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

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    We first show that given a k1k_1-letter quantum finite automata A1\mathcal{A}_1 and a k2k_2-letter quantum finite automata A2\mathcal{A}_2 over the same input alphabet Ξ£\Sigma, they are equivalent if and only if they are (n12+n22βˆ’1)∣Σ∣kβˆ’1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k-equivalent where n1n_1, i=1,2i=1,2, are the numbers of state in Ai\mathcal{A}_i respectively, and k=max⁑{k1,k2}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 k1k_1-letter measure many quantum finite automaton A1\mathcal{A}_1 and a k2k_2-letter measure many quantum finite automaton A2\mathcal{A}_2 are equivalent if and only if they are (n12+n22βˆ’1)∣Σ∣kβˆ’1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k-equivalent where nin_i, i=1,2i=1,2, are the numbers of state in Ai\mathcal{A}_i respectively, and k=max⁑{k1,k2}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 kk-letter quantum finite automata, the non-strict cut-point language equivalence problem is undecidable, i.e., it is undecidable whether Lβ‰₯Ξ»(A1)=Lβ‰₯Ξ»(A2)L_{\geq\lambda}(\mathcal{A}_1)=L_{\geq\lambda}(\mathcal{A}_2) where 0<λ≀10<\lambda\leq 1 and Ai\mathcal{A}_i are kik_i-letter quantum finite automata. Further, we show that both strict and non-strict cut-point language equivalence problem for kk-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.Comment: 30 pages, conclusion section correcte
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