Language Classes Associated With Automata Over Matrix Groups

TL;DR

This paper explores the computational power of automata over matrix groups, revealing differences based on matrix types, and introduces time complexity considerations for these automata.

Contribution

It demonstrates that automata over rational matrix groups are more powerful than those over integer matrix groups and introduces a time complexity framework for group automata.

Findings

Rational matrix group automata outperform integer matrix group automata.

Finite automata over specific matrix groups like the Heisenberg and Baumslag-Solitar groups are analyzed.

Time complexity distinctions among group automata classes are established.

Abstract

We investigate the language classes recognized by group automata over matrix groups. For the case of $2 \times 2 $ matrices, we prove that the corresponding group automata for rational matrix groups are more powerful than the corresponding group automata for integer matrix groups. Finite automata over some special matrix groups, such as the discrete Heisenberg group and the Baumslag-Solitar group are also examined. We also introduce the notion of time complexity for group automata and demonstrate some separations among related classes. The case of linear-time bounds is examined in detail throughout our repertory of matrix group automata.