Streaming algorithms for groups and semigroups

TL;DR

This paper develops and analyzes streaming algorithms, called distinguishers, for solving word problems in groups and semigroups, achieving low space complexity and establishing lower bounds for certain structures.

Contribution

It introduces the notion of distinguishers for word problems, constructs low-space algorithms for various algebraic structures, and proves lower bounds for others.

Findings

Low-space distinguishers for linear and certain semigroup classes

Achieved $ ext{O}(rac{ ext{log log } n)}$ space complexity for commutative semigroups

Proved lower bounds for free inverse monoids and Thompson's group F

Abstract

We investigate deterministic and randomized streaming algorithms for word problems in finitely generated groups and semigroups. For this we introduce the notion of a distinguisher: a randomized streaming algorithm that processes two input words in parallel and, with high probability, reaches identical memory states if the words represent the same element, and distinct states otherwise. We construct such distinguishers with low error probability using logarithmic, and in some cases doubly logarithmic, space. For example, our results apply to linear semigroups and to semigroups obtained (under suitable restrictions) via standard constructions such as graph products, wreath products, and semilattice decompositions. In case of commutative semigroups and cancellative nilpotent semigroups, we achieve space complexity $\mathcal{O}(\log \log n)$. We complement these upper bounds with lower bounds demonstrating that certain well-known semigroups do not admit sublinear-space distinguishers. This includes, for example, free inverse monoids of rank at least two and Thompson's group $F$. Finally, we study randomized streaming algorithms for subgroup membership problems in free groups and their direct products.