An Oracle Hierarchy for Small One-Way Finite Automata

Abstract

We introduce a polynomial-size oracle hierarchy for one-way finite automata. In it, a problem is in level k (resp., level 0) if itself or its complement is solved by a polynomial-size nondeterministic finite automaton with access to an oracle for a problem in level k — 1 (resp., by a polynomial-size deterministic finite automaton with no oracle access). This is a generalization of the polynomial-size alternating hierarchy for one-way finite automata, as previously defined using polynomial-size alternating finite automata with a bounded number of alternations; and relies on an original definition of what it means for a nondeterministic finite automaton to access an oracle, which we carefully justify. We prove that our hierarchy is strict; that every problem in level k is solved by a deterministic finite automaton of 2k-fold exponential size; and that level 1 already contains problems beyond the entire alternating hierarchy. We then identify five restrictions to our oracle-automaton, under which the oracle hierarchy is proved to coincide with the alternating one, thus providing an oracle-based characterization for it. We also show that, given all others, each of these restrictions is necessary for this characterization.