Two-Way Non-Uniform Finite Automata

Abstract

We consider two-tape automata where one tape contains the input word [Formula: see text] and the other an advice string [Formula: see text]. Such an automaton recognizes a language [Formula: see text] if there is an advice function for which every word on the input tape is correctly classified. This model has been introduced by Küçük et al. with the aim of modeling non-uniform computations on finite automata. So far, most results concerned automata with 1-way tapes, both for the input and the advice. First, we show that making just one of the tapes 2-way already increases the model’s power. Then we consider the case where both tapes are 2-way, which can also be viewed as a restricted version of the non-uniform families of automata used by Ibarra and Ravikumar to define the class NUDSPACE . We show this restriction to be not too significant since, e.g., [Formula: see text]DFA/poly), i.e., the languages recognized by automata with 2-way input and advice tape with polynomial advice, equals [Formula: see text]. Hence, we can show that many interesting problems concerning the state complexity of families of automata carry over to the problems concerning the advice size of non-uniform automata. In particular, the question whether there can be a more than polynomial gap in advice between determinism and nondeterminism is of great interest: The existence of a language that is recognized by a 2-way NFA with a constant number of advice heads and polynomial (resp. logarithmic) advice, while a corresponding 2-head DFA needs exponential (resp. polynomial) advice, would imply [Formula: see text] (resp. [Formula: see text]). We show that for advice of size [Formula: see text] there is no gap between determinism and nondeterminism, and that the gap is at most exponential.