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Rationality and computability of the covering radius for sofic shifts

This paper establishes that the covering radius of a primitive sofic shift is a rational number and provides an algorithm to compute it from a labeled graph presentation.

Original authors: Tom Meyerovitch, Aidan Young

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tom Meyerovitch, Aidan Young

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a secret message across a noisy radio channel. Sometimes, static interferes, and a "0" gets heard as a "1," or vice versa. In the world of data transmission, mathematicians ask a crucial question: How much noise can our system handle before we can no longer tell what the original message was?

This paper, by Tom Meyerovitch and Aidan Young, tackles a specific version of this problem involving a type of mathematical structure called a Sofic Shift. While that sounds intimidating, let's break it down using a simple story.

The Story: The "Noisy Walk" Game

1. The Setting: A City with Rules

Imagine a city where you can only walk on specific streets. You can't just go anywhere; there are rules. For example, you can't turn left twice in a row, or you must stop at every red light.

The Sofic Shift: This is the city itself. It represents all the valid paths (or messages) you are allowed to take.
The Covering Radius: This is the "safety margin." It answers: If I make a mistake and take a wrong turn (a noisy signal), how far away is the nearest valid path I could have taken?

If the covering radius is small, the system is robust; even with a few mistakes, you can easily figure out what the intended path was. If it's large, the system is fragile.

2. The Big Questions

For a long time, mathematicians knew how to calculate this safety margin for simple, specific cities. But for complex cities (called "primitive sofic shifts"), they had two big doubts:

Is the answer always a "clean" number? (Like 1.5 or 2/3, rather than a messy, infinite decimal like $\pi$ or $\sqrt{2}$ ).
Can we build a machine (an algorithm) to calculate this number for any city, no matter how complex?

Before this paper, people guessed the answer was "yes," but no one could prove it.

3. The Solution: A Two-Player Game

The authors solved this by turning the problem into a game between two players, Alice and Bob.

Alice wants to create a path through the city that is as "confusing" as possible. She wants to pick a route that is far away from any valid path, maximizing the distance (the noise).
Bob wants to be a detective. He sees Alice's confusing path and tries to find the closest valid path to correct her mistakes. He wants to minimize the distance.

The Covering Radius is the final score of this game when they play it forever. It's the point where Alice has done her best to confuse Bob, and Bob has done his best to correct her.

4. The "Tropical" Secret Sauce

To solve the game, the authors used a clever mathematical trick they call "Tropical Convolution."

Think of this like a Lego building game with special rules:

Normally, when you combine two Lego structures, you add their heights.
In this "Tropical" world, when you combine two structures, you don't add the heights; you take the minimum height and add the cost.

This sounds weird, but it's perfect for this problem. It allows the authors to break a giant, infinite path into tiny pieces, calculate the "cost" of each piece, and then snap them back together to find the total cost. It's like calculating the price of a long road trip by summing up the cheapest gas prices for every single mile, rather than trying to calculate the whole trip at once.

The Main Discoveries

Using this game and the Lego trick, the authors proved two amazing things:

The Answer is Always "Clean" (Rational):
No matter how complex the city (the Sofic Shift) is, the safety margin (Covering Radius) will always be a rational number. It will be a fraction like $3/4$ or $7/2$ . It will never be a messy, irrational number. This is huge because it means the system behaves in a predictable, orderly way.
There is a Recipe (Algorithm):
They didn't just prove the number exists; they wrote down a step-by-step recipe (an algorithm) that a computer can follow to find this number in a finite amount of time. You can feed the map of the city into the computer, and it will spit out the exact safety margin.

Why Does This Matter?

You might wonder, "Who cares about a math game?"

Data Storage: When you save a file on a hard drive or send a text message, errors happen. This math helps engineers design codes that are as efficient as possible while still being able to fix those errors.
Predictability: Knowing that the answer is always a "clean" fraction gives engineers confidence that they can build reliable systems without worrying about unpredictable, chaotic mathematical behavior.

The Takeaway

Meyerovitch and Young took a messy, infinite problem about noisy data transmission and showed that, deep down, it follows simple, orderly rules. They proved that for a huge class of data systems, the "safety margin" is always a simple fraction, and we can always calculate it exactly. They turned a chaotic puzzle into a solvable game.

1. Problem Statement

The paper addresses two fundamental questions regarding the covering radius of sofic shifts (a class of symbolic dynamical systems used to model constrained data transmission and storage):

Rationality: Is the covering radius of a primitive sofic shift always a rational number?
Computability: Is there a general algorithm to compute this covering radius from a labeled graph presentation of the shift?

While the covering radius had been computed for specific examples (like run-length-limited shifts) in previous literature, these calculations were often ad-hoc, and the general properties of the quantity for arbitrary sofic shifts were unknown.

2. Methodology

The authors employ a multi-disciplinary approach combining symbolic dynamics, combinatorics, and game theory. The core methodology involves:

Game-Theoretic Formulation: The authors model the problem as a zero-sum two-player mean-payoff game (specifically, a "non-alternating" variant).
- Player Alice chooses a walk on a graph $H$ (representing the space of all possible sequences).
- Player Bob observes Alice's choice and selects a walk on a graph $G$ (representing the shift space).
- The payoff is determined by a function $P$ measuring the "distance" (Hamming distance) between the labels of the two walks.
- The value of this game, $V(G, H, P)$ , corresponds exactly to the covering radius of the shift space defined by $G$ .
Tropical Convolution: To analyze the game, the authors introduce a binary operation called tropical convolution on the space of integer-valued functions.
- Defined as $(f * g)(u, w) = \min_{v} (f(u, v) + g(v, w))$ .
- This operation is associative and acts as the analog of classical convolution within the min-plus semiring $(\mathbb{Z} \cup \{+\infty\}, \min, +)$ .
- This algebraic structure allows the authors to analyze the concatenation of walks and the evolution of "cost" functions over time.
Analysis of "Non-Improvable" Walks: The authors define a set of non-improvable walks (optimal strategies) where the cost function is maximal in a specific partial order. They prove that the set of these optimal strategies forms a sofic shift itself.
Compactness and Periodicity: By leveraging the finiteness of the state space and the primitivity of the graphs, they show that the sequence of optimal cost functions eventually becomes periodic (up to a constant shift). This periodicity is the key to proving rationality and computability.

3. Key Contributions and Results

Main Theorems

Theorem A (Rationality): Let $G = (G, L)$ be a primitive labeled graph. The covering radius $R(X_G)$ of the corresponding edge shift is a rational number.
Theorem B (Computability): There exists an algorithm that takes a primitive labeled graph $G$ as input and computes $R(X_G)$ in finite time.

Intermediate Results

Theorem C (Game Value): For primitive graphs $G, H$ and an integer payoff function $P$ , the value of the non-alternating mean-payoff game $V(G, H, P)$ is a rational number and is computable.
Theorem F (Soficity of Optimal Strategies): The shift spaces representing the "optimal strategies" for both players in the game (denoted $X^\dagger_H$ and $Y^\dagger_{(H,G)}$ ) are themselves sofic shifts. This is a non-trivial result, as optimal strategy spaces are not always of finite type.
Corollary 5.4 (Periodic Strategies): The Nash equilibrium of the game can be achieved by periodic strategies. There exist periodic walks for both players that achieve the game value.

4. Technical Details of the Proof

The proof of Theorem C (which implies A and B) proceeds as follows:

Finite State Space: The authors define a function $M(p)$ representing the minimal cost for Bob to counter a walk $p$ chosen by Alice. They show that for sufficiently long walks, these functions take values in a bounded integer range.
Pigeonhole Principle: Because the set of possible "normalized" cost functions is finite, the sequence of these functions must eventually repeat.
Periodicity: They prove that there exist integers $N$ and $k$ such that the set of optimal cost functions at step $n+k$ is simply the set at step $n$ shifted by a constant value $V^\dagger$ .
Rationality: The game value is derived as $V^\dagger / k$ , which is necessarily rational.
Algorithm: The algorithm simply computes the sequence of optimal cost functions until the periodicity condition is detected, then outputs the ratio.

5. Significance and Implications

Theoretical Foundation: The paper resolves open questions regarding the arithmetic nature of the covering radius for a broad class of constrained systems. It establishes that despite the infinite nature of shift spaces, this specific metric is always rational.
Algorithmic Utility: The existence of a finite-time algorithm allows for the precise calculation of error-correction capabilities in constrained coding systems (e.g., magnetic recording, flash memory) modeled by sofic shifts.
Game Theory Connection: The work bridges symbolic dynamics and game theory, demonstrating that "non-alternating" mean-payoff games have rational values and periodic optimal strategies, extending the known results for alternating games.
Structural Insight: By proving that the space of optimal strategies is a sofic shift, the authors provide a structural understanding of how optimal error-correction codes behave within these systems.

6. Limitations and Open Questions

The authors note that their results rely on the primitivity of the underlying graphs.

Reducible Graphs: The results may not hold for general reducible graphs without modification (though they suggest a decomposition approach might work).
Real-Valued Payoffs: The current proofs assume integer (or rational) payoff functions. The rationality and computability of the covering radius for real-valued payoff functions remain open problems.
Algebraic Extensions: It is an open question whether the value remains in the same algebraic field (e.g., $\mathbb{Q}(\sqrt{2})$ ) if the payoff function takes values in such a field.

In summary, this paper provides a rigorous combinatorial and game-theoretic framework proving that the covering radius of primitive sofic shifts is a computable rational number, offering both theoretical clarity and practical algorithms for information theory applications.