Complexity of Promise Problems on Classical and Quantum Automata

  • Maria Paola Bianchi
  • Carlo MereghettiEmail author
  • Beatrice Palano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8808)


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.


classical and quantum automata promise problem descriptional complexity 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Maria Paola Bianchi
    • 1
  • Carlo Mereghetti
    • 1
    Email author
  • Beatrice Palano
    • 1
  1. 1.Dipartimento di InformaticaUniversitá degli Studi di Milano via Comelico 39MilanoItaly

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