Learning cycle length through finite automata

We study the space-and-time automaton-complexity of two related problems concerning the cycle length of a periodic stream of input bits. One problem is to find the exact cycle length of a periodic stream of input bits provided that the cycle length is bounded by a known parameter n. The other problem is to find a large number Κ that divides the cycle length. By "large" we mean that there is an unbounded increasing function f 4n5, such that either Κ is greater than f 4n5 or Κ is the exact cycle length. Our main results include that finding a large divisor of the cycle length can be solved in deterministic linear TIME and sub-linear SPACE, whereas finding the exact cycle length cannot be solved in deterministic TIME SPACE smaller than a constant times n squared. Results involving probabilistic automata and applications to rate-distortion theory and repeated games are also discussed.