An Oracle Hierarchy for Small One-Way Finite Automata
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.
- 7.Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: Proceedings of the Symposium on Switching and Automata Theory, pp. 125–129 (1972)Google Scholar
- 9.Sakoda, W.J., Sipser, M.: Nondeterminism and the size of two-way finite automata. In: Proceedings of STOC, pp. 275–286 (1978)Google Scholar
- 11.Yamakami, T.: Oracle pushdown automata, nondeterministic reducibilities, and the hierarchy over the family of context-free languages. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds.) SOFSEM 2014. LNCS, vol. 8327, pp. 514–525. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04298-5_45CrossRefzbMATHGoogle Scholar