Complexity of Promise Problems on Classical and Quantum Automata

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.”