Dual-Tape Perspective and Generator Independence: The Algebraic Foundation of Real Boolean Turing Machines

TL;DR

This paper introduces the Real Boolean Turing Machine (RBTM) with a dual-tape perspective, proving that computational power is independent of the specific algebraic generator used, thus clarifying the essence of non-determinism.

Contribution

It concretizes the abstract generator in CBTM as algebraic numbers, introduces the dual-tape perspective, and proves generator independence, advancing understanding of non-deterministic computation.

Findings

The generator independence theorem shows automata are isomorphic regardless of the algebraic generator used.

The dual-tape perspective decomposes tapes into real and imaginary parts, providing intuitive understanding.

The analysis of generator extraction highlights the need for dynamic dimension tracking.

Abstract

The Complex Boolean Turing Machine (CBTM) characterizes non-deterministic computation using the abstract generator $\alpha$, but the abstractness of $\alpha$ makes it difficult to understand intuitively. In this paper, by concretizing $\alpha$ as the algebraic number $\sqrt{2}$, we introduce the \textbf{Real Boolean Turing Machine (RBTM)} and propose the \textbf{dual-tape perspective}, decomposing each tape into a real tape (storing rational coefficients $a$) and an imaginary tape (storing irrational coefficients $b$). The ``1''s on the imaginary tape intuitively mark the locations of ``new dimensions,'' laying a physical foundation for subsequent dynamic dimension tracking. More importantly, we prove the \textbf{Generator Independence Theorem}: computational power is independent of the specific choice of generator, whether using $\sqrt{2}$, $\sqrt{3}$, or the imaginary unit $i$, the corresponding automata are isomorphic. This reveals that the essence of non-determinism lies in the fact of ``introducing a new element incommensurable with the base field,'' rather than the algebraic identity of the generator. Furthermore, we introduce the \textbf{generator extraction operator} and analyze its limitations within a static framework, highlighting the necessity of introducing a dynamic IVM. The RBTM serves both as a visualized instance of the CBTM and as a bridge to the subsequent dynamic dimension tracking of the Imaginary-part Verification Machine(IVM).