We study the exact decision problem for feedback capacity of finite-state channels (FSCs). Given an encoding $e$ of a binary-input binary-output rational unifilar FSC with specified rational initial distribution, and a rational threshold $q$, we ask whether the feedback capacity satisfies $C_{fb}(W_e, π_{1,e}) \ge q$. We prove that this exact threshold problem is undecidable, even when restricted to a severely constrained class of rational unifilar FSCs with bounded state space. The reduction is effective and preserves rationality of all channel parameters. As a structural consequence, the exact threshold predicate does not lie in the existential theory of the reals ($\exists\mathbb{R}$), and therefore cannot admit a universal reduction to finite systems of polynomial equalities and inequalities over the real numbers. In particular, there is no algorithm deciding all instances of the exact feedback-capacity threshold problem within this class. These results do not preclude approximation schemes or solvability for special subclasses; rather, they establish a fundamental limitation for exact feedback-capacity reasoning in general finite-state settings. At the metatheoretic level, the undecidability result entails corresponding Gödel-Tarski-Löb incompleteness phenomena for sufficiently expressive formal theories capable of representing the threshold predicate.

翻译：本文研究有限状态信道反馈容量的精确判定问题。给定具有指定有理初始分布的二元输入二元输出有理单峰有限状态信道的编码$e$，以及有理阈值$q$，我们探讨其反馈容量是否满足$C_{fb}(W_e, π_{1,e}) \ge q$。我们证明该精确阈值问题是不可判定的，即使将其限制在状态空间有界的强约束有理单峰有限状态信道类中亦然。该归约是有效的，且保持所有信道参数的有理性。作为结构推论，精确阈值谓词不属于实数存在理论（$\exists\mathbb{R}$），因此无法通过有限实数多项式等式与不等式系统进行普适归约。特别地，不存在能够判定此类信道所有精确反馈容量阈值实例的算法。这些结果并不排除近似方案或特定子类的可解性，而是确立了广义有限状态场景中精确反馈容量推理的根本性局限。在元理论层面，该不可判定性结果意味着：对于足以表达该阈值谓词的充分形式化理论，将对应产生哥德尔-塔斯基-勒布型不完备现象。