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.
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A function is elementary if it is the composition of finitely many arithmetic operations, exponentials, logarithms, constants, and solutions of algebraic equations.
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Anabtawi, M., Hassan, S., Kapoutsis, C., Zakzok, M. (2019). An Oracle Hierarchy for Small One-Way Finite Automata. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2019. Lecture Notes in Computer Science(), vol 11417. Springer, Cham. https://doi.org/10.1007/978-3-030-13435-8_4
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DOI: https://doi.org/10.1007/978-3-030-13435-8_4
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