Abstract
We consider the promise problem \(A^{N,r_1,r_2}\) on a unary alphabet \({\left\{ \sigma \right\} }\) studied by Gruska et al. in [21]. This problem is formally defined as the pair \(A^{N,r_1,r_2}=(A^{N,r_1}_{yes},A^{N,r_2}_{no})\), with \(0\le r_1\ne r_2<N\), \(A^{N,r_1}_{yes}={\left\{ \sigma ^n \ \mid \ n\equiv r_1 \mod N\right\} }\) and \(A^{N,r_2}_{no}={\left\{ \sigma ^n \ \mid \ n \equiv r_2 \mod N\right\} }\). There, it is shown that a measure-once one-way quantum automaton can solve exactly \(A^{N,r_1,r_2}\) with only \(3\) basis states, while any one-way deterministic finite automaton requires \(d\) states, \(d\) being the smallest integer such that \(d\mid N\) and \(d \not \mid (r_2-r_1) \mod N\). Here, we introduce the promise problem \({\textsc {Diof}}^{\,{a},N}_{r_1,r_2}\) as an extension of \(A^{N,r_1,r_2}\) to general alphabets. Even for this problem, we show the same descriptional superiority of the quantum paradigm over one-way deterministic automata. Moreover, we prove that even by adding features to classical automata, namely nondeterminism, probabilism, two-way motion, we cannot obtain automata for \(A^{N,r_1,r_2}\) and \({\textsc {Diof}}^{\,{a},N}_{r_1,r_2}\) smaller than one-way deterministic.
Partially supported by MIUR under the project “PRIN: Automi e Linguaggi Formali: Aspetti Matematici e Applicativi.”
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Bianchi, M.P., Mereghetti, C., Palano, B. (2014). Complexity of Promise Problems on Classical and Quantum Automata. In: Calude, C., Freivalds, R., Kazuo, I. (eds) Computing with New Resources. Lecture Notes in Computer Science(), vol 8808. Springer, Cham. https://doi.org/10.1007/978-3-319-13350-8_12
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