Affine automata provide a finite-state computational model that preserves the linear-algebraic structure of quantum computation while operating entirely over the reals. Recent work has shown that affine automata can far surpass classical probabilistic finite-state verifiers. However, prior constructions relied on arbitrary real-valued transition matrices, leaving open whether the observed power stems from the affine mechanism itself or from computational resources implicitly encoded in irrational or infinite-precision parameters. This paper studies one-way and two-way automata with deterministic and affine states as verifiers in Arthur--Merlin proof systems under the restriction that every affine transition matrix has rational entries, and shows that the resulting rational model still supports the main verification advantages of affine finite-state verification. At the one-way level, we verify benchmark nonregular languages that are provably hard or impossible for classical two-way probabilistic verifiers. At the two-way level, we achieve weak verification of every Turing-recognizable language, strong bounded-error verification for every language in $\mathbf{ATIME}(2^{O(n)})$, and perfect-completeness strong verification for every language in $\mathbf{PSPACE}$. These results establish that the remarkable verification power of affine finite-state automata is structural.

翻译：仿射自动机提供了一种有限状态计算模型，它保留了量子计算的线性代数结构，同时完全在实数域上运行。近期研究表明，仿射自动机在验证能力上可远超经典概率有限状态验证者。然而，先前的构造依赖于任意实数值转移矩阵，由此留下一个开放性问题：所观察到的强大能力究竟源于仿射机制本身，还是隐式编码在无理数或无限精度参数中的计算资源。本文研究在阿瑟-梅林证明系统中，使用具有确定性和仿射状态的一向及双向自动机作为验证者，并施加每条仿射转移矩阵元素均为有理数的限制，证明该有理数模型仍能支撑仿射有限状态验证的主要优势。在一向层面上，我们能够验证那些被证明对经典双向概率验证者而言困难或不可能的基准非正则语言。在双向层面上，我们实现了对每个图灵可识别语言的弱验证、对$\mathbf{ATIME}(2^{O(n)})$中每个语言的有界错误强验证，以及对$\mathbf{PSPACE}$中每个语言的完美完备性强验证。这些结果确立了仿射有限状态自动机的非凡验证能力源于其结构性。