Abstract
Theory of relativization provides profound insights into the structural properties of various collections of mathematical problems by way of constructing desirable oracles that meet numerous requirements of the problems. This is a meaningful way to tackle unsolved questions on relationships among computational complexity classes induced by machine-based computations that can relativize. Slightly different from an early study on relativizations of uniform models of finite automata in [Tadaki, Yamakami, and Li (2010); Yamakami (2014)], we intend to discuss relativizations of state complexity classes (particularly, \(1\mathrm {BQ}\) and \(2\mathrm {BQ}\)) defined in terms of nonuniform families of time-unbounded quantum finite automata with polynomially many inner states. We create various relativized worlds where certain nonuniform state complexity classes become equal or different. By taking a nonuniform family of promise decision problems as an oracle, we can define a Turing reduction witnessed by a certain nonuniform finite automata family. We demonstrate closure properties of certain nonuniform state complexity classes under such reductions. Turing reducibility further enables us to define a hierarchy of nonuniform nondeterministic state complexity classes.
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Notes
- 1.
In some literature, a quantum finite automaton is allowed to use “classical states” besides “quantum states.” Such an automaton is often abbreviated as a qcfa. If we use a garbage tape and a measurement, then we essentially do not need to introduce classical states. To simplify our description of 2qfa’s, we use no classical states in this exposition.
- 2.
A tape is circular if both ends of the tape are glued together so that a tape head moves out of \(\$\) to the right, it reaches
, and the vice versa. - 3.
A tape is said to be write only if a tape head always moves to the next blank cell just after writing any non-blank symbol.
, and the vice versa.References
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Yamakami, T. (2019). Relativizations of Nonuniform Quantum Finite Automata Families. In: McQuillan, I., Seki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2019. Lecture Notes in Computer Science(), vol 11493. Springer, Cham. https://doi.org/10.1007/978-3-030-19311-9_20
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