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

Tianrong Lin

Published Aug 15, 2013• 2 citations
OpenAlex

Abstract

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}

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Language recognition by generalized quantum finite automata with unbounded error (abstract & poster)

Abuzer Yakaryılmaz, A. C. Cem Say

In this note, we generalize the results of arXiv:0901.2703v1 We show that all one-way quantum finite automaton (QFA) models that are at least as general as Kondacs-Watrous QFA's are equivalent in power to classical probabilistic finite automata in this setting. Unlike their probabilistic counterparts, allowing the tape head to stay put for some steps during its traversal of the input does enlarge the class of languages recognized by such QFA's with unbounded error. (Note that, the proof of Theorem 1 in the abstract was presented in the previous version (arXiv:0901.2703v1).)

arXiv:0901.2703•Jan 18, 2009•2 citations
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}
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.
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}
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.
Abstract. We prove that several variants of the quantum finite automaton (QFA) model are strictly superior to their probabilistic counterparts, in terms of both language recognition power and succinctness, in the onesided unbounded error setting. This is the first case where one-way QFA models have been shown to outperform the corresponding probabilistic automata in both these respects. We give a characterization of the class of languages recognized by these QFA’s with cutpoint 0, demonstrating that it is equal to the class of exclusive stochastic languages. We prove several closure properties of the related classes. 1
In this note, we generalize the results of arXiv:0901.2703v1 We show that all one-way quantum finite automaton (QFA) models that are at least as general as Kondacs-Watrous QFA's are equivalent in power to classical probabilistic finite automata in this setting. Unlike their probabilistic counterparts, allowing the tape head to stay put for some steps during its traversal of the input does enlarge the class of languages recognized by such QFA's with unbounded error. (Note that, the proof of Theorem 1 in the abstract was presented in the previous version (arXiv:0901.2703v1).)