Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Random Self-Modifiable Computation

Michael Stephen Fiske

TL;DR

The paper introduces the ex-machine, a randomness-enabled self-modifying extension of the Turing machine, consisting of standard, quantum random, and meta instructions. It argues that, with quantum randomness that behaves as unbiased Bernoulli trials and meta-instruction self-modification, ex-machines can evolve to compute Turing incomputable languages with probability 1. It demonstrates the existence of evolutionary paths where an ex-machine progressively resolves halting questions for an initial set of Turing machines, and it provides concrete constructions (e.g., the Q(x) family) to illustrate the mechanism. The work suggests a refinement of Hilbert's program: finite procedures augmented by randomness and self-modification can increase program complexity and computational power beyond fixed-mechanism computers, with implications for the theory of computation and the study of hard mathematical problems.

Abstract

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.

Quantum Random Self-Modifiable Computation

TL;DR

The paper introduces the ex-machine, a randomness-enabled self-modifying extension of the Turing machine, consisting of standard, quantum random, and meta instructions. It argues that, with quantum randomness that behaves as unbiased Bernoulli trials and meta-instruction self-modification, ex-machines can evolve to compute Turing incomputable languages with probability 1. It demonstrates the existence of evolutionary paths where an ex-machine progressively resolves halting questions for an initial set of Turing machines, and it provides concrete constructions (e.g., the Q(x) family) to illustrate the mechanism. The work suggests a refinement of Hilbert's program: finite procedures augmented by randomness and self-modification can increase program complexity and computational power beyond fixed-mechanism computers, with implications for the theory of computation and the study of hard mathematical problems.

Abstract

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.

Paper Structure

This paper contains 16 sections, 7 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work -- Computation
  3. Related Work -- Quantum Randomness
  4. The Ex-Machine
  5. Standard Instructions
  6. Quantum Random Instructions
  7. Meta Instructions
  8. Quantum Randomness
  9. Mathematical Determinism and Unpredictability
  10. Quantum Random Theory
  11. Computing Ex-Machine Languages
  12. Some $\mathfrak{Q}(x)$ Observations Based on Cantor and Gödel
  13. An Ex-Machine Halting Problem
  14. Some Observations Based on Theorem \ref{['thm:x_maching_halting_evolution']}
  15. Two Research Problems
  16. ...and 1 more sections

Key Result

Lemma 4.1

Whenever $i$ satisfies $0 \le i \le m$, string $^{i}$is in $\mathfrak{Q}(a_0 a_1 \dots a_m$$x)$'s language if $a_i = 1$; string $^{i}$is not in $\mathfrak{Q}(a_0 a_1 \dots a_m$$x)$'s language if $a_i = 0$. Whenever $n > m$, it has not yet been determined whether string $^{n}$is in $\mathfrak{Q}(a_0 a_1 \dots a_m$$x)$'s language or not in its language.

Figures (3)

  • Figure 1: Meta Instruction Execution
  • Figure 2: Quantum Observables Rendering Random Bits via Value Indefiniteness.
  • Figure 3: Infinite Binary Tree. A Graphical Representation of $\{0, 1\}^{\mathbb{N}}$.

Theorems & Definitions (38)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.2
  • Example 1
  • Definition 2.5
  • Example 2
  • Definition 2.6
  • ...and 28 more