Shannon meets Gödel-Tarski-Löb: Undecidability of Shannon Feedback Capacity for Finite-State Channels

2026-03-17

AI Summary

This study addresses the exact decidability of whether the feedback capacity of a rational unidirectional finite-state channel with binary input and output exceeds a given rational threshold, under a rational initial state distribution. By leveraging computability theory and effective reduction techniques—while preserving the rationality of all channel parameters—the authors construct instances for which this decision problem is undecidable. They establish, for the first time, that the problem is fundamentally undecidable and reveal its deep connection to Gödel–Tarski–Löb incompleteness phenomena, demonstrating that it cannot be reduced to solving polynomial systems over the real numbers. This result underscores an intrinsic theoretical limitation: even within this restricted class of channels, precisely determining whether feedback capacity meets a specified rational threshold is algorithmically impossible.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Decision/Utility Theory
• Machine Learning: Information Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 $C_{fb}(W_e, π_{1,e}) \ge q$. 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.

Problem

• feedback capacity
• finite-state channels
• undecidability
• threshold problem
• exact decision problem

Innovation

• undecidability
• feedback capacity
• finite-state channels
• existential theory of the reals
• Gödel-Tarski-Löb incompleteness