Concurrency Counts

One of the natural examples to the power of concurrency is the ability to count to n with log n concurrent processes. In this work we study the cost of counting within weaker types of concurrency, where no cooperation takes place between the concurrent processes, except when time comes to decide whether the input should be accepted. We first study the number of states required for deterministic, nondeterministic, universal, and alternating automata to count to n, namely to accept the unary language {1n}. We then study settings in which the task of counting should be accomplished before the whole input is read (e.g., in the language Σn ·0 ·Σ∗). We show that then, the weaker types of concurrency are not helpful. Our results explain the strength of synchronized concurrent systems, where synchronization between the underlying components is allowed, and imply that semi-extended regular expressions, which can be efficiently translated to partially input-synchronized alternating automata, cannot be efficiently translated to alternating automata.