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

Tianrong Lin

Published Mar 1, 2012• 0 citations
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Abstract

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.

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

Tianrong Lin

We first show that given a $k_1$-letter quantum finite automata $\mathcal{A}_1$ and a $k_2$-letter quantum finite automata $\mathcal{A}_2$ over the same input alphabet $Σ$, they are equivalent if and only if they are $(n_1^2+n_2^2-1)|Σ|^{k-1}+k$-equivalent where $n_1$, $i=1,2$, are the numbers of state 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 $(n_1^2+n_2^2-1)|Σ|^{k-1}+k$-equivalent where $n_i$, $i=1,2$, are the numbers of state in $\mathcal{A}_i$ respectively, and $k=\max\{k_1,k_2\}$. Next, we study the language equivalence problem of those two kinds of quantum finite automata. We show that for $k$-letter quantum finite automata, the non-strict cut-point language equivalence problem is undecidable, i.e., it is undecidable whether $L_{\geqλ}(\mathcal{A}_1)=L_{\geqλ}(\mathcal{A}_2)$ where $0<λ\leq 1$ and $\mathcal{A}_i$ are $k_i$-letter quantum finite automata. Further, we show that both strict and non-strict cut-point language equivalence problem for $k$-letter measure many quantum finite automata are undecidable. The direct consequences of the above outcomes are summarized in the paper. Finally, we comment on existing proofs about the minimization problem of one way quantum finite automata not only because we have been showing great interest in this kind of problem, which is very important in classical automata theory, but also due to that the problem itself, personally, is a challenge. This problem actually remains open.

arXiv:1203.0113•Mar 1, 2012•0 citations

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, 2013•2 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:1308.3301•Aug 15, 2013•0 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:0812.1061•Dec 5, 2008•4 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:1106.5223•Jun 26, 2011•0 citations

Decidable and Undecidable Problems about Quantum Automata

Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran +1 more

We study the following decision problem: is the language recognized by a quantum finite automaton empty or nonempty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata, for which it is known that strict andnonstrict thresholds both lead to undecidable problems.

arXiv:quant-ph/0304082•SIAM Journal on Computing•Jan 1, 2005•76 citations
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 k1k_1k1​-letter quantum finite automaton A1\mathcal{A}_1A1​ and a k2k_2k2​-letter quantum finite automaton A2\mathcal{A}_2A2​ over the same input alphabet Σ\SigmaΣ, they are equivalent if and only if they are (n12+n22−1)∣Σ∣k−1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k(n12​+n22​−1)∣Σ∣k−1+k-equivalent where nin_ini​, i=1,2i=1,2i=1,2, are the number of states in Ai\mathcal{A}_iAi​ respectively, and k=max⁡{k1,k2}k=\max\{k_1,k_2\}k=max{k1​,k2​}. 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 k1k_1k1​-letter measure many quantum finite automaton A1\mathcal{A}_1A1​ and a k2k_2k2​-letter measure many quantum finite automaton A2\mathcal{A}_2A2​ are equivalent if and only if they are (n12+n22−1)∣Σ∣k−1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k(n12​+n22​−1)∣Σ∣k−1+k-equivalent where nin_ini​, i=1,2i=1,2i=1,2, are the number of states in Ai\mathcal{A}_iAi​ respectively, and k=max⁡{k1,k2}k=\max\{k_1,k_2\}k=max{k1​,k2​}.
Next, we study the emptiness problem of those two kinds of quantum finite automata. We show that whether the language recognized by a kkk-letter quantum finite automaton with non-strict cut-point is empty is undecidable, but we leave open the emptiness of language reorganized by a kkk-letter quantum finite automaton with strict cutpoint. We also show that whether the languages recognized by a kkk-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.
We first show that given a k1k_1k1​-letter quantum finite automata A1\mathcal{A}_1A1​ and a k2k_2k2​-letter quantum finite automata A2\mathcal{A}_2A2​ over the same input alphabet ΣΣΣ, they are equivalent if and only if they are (n12+n22−1)∣Σ∣k−1+k(n_1^2+n_2^2-1)|Σ|^{k-1}+k(n12​+n22​−1)∣Σ∣k−1+k-equivalent where n1n_1n1​, i=1,2i=1,2i=1,2, are the numbers of state in Ai\mathcal{A}_iAi​ respectively, and k=max⁡{k1,k2}k=\max\{k_1,k_2\}k=max{k1​,k2​}. 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 k1k_1k1​-letter measure many quantum finite automaton A1\mathcal{A}_1A1​ and a k2k_2k2​-letter measure many quantum finite automaton A2\mathcal{A}_2A2​ are equivalent if and only if they are (n12+n22−1)∣Σ∣k−1+k(n_1^2+n_2^2-1)|Σ|^{k-1}+k(n12​+n22​−1)∣Σ∣k−1+k-equivalent where nin_ini​, i=1,2i=1,2i=1,2, are the numbers of state in Ai\mathcal{A}_iAi​ respectively, and k=max⁡{k1,k2}k=\max\{k_1,k_2\}k=max{k1​,k2​}. Next, we study the language equivalence problem of those two kinds of quantum finite automata. We show that for kkk-letter quantum finite automata, the non-strict cut-point language equivalence problem is undecidable, i.e., it is undecidable whether L≥λ(A1)=L≥λ(A2)L_{\geqλ}(\mathcal{A}_1)=L_{\geqλ}(\mathcal{A}_2)L≥λ​(A1​)=L≥λ​(A2​) where 0<λ≤10<λ\leq 10<λ≤1 and Ai\mathcal{A}_iAi​ are kik_iki​-letter quantum finite automata. Further, we show that both strict and non-strict cut-point language equivalence problem for kkk-letter measure many quantum finite automata are undecidable. The direct consequences of the above outcomes are summarized in the paper. Finally, we comment on existing proofs about the minimization problem of one way quantum finite automata not only because we have been showing great interest in this kind of problem, which is very important in classical automata theory, but also due to that the problem itself, personally, is a challenge. This problem actually remains open.
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}
Let L>λ(A)L_{>\lambda}(\mathcal{A})L>λ​(A) and L≥λ(A)L_{\geq\lambda}(\mathcal{A})L≥λ​(A) be the languages recognized by {\em measure many 1-way quantum finite automata (MMQFA)} (or,{\em enhanced 1-way quantum finite automata(EQFA)}) A\mathcal{A}A with strict, resp. non-strict cut-point λ\lambdaλ. 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}
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(a+b)^{*}b(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 k1k_{1}k1​-letter QFA A1{\cal A}_1A1​ and a k2k_{2}k2​-letter QFA A2{\cal A}_2A2​, over the same input alphabet ΣΣΣ are equivalent if and only if they are (n2mk−1−mk−1+k)(n^2m^{k-1}-m^{k-1}+k)(n2mk−1−mk−1+k)-equivalent, where m=∣Σ∣m=|Σ|m=∣Σ∣ is the cardinality of ΣΣΣ, k=max⁡(k1,k2)k=\max(k_{1},k_{2})k=max(k1​,k2​), and n=n1+n2n=n_{1}+n_{2}n=n1​+n2​, with n1n_{1}n1​ and n2n_{2}n2​ being the numbers of states of A1{\cal A}_{1}A1​ and A2{\cal A}_{2}A2​, respectively. When k=1k=1k=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=1m=1m=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(n6mn2mk−1−mk−1+2k−1)O(n^6m^{n^2m^{k-1}-m^{k-1}+2k-1})O(n6mn2mk−1−mk−1+2k−1). Therefore, we design a polynomial-time O(m2k−1n8+kmkn6)O(m^{2k-1}n^{8}+km^kn^{6})O(m2k−1n8+kmkn6) 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 kkk is thought of as a constant.
Two quantum finite automata are equivalent if for all input string ωωω over the input alphabet the two automata accept ωωω with equal probability. In [Theoret. Comput. Sci. 410 (2009) 3006-3017], it was shown that a k1k_1k1​-letter QFA A1\mathcal{A}_1A1​ and a k2k_2k2​-letter QFA A2\mathcal{A}_2A2​ over Σ={σ}Σ=\{σ\}Σ={σ}, are equivalent if and only if they are (n1+n2)4+k−1(n_1+n_2)^4+k-1(n1​+n2​)4+k−1-equivalent where nin_ini​ is the number of states of Ai\mathcal{A}_iAi​, i=1,2i=1,2i=1,2, and k=max⁡{k1,k2}k=\max\{k_1,k_2\}k=max{k1​,k2​}. In this letter, we improve the above upper-bound to (n12+n22−1)+k(n_1^2+n_2^2-1)+k(n12​+n22​−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={σ1,...,σt}Σ=\{σ_1,...,σ_t\}Σ={σ1​,...,σt​} with 2≤t<∞2\leq t<\infty2≤t<∞, there exists an integer zzz such that A1\mathcal{A}_1A1​ and A2\mathcal{A}_2A2​ are equivalent if and only if they satisfy zzz-equivalent.