Complex Boolean Turing Machines: An Algebraic Semantic Framework for Computational Complexity

TL;DR

This paper introduces the Complex Boolean Turing Machine (CBTM), an algebraic semantic framework that models nondeterminism through field extensions in GF(4), providing a new perspective on computational complexity.

Contribution

It develops an algebraic semantics for Turing machines using GF(4), establishing polynomial equivalence with classical models and offering a novel algebraic understanding of nondeterminism.

Findings

CBTM is polynomially equivalent to classical Turing machines.

Supports arbitrary k-way nondeterminism via algebraic field extensions.

Provides an algebraic perspective on the nature of nondeterminism.

Abstract

Traditional Turing machines are semantically poor, they only concern the syntactic manipulation of symbols, discarding the mathematical semantics behind the symbols. This semantic deficiency is considered the root cause of the three major barriers: relativization, natural proofs, and algebrization. This paper proposes the Complex Boolean Turing Machine (CBTM), elevating computational symbols to algebraic elements in $\mathrm{GF}(4)$, so that each operation has a clear mathematical interpretation. The core insight of the CBTM is: \textbf{Non-deterministic computation corresponds to algebraic field extension}, when reading a symbol representing a new dimension, the computation must branch into two paths, just as introducing a new element $\alpha$ into the field $\mathbb{Q}$ yields the extension $\mathbb{Q}(\alpha)$. We separate old data from new dimensions via the projection operators $\mathfrak{Re}$ and $\mathfrak{Im}$, and introduce a dual-tape perspective to intuitively decompose abstract algebraic symbols into a real tape (deterministic computation) and an imaginary tape (non-deterministic control). Moreover, the algebraic semantics of the CBTM naturally support arbitrary $k$-way non-determinism: by introducing multiple new dimensions, we can generate high-dimensional algebraic extensions $\mathbb{Q}(\alpha_1,\dots,\alpha_d)$, whose dimension $2^d$ corresponds exactly to the number of branches. We prove that the CBTM is polynomially equivalent to classical Turing machines and non-deterministic Turing machines, with $\mathbf{P}_{cb}=\mathbf{P}$ and $\mathbf{NP}_{cb}=\mathbf{NP}$. Thus, the CBTM does not introduce hyper-computation but provides a new algebraic perspective for understanding the essence of non-determinism. This work serves as the computational model foundation for the series of papers.