Cellular Devices and Unary Languages

Devices of interconnected parallel acting sequential automata are investigated from a language theoretic point of view. Starting with the well-known result that each unary language accepted by a deterministic one-way cellular automaton (OCA) in real time has to be a regular language, we will answer the three natural questions ‘How much time do we have to provide?’ ‘How much power do we have to plug in the single cells (i.e., how complex has a single cell to be)?’ and ‘How can we modify the mode of operation (i.e., how much nondeterminism do we have to add)?’ in order to accept non-regular unary languages. We show the surprising result that for classes of generalized interacting automata parallelism does not yield to more computational capacity than obtained by a single sequential cell. Moreover, it is proved that there exists a unary complexity class in between the real-time and linear-time OCA languages, and that there is a gap between the unary real-time OCA languages and that class. Regarding nondeterminism as limited resource it is shown that a slight increase of the degree of nondeterminism as well as adding two-way communication reduces the time complexity from linear time to real time. Furthermore, by adding a wee bit nondeterminism an infinite hierarchy of unary language families dependent on the degree of nondeterminism is derived.