[Paper Review] Computable Følner monotilings and a theorem of Brudno I

Nikita Moriakov|arXiv (Cornell University)|Sep 25, 2015

Computability, Logic, AI Algorithms6 references3 citations

TL;DR

This paper extends Brudno's theorem linking topological entropy and Kolmogorov complexity to subshifts over computable amenable groups by introducing computable Følner monotilings—a constructive, algorithmic version of Weiss’s Følner monotilings. It proves that for such groups, including ℤᵈ and UT_d(ℤ), the topological entropy of a subshift equals the asymptotic Kolmogorov complexity of its word presheaf, generalizing Brudno’s original result to a broad class of groups with effective structure.

ABSTRACT

The purpose of this article is to extend the earliest results of A.A. Brudno, connecting topological entropy of a subshift X over $\mathbb{N}$ to the Kolmogorov complexity of words in X, to subshifts over computable groups that posses computable Følner monotilings, which we introduce in this work. The classical examples of such groups are the groups $\mathbb{Z}^d$ and the groups of upper-triangular matrices with integer entries. Following the work of B. Weiss we show that the class of such groups is closed under group extensions.

Motivation & Objective

• To generalize A.A. Brudno’s result connecting topological entropy and Kolmogorov complexity from ℕ to subshifts over computable amenable groups.
• To introduce and formalize the concept of computable Følner monotilings as an effective, algorithmic counterpart to Weiss’s Følner monotilings.
• To establish that the class of computable groups admitting computable Følner monotilings is closed under computable group extensions.
• To prove that for subshifts over computable groups with computable normal Følner monotilings, topological entropy equals the asymptotic Kolmogorov complexity of the associated word presheaf.
• To provide explicit algorithms for computable Følner monotilings in key examples, including ℤᵈ and UT_d(ℤ), demonstrating their constructibility.

Proposed method

• Introduces computable spaces, morphisms between them, and word presheaves to formalize the structure of subshifts over groups.
• Defines the Kolmogorov complexity of sections of word presheaves and introduces asymptotic Kolmogorov complexity as a key measure.
• Develops the notion of a computable Følner monotiling as a computable, normal, and effective tiling of a group using Følner sets.
• Constructs a decompressor algorithm that encodes group-labeled words using dictionary-based encoding on interior sets and remainder encoding on boundary sets.
• Uses Følner sequences to bound the fraction of boundary elements, ensuring that the complexity of encoded words approaches the entropy rate.
• Applies the decompressor to show that the limsup of normalized Kolmogorov complexity is bounded by the entropy, leading to equality in the limit.

Experimental results

Research questions

• RQ1 Can Brudno’s entropy-complexity equivalence be extended from ℕ to subshifts over more general amenable groups, such as ℤᵈ and nilpotent groups?
• RQ2 What effective, algorithmic structure is needed to ensure that Kolmogorov complexity and topological entropy remain asymptotically equivalent in higher-dimensional or non-abelian group actions?
• RQ3 How can the notion of Følner monotilings be made computable in a way that preserves the essential properties needed for entropy-complexity duality?
• RQ4 Do computable Følner monotilings exist for important classes of groups such as UT_d(ℤ), and can they be explicitly constructed?
• RQ5 Is the class of computable groups admitting computable Følner monotilings closed under computable group extensions?

Key findings

• The topological entropy of a subshift over a computable group with a computable normal Følner monotiling equals the asymptotic Kolmogorov complexity of its associated word presheaf.
• For every d ∈ ℕ, the group ℤᵈ admits a computable Følner monotiling with explicitly constructible algorithms.
• The group UT_d(ℤ) of upper-triangular integer matrices also admits a computable Følner monotiling, extending the result to non-abelian nilpotent groups.
• The class of computable groups admitting computable Følner monotilings is closed under computable group extensions, ensuring broad applicability.
• The proof establishes that the limsup of normalized Kolmogorov complexity is bounded by any ε > 0 and a term vanishing as |F_k| → ∞, leading to equality in the limit.
• The requirement of normality in the Følner monotiling is non-restrictive, as every computable Følner monotiling can be transformed into a computable normal one.