Determinism without Determinization

A nondeterministic semiautomaton S is predictable if there exists k ≥ 0 such that, if S knows the current input a and the next k inputs, the transition under a is deterministic. Nondeterminism may occur only when the length of the unread input is ≤ k. We develop a theory of predictable semiautomata. If a semiautomaton S with n states is k-predictable, but not (k−1)-predictable, then k ≤ (n −n)/2, and this bound can be reached for a suitable input alphabet. We characterize k-predictable semiautomata. We introduce the predictor semiautomaton, based on a look-ahead semiautomaton. The predictor is essentially deterministic and simulates a nondeterministic semiautomaton by finding the set of states reachable by a word w, if it belongs to the language L of the semiautomaton (i.e., if it defines a path from an initial state to some state), or by stopping as soon as it infers that w 6∈ L. Membership in L can be decided deterministically.