Reductions among prediction problems: on the difficulty of predicting automata

Consider the following question: Given examples of words accepted and rejected by an unknown automaton, is there an algorithm that in a feasible amount of time will learn to predict which words will be accepted by the automaton? We develop a notion of prediction-preserving reducibility, and show that if DFAs are predictable, then so are all languages in logspace. In particular, the predictability of DFAs implies the predictability of all boolean formulas, Similarly, predictability of NFAs and PDAs (or CFGs) implies predictability of all languages in nondeterministic logspace and mC3%, respectively. We give relationships between the complexity of the membership problem for a class of automata, and the complexity of the prediction problem. Examples are given of prediction problems whose predictability would imply the predictability of all languages in P. Assuming the existence of 1-way functions, it follows from 1131 that these problems are not predictable even in an extremely weak sense.