On the Finiteness Problem for Automaton (Semi)groups
  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. 
Ali Akhavi, Ines Klimann, Sylvain Lombardy, Jean Mairesse and Matthieu   Picantin
cs.FL math.GR