Determinism and Nondeterminism in Finite Automata with Advice.

We consider the model of finite automata with advice introduced by Kucuk et al. We show that there are languages, in particular the language of palindromes, that cannot be recognized by \(\text {DFA}\) regardless of the size of advice. Also, we show that a \(\text {DFA}\) cannot utilize more than exponential advice. We initiate the study of \(\text {NFA}\) with advice: we show that, unlike the \(\text {DFA}\), \(\text {NFA}\) can recognize all languages with advice of exponential size. On the other side of the spectrum, with constant advice, \(\text {DFA}\) are as powerful as \(\text {NFA}\). We show that for any growing function f, there are languages that can be recognized by \(\text {NFA}\) with advice f(n), but cannot be recognized by \(\text {DFA}\) regardless of advice size. We also ask what languages can be recognized with polynomial advice. For \(\text {NFA}\) we show that this class is not closed under complementation, and that it contains all bounded languages. Bounded languages over one-letter words can even be recognized by \(\text {DFA}\) with polynomial advice. We also give examples of languages that cannot be recognized by \(\text {NFA}\) with polynomial advice. Finally, we show that increasing advice helps for \(\text {NFA}\), and for any advice of size \(f(n)\le n\) we show that there is a language that can be recognized by a \(\text {DFA}\) with advice f(n), but cannot be recognized by an \(\text {NFA}\) with advice o(f(n)).