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Streaming word problems

Markus Lohrey, Lukas Lück, Julio Xochitemol

TL;DR

The paper investigates streaming algorithms for the word problem in finitely generated groups, introducing a framework that combines randomized streaming with group growth and structure. It shows that for several important classes—finitely generated linear groups, metabelian groups, and free solvable groups—WP can be solved in logarithmic space using randomized streaming algorithms, and it establishes closure properties under finite extensions, graph products, and wreath products by free abelian groups. It identifies strong lower bounds in cases such as Thompson's group F and analyzes the Grigorchuk group to illustrate gaps between deterministic and randomized space complexities. The work also extends to subgroup membership problems in free groups and direct products, and it outlines open questions related to hyperbolic groups, residual finiteness, and graphs of groups. Overall, the paper connects growth, algebraic structure, and streaming complexity to derive both upper and lower bounds and to guide future explorations in group-theoretic streaming computation.

Abstract

We study deterministic and randomized streaming algorithms for word problems of finitely generated groups. For finitely generated linear groups, metabelian groups and free solvable groups we show the existence of randomized streaming algorithms with logarithmic space complexity for their word problems. We also show that the class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group theoretical constructions: finite extensions, graph products and wreath products by free abelian groups. We contrast these results with several lower bound. An example of a finitely presented group, where the word problem has only a linear space randomized streaming algorithm, is Thompson's group $F$. Finally, randomized streaming algorithms for subgroup membership problems in free groups and direct products of free groups are studied.

Streaming word problems

TL;DR

The paper investigates streaming algorithms for the word problem in finitely generated groups, introducing a framework that combines randomized streaming with group growth and structure. It shows that for several important classes—finitely generated linear groups, metabelian groups, and free solvable groups—WP can be solved in logarithmic space using randomized streaming algorithms, and it establishes closure properties under finite extensions, graph products, and wreath products by free abelian groups. It identifies strong lower bounds in cases such as Thompson's group F and analyzes the Grigorchuk group to illustrate gaps between deterministic and randomized space complexities. The work also extends to subgroup membership problems in free groups and direct products, and it outlines open questions related to hyperbolic groups, residual finiteness, and graphs of groups. Overall, the paper connects growth, algebraic structure, and streaming complexity to derive both upper and lower bounds and to guide future explorations in group-theoretic streaming computation.

Abstract

We study deterministic and randomized streaming algorithms for word problems of finitely generated groups. For finitely generated linear groups, metabelian groups and free solvable groups we show the existence of randomized streaming algorithms with logarithmic space complexity for their word problems. We also show that the class of finitely generated groups with a logspace randomized streaming algorithm for the word problem is closed under several group theoretical constructions: finite extensions, graph products and wreath products by free abelian groups. We contrast these results with several lower bound. An example of a finitely presented group, where the word problem has only a linear space randomized streaming algorithm, is Thompson's group . Finally, randomized streaming algorithms for subgroup membership problems in free groups and direct products of free groups are studied.
Paper Structure (23 sections, 40 theorems, 105 equations, 2 figures, 3 algorithms)

This paper contains 23 sections, 40 theorems, 105 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Communication complexity
  4. Probabilistic finite automata
  5. Sequential transducer
  6. Streaming algorithms: definitions
  7. Streaming algorithms: general results
  8. Groups and word problems
  9. Streaming algorithms for word problems and growth
  10. Comparison to sofic groups
  11. Distinguishers for groups
  12. Randomized streaming algorithms for linear groups
  13. Closure properties for the space complexity of distinguishers
  14. Finitely generated subgroups, finite extensions, direct products
  15. Randomized streaming algorithms for graph products
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $\delta > 0$, $p \in [0,1]_{\mathbb{R}}$, and $X_1, X_2, \ldots X_k$ be independent identically distributed Bernoulli random variables with $\mathop{\mathsf{Prob}}[X_i = 1] = \epsilon$ and $\mathop{\mathsf{Prob}}[X_i = 0] = 1-\epsilon$ for all $i$. Then we have:

Figures (2)

  • Figure 1: An $\mathcal{A}_G$-factorization of type (i) for $k=4$. The red-blue loops outside of $\mathcal{A}_G$ are loops in the Cayley graph of the free group $F(\Gamma)$.
  • Figure 2: An $\mathcal{A}_G$-factorization of type (ii) for $k=4$. The red-blue loops outside of $\mathcal{A}_G$ are loops in the Cayley graph of the free group $F(\Gamma)$.

Theorems & Definitions (75)

  • Theorem 2.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof
  • Lemma 5.1
  • ...and 65 more