Message complexity for unary multiautomata systems

Finitely many two-way automata work independently and synchronously on a unary input. Some of their states are broadcasting, i.e., dispatched to all other automata. At each step of the computation, each automaton changes state and moves right, left or stay in place according to the current state and the possible messages dispatched. The input is recognized if the following occurs: starting from the initial configuration (the heads of all automata are positioned to the left end of the tape) one automaton reaches a final state when its head is positioned to the right end of the tape. We show that if the number of messages sent during the computation is bounded by some integer which is independent of the length of the input, then the language recognized is regular,