Two-tape finite automata with quantum and classical states

{\it Two-way finite automata with quantum and classical states} (2QCFA) were introduced by Ambainis and Watrous, and {\it two-way two-tape deterministic finite automata} (2TFA) were introduced by Rabin and Scott. In this paper we study 2TFA and propose a new computing model called {\it two-way two-tape finite automata with quantum and classical states} (2TQCFA). First, we give efficient 2TFA algorithms for recognizing languages which can be recognized by 2QCFA. Second, we give efficient 2TQCFA algorithms to recognize several languages whose status vis-a-vis 2QCFA have been posed as open questions, such as $L_{square}=\{a^{n}b^{n^{2}}\mid n\in \mathbf{N}\}$. Third, we show that $\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\}$ can be recognized by {\it $(k+1)$-tape deterministic finite automata} ($(k+1)$TFA). Finally, we introduce {\it $k$-tape automata with quantum and classical states} ($k$TQCFA) and prove that $\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\}$ can be recognized by $k$TQCFA.Comment: 25 page

Another approach to the equivalence of measure-many one-way quantum finite automata and its application

In this paper, we present a much simpler, direct and elegant approach to the equivalence problem of {\it measure many one-way quantum finite automata} (MM-1QFAs). The approach is essentially generalized from the work of Carlyle [J. Math. Anal. Appl. 7 (1963) 167-175]. Namely, we reduce the equivalence problem of MM-1QFAs to that of two (initial) vectors. As an application of the approach, we utilize it to address the equivalence problem of {\it Enhanced one-way quantum finite automata} (E-1QFAs) introduced by Nayak [Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp.~369-376]. We prove that two E-1QFAs $\mathcal{A}_1$ and $\mathcal{A}_2$ over $\Sigma$ are equivalence if and only if they are $n_1^2+n_2^2-1$-equivalent where $n_1$ and $n_2$ are the numbers of states in $\mathcal{A}_1$ and $\mathcal{A}_2$, respectively.Comment: V 10: Corollary 3 is deleted, since it is folk. (V 9: Revised in terms of the referees's comments) All comments, especially the linguistic comments, are welcom