A cyclic graph language is accepted by a Cayley automaton  with pebbles if and only if their orders can be recognized in logspace by an ordinary Turing machine. There is an infinite hierarchy of cyclic graph languages and although most decision problems are decidable for 1-pebble cyclic automata, they are recursively unsolvable for 2-pebble automata. © 1987.
Theoretical Computer Science
Garzon, M. (1987). Cyclic automata. Theoretical Computer Science, 53 (2022-02-03), 307-317. https://doi.org/10.1016/0304-3975(87)90071-5