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

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 $\Sigma$, they are equivalent if and only if they are $(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k$-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 $(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k$-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 $L_{\geq\lambda}(\mathcal{A}_1)=L_{\geq\lambda}(\mathcal{A}_2)$ where $0<\lambda\leq 1$ 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.Comment: 30 pages, conclusion section correcte

From Quantum Query Complexity to State Complexity

State complexity of quantum finite automata is one of the interesting topics in studying the power of quantum finite automata. It is therefore of importance to develop general methods how to show state succinctness results for quantum finite automata. One such method is presented and demonstrated in this paper. In particular, we show that state succinctness results can be derived out of query complexity results.Comment: Some typos in references were fixed. To appear in Gruska Festschrift (2014). Comments are welcome. arXiv admin note: substantial text overlap with arXiv:1402.7254, arXiv:1309.773

Quantum Pushdown Automata

Quantum finite automata, as well as quantum pushdown automata (QPA) were first introduced by C. Moore and J. P. Crutchfield. In this paper we introduce the notion of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of Kondacs and Watrous. It is established that the unitarity criteria of QPA are not equivalent to the corresponding unitarity criteria of quantum Turing machines. We show that QPA can recognize every regular language. Finally we present some simple languages recognized by QPA, not recognizable by deterministic pushdown automata.Comment: Conference SOFSEM 2000, extended version of the pape