In this paper, we settle a problem in probabilistic verification of infinite--state process (specifically, {it probabilistic pushdown process}). We show that model checking {it stateless probabilistic pushdown process} (pBPA) against {it probabilistic computational tree logic} (PCTL) is undecidable.
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 $L_{geqlambda}(mathcal{A}_1)=L_{geqlambda}(mathcal{A}_2)$ where $0<lambdaleq 1$ 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.
