem to a discrete autonomous dynamical systems problem in $\mathbb {C}$. If machine configuration (q, k, T) halts after n computational steps, then the orbit of ϕ(q, k, T) exits one of the domains U _j,k or V _j,k on the nth iteration and enters the halting attractor H. If machine configuration (r, j, S) never halts, then the orbit of ϕ(r, j, S) never reaches the halting attractor.