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 $α$ into the field $\mathbb{Q}$ yields the extension $\mathbb{Q}(α)$. 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 $\mathbb{Q}(α_1,\dots,α_d)$, 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.

翻译：传统图灵机的语义贫乏，它们仅关注符号的语法操作，而忽略了符号背后的数学语义。这种语义缺陷被认为是三大障碍（相对化、自然证明和代数化）的根本原因。本文提出复数布尔图灵机（CBTM），将计算符号提升为$\mathrm{GF}(4)$中的代数元素，使得每个操作都具有清晰的数学解释。CBTM的核心洞见是：\textbf{非确定性计算对应代数域扩张}，当读取代表新维度的符号时，计算必须分支为两条路径，正如向域$\mathbb{Q}$引入新元素$α$得到扩张$\mathbb{Q}(α)$。我们通过投影算子$\mathfrak{Re}$和$\mathfrak{Im}$将旧数据与新维度分离，并引入双带视角，将抽象代数符号直观分解为实带（确定性计算）和虚带（非确定性控制）。此外，CBTM的代数语义天然支持任意$k$路非确定性：通过引入多个新维度，可生成高维代数扩张$\mathbb{Q}(α_1,\dots,α_d)$，其维度$2^d$恰好对应分支数量。我们证明CBTM与经典图灵机及非确定性图灵机多项式等价，有$\mathbf{P}_{cb}=\mathbf{P}$且$\mathbf{NP}_{cb}=\mathbf{NP}$。因此，CBTM并未引入超计算，而是为理解非确定性的本质提供了新的代数视角。本工作是该系列论文的计算模型基础。