On the Finiteness Problem for Automaton (Semi)groups

TL;DR

This paper investigates the finiteness problem for automaton (semi)groups, providing new effective criteria for semigroups and groups, and testing these criteria on small automata to evaluate their efficiency.

Contribution

It introduces effective sufficient and necessary conditions for finiteness in automaton semigroups and groups, with a comprehensive analysis and testing on small automata.

Findings

Effective sufficient condition for semigroup finiteness.

Effective necessary condition for group finiteness.

Testing criteria on small automata demonstrates their practical efficiency.

Abstract

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskii, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.