Efficient Compression in Semigroups

TL;DR

This paper characterizes which classes of finite semigroups allow efficient data compression using straight-line programs, improving bounds and solving a longstanding conjecture on solvable groups' membership problem complexity.

Contribution

It completes the classification of pseudovarieties of finite semigroups with efficient compression and improves bounds on straight-line program parameters.

Findings

Characterized pseudovarieties of finite semigroups with efficient compression.

Improved bounds on straight-line program length and width.

Proved membership problem for all solvable groups is in FOLL.

Abstract

Straight-line programs are a central tool in several areas of computer science, including data compression, algebraic complexity theory, and the algorithmic solution of algebraic equations. In the algebraic setting, where straight-line programs can be interpreted as circuits over algebraic structures such as semigroups or groups, they have led to deep insights in computational complexity. A key result by Babai and Szemer\'edi (1984) showed that finite groups afford efficient compression via straight-line programs, enabling the design of a black-box computation model for groups. Building on their result, Fleischer (2019) placed the Cayley table membership problem for certain classes (pseudovarieties) of finite semigroups in NPOLYLOGTIME, and in some cases even in FOLL. He also provided a complete classification of pseudovarieties of finite monoids affording efficient compression. In this work, we complete this classification program initiated by Fleischer, characterizing precisely those pseudovarieties of finite semigroups that afford efficient compression via straight-line programs. Along the way, we also improve several known bounds on the length and width of straight-line programs over semigroups, monoids, and groups. These results lead to new upper bounds for the membership problem in the Cayley table model: for all pseudovarieties that afford efficient compression and do not contain any nonsolvable group, we obtain FOLL algorithms. In particular, we resolve a conjecture of Barrington, Kadau, Lange, and McKenzie (2001), showing that the membership problem for all solvable groups is in FOLL.