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

Angshul Majumdar

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.

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

Abstract

We study the exact decision problem for feedback capacity of finite-state channels (FSCs). Given an encoding of a binary-input binary-output rational unifilar FSC with specified rational initial distribution, and a rational threshold , we ask whether the feedback capacity satisfies . 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 (), 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.
Paper Structure (62 sections, 26 theorems, 96 equations)

This paper contains 62 sections, 26 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Exact feedback-capacity thresholding as a decision problem
  3. Historical context and the FSC feedback-capacity literature
  4. What is proved here and how it is positioned
  5. (1) Undecidability of exact thresholding.
  6. (2) A barrier to universal real-algebraic exact formulations.
  7. (3) Computability-theoretic consequences for exact certification.
  8. (4) Gödel--Tarski--Löb consequences.
  9. What this paper does not claim
  10. An inconspicuous logic bridge
  11. Preliminaries and notation
  12. Basic notation and encodings
  13. Finite-state channels and the unifilar subclass
  14. Feedback model.
  15. Unifilar FSCs.
  16. ...and 47 more sections

Key Result

Lemma 4.1

Fix $N\ge 1$. For either channel $\mathsf{Ch}^{(N)}_{\mathrm{good}}$ or $\mathsf{Ch}^{(N)}_{\mathrm{bad}}$, and for any causal strategy $p(x^n\Vert y^{n-1})$, we have Consequently, and hence

Theorems & Definitions (73)

  • Definition 2.1: Restricted channel class
  • Definition 2.2: Unifilar FSC
  • Definition 2.3: Computable function
  • Definition 2.4: Recursively enumerable set
  • Definition 2.5: Recursively axiomatizable theory
  • Definition 2.6: Provability predicate
  • Definition 3.1: Feedback capacity threshold problem
  • Definition 3.2: Directed information
  • Definition 3.3: Finite-horizon directed-information value
  • Remark 3.4: Compact semialgebraic policy space
  • ...and 63 more