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On equivalence and emptiness problems of multi-letter (measure many) quantum finite automata

Tianrong Lin

In this paper, we study some decision problems both for {\it multi-letter quantum finite automata} and {\it measure many multi-letter quantum finite automata}. We first show that given a $k_1$-letter quantum finite automaton $\mathcal{A}_1$ and a $k_2$-letter quantum finite automaton $\mathcal{A}_2$ over the same input alphabet $\Sigma$, they are equivalent if and only if they are $(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k$-equivalent where $n_i$, $i=1,2$, are the number of states in $\mathcal{A}_i$ respectively, and $k=\max\{k_1,k_2\}$. By applying a method, due to the author, used to deal with the equivalence problem of {\it measure many one-way quantum finite automata}, we also show that a $k_1$-letter measure many quantum finite automaton $\mathcal{A}_1$ and a $k_2$-letter measure many quantum finite automaton $\mathcal{A}_2$ are equivalent if and only if they are $(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k$-equivalent where $n_i$, $i=1,2$, are the number of states in $\mathcal{A}_i$ respectively, and $k=\max\{k_1,k_2\}$. Next, we study the emptiness problem of those two kinds of quantum finite automata. We show that whether the language recognized by a $k$-letter quantum finite automaton with non-strict cut-point is empty is undecidable, but we leave open the emptiness of language reorganized by a $k$-letter quantum finite automaton with strict cutpoint. We also show that whether the languages recognized by a $k$-letter measure many quantum finite automaton with both nonstrict and strict cutpoints are undecidable. And the direct consequences of the above outcomes are summarized in the paper.

Mar 1, 20120 citations

Decidability of the Equivalence of Multi-Letter Quantum Finite Automata

Daowen Qiu, Xiangfu Zou, Lvzhou Li +1 more

Multi-letter {\it quantum finite automata} (QFAs) were a quantum variant of classical {\it one-way multi-head finite automata} (J. Hromkovič, Acta Informatica 19 (1983) 377-384), and it has been shown that this new one-way QFAs (multi-letter QFAs) can accept with no error some regular languages $(a+b)^{*}b$ that are unacceptable by the previous one-way QFAs. In this paper, we study the decidability of the equivalence of multi-letter QFAs, and the main technical contributions are as follows: (1) We show that any two automata, a $k_{1}$-letter QFA ${\cal A}_1$ and a $k_{2}$-letter QFA ${\cal A}_2$, over the same input alphabet $Σ$ are equivalent if and only if they are $(n^2m^{k-1}-m^{k-1}+k)$-equivalent, where $m=|Σ|$ is the cardinality of $Σ$, $k=\max(k_{1},k_{2})$, and $n=n_{1}+n_{2}$, with $n_{1}$ and $n_{2}$ being the numbers of states of ${\cal A}_{1}$ and ${\cal A}_{2}$, respectively. When $k=1$, we obtain the decidability of equivalence of measure-once QFAs in the literature. It is worth mentioning that our technical method is essentially different from that for the decidability of the case of single input alphabet (i.e., $m=1$). (2) However, if we determine the equivalence of multi-letter QFAs by checking all strings of length not more than $ n^2m^{k-1}-m^{k-1}+k$, then the worst time complexity is exponential, i.e., $O(n^6m^{n^2m^{k-1}-m^{k-1}+2k-1})$. Therefore, we design a polynomial-time $O(m^{2k-1}n^{8}+km^kn^{6})$ algorithm for determining the equivalence of any two multi-letter QFAs. Here, the time complexity is concerning the number of states in the multi-letter QFAs, and $k$ is thought of as a constant.

arXiv:0812.1061Dec 5, 20084 citations

Emptiness and equivalence of language on measure many quantum automata

Tianrong Lin

Let $L_{>\lambda}(\mathcal{A})$ and $L_{\geq\lambda}(\mathcal{A})$ be the languages recognized by {\em measure many 1-way quantum finite automata (MMQFA)} (or,{\em enhanced 1-way quantum finite automata(EQFA)}) $\mathcal{A}$ with strict, resp. non-strict cut-point $\lambda$. We consider the languages equivalence problem, showing that \begin{itemize} \item {both strict and non-strict languages equivalence are undecidable;} \item {to do this, we provide an additional proof of the undecidability of non-strict and strict emptiness of MMQFA(EQFA), and then reducing the languages equivalence problem to emptiness problem;} \item{Finally, some other Propositions derived from the above results are collected.} \end{itemize}

arXiv:1308.3301Aug 15, 20130 citations

Some results on equivalence of multi-letter quantum finite automata

Tianrong Lin

Two quantum finite automata are equivalent if for all input string $ω$ over the input alphabet the two automata accept $ω$ with equal probability. In [Theoret. Comput. Sci. 410 (2009) 3006-3017], it was shown that a $k_1$-letter QFA $\mathcal{A}_1$ and a $k_2$-letter QFA $\mathcal{A}_2$ over $Σ=\{σ\}$, are equivalent if and only if they are $(n_1+n_2)^4+k-1$-equivalent where $n_i$ is the number of states of $\mathcal{A}_i$, $i=1,2$, and $k=\max\{k_1,k_2\}$. In this letter, we improve the above upper-bound to $(n_1^2+n_2^2-1)+k$. This also answers an open problem of Qiu et al. [Acta Informatica 48 (2011) 271-290]. Further, we show that, in the case of $Σ=\{σ_1,...,σ_t\}$ with $2\leq t<\infty$, there exists an integer $z$ such that $\mathcal{A}_1$ and $\mathcal{A}_2$ are equivalent if and only if they satisfy $z$-equivalent.

arXiv:1106.5223Jun 26, 20110 citations

Determining the equivalence for 1-way quantum finite automata

Lvzhou Li, Daowen Qiu

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.

arXiv:quant-ph/0703087Mar 11, 20071 citation

Note on equivalence of cutpoint languages recognized by measure many quantum finite automata

Tianrong Lin

This note revisits the equivalence of languages recognized by measure many one way quantum finite automata with non/strict cutpoint. The main contributions are as follows: \begin{enumerate}[(1)] \item {We provide an additional proof of the undecidability of non/strict emptiness of measure many one way quantum finite automata;} \item {By the undecidability of non/strict emptiness of measure many one way quantum finite automata, we show that the equivalence of languages recognized by measure many one way quantum finite automata with non/strict cutpoint is undecidable, implying the undecidability of containment problem of measure many one way quantum finite automata} \end{enumerate}

Aug 15, 20132 citations

On equivalence, languages equivalence and minimization of multi-letter and multi-letter measure-many quantum automata

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