On tally languages and generalized interacting automata

Devices of interconnected parallel acting sequential automata are investigated from a language theoretic point of view. Starting with the well­known result that each tally language acceptable by a classical one­way cellular automaton (OCA) in real­time has to be a regular language we will answer the three natural questions \u27How much time do we have to provide?\u27 \u27How much power do we have to plug in the single cells (i.e., how complex has a single cell to be)?\u27 and \u27How can we modify the mode of operation (i.e., how much nondeterminism do we have to add)?\u27 in order to accept non­regular tally languages. We show the surprising result that for some classes of generalized interacting automata parallelism does not lead to more accepting power than obtained by a single sequential cell. Adding a wee bit of nondeterminism an infinite hierarchy of unary language families can be shown by allowing more and more nondeterminism