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Word problems and embedding-obstructions in cellular automata groups on groups

Ville Salo

TL;DR

This work analyzes word problems and embeddability for automorphism groups of subshifts on countable groups, revealing a growth-driven complexity landscape: polynomial-growth bases place word problems in co-NP, while faster growth (under the Gap Conjecture) yields PSPACE-hardness; free and surface groups admit PSPACE upper bounds, and the lamplighter group attains co-NEXPTIME-completeness in natural settings. The authors introduce ripple catching, a powerful construction that uses cone-based propagation to encode computations within automorphism actions, and develop arboreous graph frameworks to realize PSPACE-hardness in broad classes of groups. They establish key non-embedding results (e.g., dimension and center-related obstructions) and derive upper bounds via polynomial splitting schemes for free and surface groups, yielding a cohesive complexity map for $ ext{Aut}(X)$ across diverse groups and subshifts. The work also identifies several open questions on embeddings, growth-type limitations, and the reach of the splitting-scheme methodology, pointing to rich interactions between symbolic dynamics, group theory, and computational complexity.

Abstract

We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are in co-NP, and can be co-NP-hard. We show that under the Gap Conjecture of Grigorchuk, their word problems are PSPACE-hard on all other groups. On free and surface groups, we show that they are indeed always in PSPACE. On a group with co-NEXPTIME word problem, CA groups themselves have co-NEXPTIME word problem, and on the lamplighter group (which itself has polynomial-time word problem) we show they can be co-NEXPTIME-hard. We show also nonembeddability results: the group of cellular automata on a non-cyclic free group does not embed in the group of cellular automata on the integers (this solves a question of Barbieri, Carrasco-Vargas and Rivera-Burgos); and the group of cellular automata in dimension $D$ does not embed in a group of cellular automata in dimension $d$ if $D > d$ (this solves a question of Hochman).

Word problems and embedding-obstructions in cellular automata groups on groups

TL;DR

This work analyzes word problems and embeddability for automorphism groups of subshifts on countable groups, revealing a growth-driven complexity landscape: polynomial-growth bases place word problems in co-NP, while faster growth (under the Gap Conjecture) yields PSPACE-hardness; free and surface groups admit PSPACE upper bounds, and the lamplighter group attains co-NEXPTIME-completeness in natural settings. The authors introduce ripple catching, a powerful construction that uses cone-based propagation to encode computations within automorphism actions, and develop arboreous graph frameworks to realize PSPACE-hardness in broad classes of groups. They establish key non-embedding results (e.g., dimension and center-related obstructions) and derive upper bounds via polynomial splitting schemes for free and surface groups, yielding a cohesive complexity map for across diverse groups and subshifts. The work also identifies several open questions on embeddings, growth-type limitations, and the reach of the splitting-scheme methodology, pointing to rich interactions between symbolic dynamics, group theory, and computational complexity.

Abstract

We study groups of reversible cellular automata, or CA groups, on groups. More generally, we consider automorphism groups of subshifts of finite type on groups. It is known that word problems of CA groups on virtually nilpotent groups are in co-NP, and can be co-NP-hard. We show that under the Gap Conjecture of Grigorchuk, their word problems are PSPACE-hard on all other groups. On free and surface groups, we show that they are indeed always in PSPACE. On a group with co-NEXPTIME word problem, CA groups themselves have co-NEXPTIME word problem, and on the lamplighter group (which itself has polynomial-time word problem) we show they can be co-NEXPTIME-hard. We show also nonembeddability results: the group of cellular automata on a non-cyclic free group does not embed in the group of cellular automata on the integers (this solves a question of Barbieri, Carrasco-Vargas and Rivera-Burgos); and the group of cellular automata in dimension does not embed in a group of cellular automata in dimension if (this solves a question of Hochman).

Paper Structure

This paper contains 28 sections, 65 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Complexity-theoretic results
  3. Non-embeddability results
  4. Definitions, conventions and some standard observations
  5. Preliminaries
  6. Groups
  7. Symbolic dynamics
  8. $\mathrm{PAut}$
  9. Ripple catching
  10. Nonembedding results
  11. Dropping the centers
  12. Embeddings into other groups
  13. Groups with free submonoids
  14. Simplistic growth lower bound
  15. PSPACE-hardness of the word problem
  16. ...and 13 more sections

Key Result

Theorem A

Let $G$ be an f.g. group with polynomial growth, and let $X \subset A^G$ be any subshift of finite type with language in NP. Then every finitely-generated subgroup of $\mathrm{Aut}(X)$ has word problem in co-NP. Furthermore, for any infinite f.g. group $G$ with polynomial growth, $\mathrm{Aut}(A^G)$

Theorems & Definitions (129)

  • Theorem A
  • Theorem B
  • Conjecture 1.1
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Theorem H
  • Lemma 2.1
  • ...and 119 more