Language complexity on the synchronous anonymous ring

A set of n nondistinct processors, organized as a ring and operating synchronously, have to compute a function of their initial values. Every computable function can be computed with O(n log n) messages, while some functions can be computed with as few as O(n) messages. We prove a necessary and sufficient condition for a regular language to be recognized with O(n) messages. Languages that do not satisfy this condition are 'hard' to compute, i.e., their recognition requires Ω(n log n) message. The condition is an extension of the notion of counter-free regular languages. These results give a gap theorem for recognizing regular languages on the synchronous anonymous ring. In contrast, we show a family of nonregular languages, computing thresholds, that obtain any intermediate complexity in the range Θ{round}(n) to Θ{round}(n log n).