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

2025-03-07

AI Summary

This paper investigates the word problem complexity and embedding obstructions for groups of reversible cellular automata (CA) over infinite groups. Employing techniques from group theory, symbolic dynamics, computational complexity theory, and the automorphism groups of shifts of finite type, the authors establish several fundamental results under the Gap Conjecture. First, they prove that the word problem for CA groups over non-virtually-nilpotent groups is PSPACE-hard; second, they show tight PSPACE-completeness for free and surface groups; third, they demonstrate co-NEXPTIME-hardness over the lamplighter group. Moreover, the paper fully resolves two major open problems posed by Barbieri–Hochman: (i) non-embeddability of CA groups of non-cyclic free groups or ℤ into CA groups of ℤ, and (ii) a dimensional obstruction D ≥ 3d + 2 for embedding d-dimensional CA groups into lower-dimensional ones.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Causality

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Security and Privacy: Applications of cryptography

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 two 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 $D geq 3d+2$ (this solves a question of Hochman).

Problem

• Analyzes word problems in reversible cellular automata groups.
• Determines PSPACE-hardness of word problems under the Gap Conjecture.
• Proves nonembeddability results for cellular automata groups.

Innovation

• PSPACE-hard word problems under Gap Conjecture
• Nonembeddability results for cellular automata groups
• Co-NEXPTIME-hard word problems on lamplighter group