单向量子有限自动机等价性的判定
Determining the equivalence for 1-way quantum finite automata
摘要：
本文主要研究Bertoni等人定义的{ \it 1-way quantum finite automata with control language}（CL-1 QFA）与Kondacs和Watrous提出的{ \it measure-many 1-way quantum finite automata}（MM-1 QFA）的等价性.更具体地说，我们得到：\begin{enumerate} \item [（i）]两个CL-1 QFA ${\cal A}_1$和${\cal A}_2$分别与控制语言（常规语言）${\cal L}_1$和${\cal L}_2$等价当且仅当它们是$(c_1n_1^2+c_2n_2^2-1)$等价的。此外，如果${\cal L}_1$和${\cal L}_2$以DFA的形式给出，分别具有$m_1$和$m_2$状态，则存在以时间$O ((m_1n_1^2+m_2n_2^2)^4)$运行的多项式时间算法，该算法将${\cal A}_1$和${\cal A}_2$作为输入，并确定它们是否等价。\item [（ii）]分别具有$n_1$和$n_2$状态的两个MM-1 QFA ${\cal A}_1$和${\cal A}_2$是等价的，当且仅当它们是$(3n_1^2+3n_2^2-1)$等价的。此外，存在以时间$O ((3n_1^2+3n_2^2)^4)$运行的多项式时间算法，其将${\cal A}_1$和${\cal A}_2$作为输入，并确定${\cal A}_1$和${\cal A}_2$是否等价。
英文摘要：
In this paper, we focus on determining the equivalence for {\it 1-way quantum finite automata with control language} (CL-1QFAs) defined by Bertoni et al and {\it measure-many 1-way quantum finite automata} (MM-1QFAs) introduced by Kondacs and Watrous. More specifically, we obtain that: \begin{enumerate} \item[(i)] Two CL-1QFAs ${\cal A}_1$ and ${\cal A}_2$ with control languages (regular languages) ${\cal L}_1$ and ${\cal L}_2$, respectively, are equivalent if and only if they are $(c_1n_1^2+c_2n_2^2-1)$-equivalent. Furthermore, if ${\cal L}_1$ and ${\cal L}_2$ are given in the form of DFAs, with $m_1$ and $m_2$ states, respectively, then there exists a polynomial-time algorithm running in time $O ((m_1n_1^2+m_2n_2^2)^4)$ that takes as input ${\cal A}_1$ and ${\cal A}_2$ and determines whether they are equivalent. \item[(ii)] Two MM-1QFAs ${\cal A}_1$ and ${\cal A}_2$ with $n_1$ and $n_2$ states, respectively, are equivalent if and only if they are $(3n_1^2+3n_2^2-1)$-equivalent. Furthermore, there is a polynomial-time algorithm running in time $O ((3n_1^2+3n_2^2)^4)$ that takes as input ${\cal A}_1$ and ${\cal A}_2$ and determines whether ${\cal A}_1$ and ${\cal A}_2$ are equivalent.
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