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

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.