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 k that divides the cycle length. By “large” we mean that there is an unbounded increasing function f(n), such that either k is greater than f(n) or k is the exact cycle length. Our main results include that finding a large divisor of the cycle length can be solved in deterministic SPACE o(n) and TIME O(n), whereas finding the exact cycle length cannot be solved in deterministic TIME × SPACE smaller than Ω(n2). Results involving probabilistic automata and applications to rate-distortion theory and repeated games are also discussed.