Irrational numbers like the square root of two provide the same computational power as any other irrational number when used as a generator for a Turing machine.
April 23, 2026
Original Paper
Dual-Tape Perspective and Generator Independence: The Algebraic Foundation of Real Boolean Turing Machines
arXiv · 2604.16390
The Takeaway
Non-deterministic computation relies on the presence of an incommensurable element rather than the specific identity of a number. Mathematicians previously wondered if different irrational constants would change the fundamental capabilities of these machines. This proof shows that the magic of the calculation comes from the gap between numbers themselves. It places the foundation of certain computing types squarely within number theory. Hardware designers can choose any irrational source without worrying about losing performance or logic. This clarifies the algebraic limits of what future computers can actually do.
From the 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 irrat