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Avoshifts

Ville Salo

TL;DR

This work introduces avoshifts, a broad class of subshifts defined by the avo property, and situates them among known classes such as group shifts, k-TEP subshifts, and SFTs with safe symbols. Focusing on polycyclic groups and inductive intervals, it proves that avoshifts are recursively enumerable within SFTs, and provides effective methods for computing projective subdynamics, avofactors, and for deciding equality and inclusion. The framework unifies and extends decidability results to non-algebraic examples and yields new proofs for decidability of languages and inclusion for SFTs on free groups. Central to the theory are notions of equal extension counts, uniform avo, and the constructible, well-behaved family of inductive intervals, which collectively enable SFT-ness and computability in this broad class. The paper also discusses practical implications, potential SAT-based implementations, and directions for extending the theory to broader classes of groups and shape families.

Abstract

An avoshift is a subshift where for each set $C$ from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on $C$ to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets $C$) by for example cellwise quasigroup shifts, TEP subshifts, and subshifts of finite type with a safe symbol. In this paper we concentrate on avoshifts on polycyclic groups, when the sets $C$ are what we call ``inductive intervals''. We show that then avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.

Avoshifts

TL;DR

This work introduces avoshifts, a broad class of subshifts defined by the avo property, and situates them among known classes such as group shifts, k-TEP subshifts, and SFTs with safe symbols. Focusing on polycyclic groups and inductive intervals, it proves that avoshifts are recursively enumerable within SFTs, and provides effective methods for computing projective subdynamics, avofactors, and for deciding equality and inclusion. The framework unifies and extends decidability results to non-algebraic examples and yields new proofs for decidability of languages and inclusion for SFTs on free groups. Central to the theory are notions of equal extension counts, uniform avo, and the constructible, well-behaved family of inductive intervals, which collectively enable SFT-ness and computability in this broad class. The paper also discusses practical implications, potential SAT-based implementations, and directions for extending the theory to broader classes of groups and shape families.

Abstract

An avoshift is a subshift where for each set from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets ) by for example cellwise quasigroup shifts, TEP subshifts, and subshifts of finite type with a safe symbol. In this paper we concentrate on avoshifts on polycyclic groups, when the sets are what we call ``inductive intervals''. We show that then avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.
Paper Structure (19 sections, 44 theorems, 20 equations, 1 figure)

This paper contains 19 sections, 44 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Avoshifts and basic properties
  4. Equal extension counts
  5. Examples of avoshifts
  6. Being SFT in an area
  7. Connections between the (uniform) avo and uniform SFT properties
  8. Polycyclic groups and inductive intervals
  9. Main properties of inductive intervals
  10. SFTness and the avo property in concrete examples
  11. Polycyclic groups
  12. Free group SFTs are avo
  13. More applications of Lemma \ref{['lem:UniformSFT']}
  14. Decidability results
  15. Avofactors
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $G$ be polycyclic, and let $X \subset A^G$ be an avoshift for inductive intervals. Then

Figures (1)

  • Figure 1: The group $\mathbb{Z}^3$ visualized in Minecraft 1.21.1 minecraft, with the first axis pointing right, the second axis forward, and the third axis upward. The inductive interval with axis intervals $([-8, -1], [1, 5], [-64, -1])$ is filled with blocks: a block of birch marks the origin, glass is used to fill on the first axis, cherry tree trunks on the second, and desert sand on the last. Plants and camels appear organically, and serve no mathematical purpose.

Theorems & Definitions (109)

  • Theorem 1.1
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 99 more