Shannon meets G\"{o}del-Tarski-L\"{o}b: Undecidability of Shannon Feedback Capacity for Finite-State Channels

TL;DR

This paper proves that determining the exact feedback capacity threshold for finite-state channels is undecidable, revealing fundamental limits in information theory and formal logic.

Contribution

It establishes the undecidability of the exact feedback capacity threshold problem for finite-state channels, a significant theoretical limitation.

Findings

Exact threshold problem is undecidable for rational unifilar FSCs.

The problem does not lie in the existential theory of the reals.

No universal algorithm exists for all instances of the problem.

Abstract

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 $C_{fb}(W_e, \pi_{1,e}) \ge q$. 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 G\"odel-Tarski-L\"ob incompleteness phenomena for sufficiently expressive formal theories capable of representing the threshold predicate.