[Paper Review] A note on the definition of sliding block codes and the Curtis-Hedlund-Lyndon Theorem

Marcelo Sobottka, Daniel Gonçalves|arXiv (Cornell University)|Jul 8, 2015

Cellular Automata and Applications5 references6 citations

TL;DR

This paper proposes a generalized definition of sliding block codes for shift spaces over countable (possibly infinite) alphabets, where the local rule depends on a variable neighborhood that adapts to the configuration around each site. The key contribution is a revised version of the Curtis-Hedlund-Lyndon Theorem that characterizes generalized sliding block codes as precisely the continuous, translation-commuting maps—restoring the classical equivalence in the infinite alphabet setting.

ABSTRACT

In this note we propose an alternative definition for sliding block codes between shift spaces. This definition coincides with the usual definition in the case that the shift space is defined on a finite alphabet, but it encompass a larger class of maps when the alphabet is infinite. In any case, the proposed definition keeps the idea that a sliding block code is a map with a local rule. Using this new definition we prove that the Curtis-Hedlund-Lyndon Theorem always holds for shift spaces over countable alphabets.

Motivation & Objective

• To address the failure of the classical Curtis-Hedlund-Lyndon Theorem in shift spaces over infinite alphabets due to lack of compactness.
• To resolve the mismatch between continuous, shift-commuting maps and classical sliding block codes when the alphabet is infinite.
• To propose a generalized definition of sliding block codes that retains locality but allows variable neighborhoods depending on the configuration.
• To prove that continuity and translation-commuting properties fully characterize the generalized sliding block codes, restoring the classical theorem's equivalence in the infinite alphabet case.

Proposed method

• Define generalized sliding block codes via a family of finite neighborhoods that depend on the local configuration of the input sequence.
• Re-express classical sliding block codes as maps where each output symbol depends on a fixed finite neighborhood via a local rule.
• Use the topological structure of shift spaces—specifically, clopen cylinders and translation invariance—to analyze continuity and shift-commuting properties.
• Prove that a map is a generalized sliding block code if and only if it is continuous and commutes with all translations, using inverse images of cylinders and their decomposition into unions of cylinders.
• Leverage the fact that preimages of cylinders under continuous, shift-commuting maps are unions of cylinders, ensuring openness and continuity.
• Show that the generalized definition reduces to the classical one when the alphabet is finite, preserving consistency with prior theory.

Experimental results

Research questions

• RQ1 Can the classical definition of sliding block codes be extended to shift spaces over infinite alphabets while preserving the locality principle?
• RQ2 Why does the classical Curtis-Hedlund-Lyndon Theorem fail for infinite alphabets, and what structural property is missing?
• RQ3 Is there a broader class of maps that still satisfies the continuity and shift-commuting conditions but cannot be captured by the classical definition?
• RQ4 Can a revised version of the Curtis-Hedlund-Lyndon Theorem be established for infinite alphabets using a generalized definition of sliding block codes?
• RQ5 What is the relationship between continuity, translation invariance, and the existence of a local rule in the infinite alphabet setting?

Key findings

• The proposed generalized sliding block code definition extends the classical one and applies naturally to shift spaces over countably infinite alphabets.
• The generalized definition maintains the intuitive idea of a local rule, but allows the neighborhood size and shape to vary depending on the configuration around each site.
• A map between shift spaces is a generalized sliding block code if and only if it is continuous and commutes with all translations.
• This revised characterization restores the equivalence of the Curtis-Hedlund-Lyndon Theorem in the infinite alphabet case, where the classical version fails due to lack of uniform continuity.
• The proof relies on the fact that preimages of cylinders under continuous, shift-commuting maps are unions of cylinders, hence open, ensuring continuity.
• The example of the map $\Phi(\mathbf{x})_j = \max_{i \geq j} x_i$ is continuous and shift-commuting but not a generalized sliding block code, illustrating the necessity of the new definition.