REDUCTIONS AMONG PREDICTION PROBLEMS: ON THE DIFFICULTY OF PREDICTING AUTOMATA.

Given examples of words accepted and rejected by an unknown automaton, the question of whether there is an algorithm that in a feasible amount of time will learn to predict which words will be accepted by the automaton is examined. A notion of prediction-preserving reducibility is developed, and it is shown 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. Similar results hold for NFAs and PDAs (or CFGs). Relationships between the complexity of the membership problem for a class of automata and the complexity of the prediction problem are obtained. Examples are give of prediction problems in which predictability implies the predictability of all langages in P. Assuming the existence of one-way functions, it follows that these problems are not predictable, even in an extremely weak sense.