Decidability of the Equivalence of Multi-Letter Quantum Finite Automata

Multi-letter quantum finite automata (QFAs) were a new one-way QFA model proposed recently by Belovs, Rosmanis, and Smotrovs (LNCS, Vol. 4588, Springer, Berlin, 2007, pp. 60-71), and they showed that multi-letter QFAs can accept with no error some regular languages (a + b)b that are unacceptable by the one-way QFAs. In this paper, we study the decidability of the equivalence of multi-letter QFAs, and the main technical contributions are as follows: (1) We show that any two automata, a k1-letter QFA A1 and a k2-letter QFA A2, over the same input alphabet Σ are equivalent if and only if they are (nm −m + k)-equivalent, where m = |Σ| is the cardinality of Σ, k = max(k1, k2), and n = n1 +n2, with n1 and n2 being the numbers of states of A1 and A2, respectively. When k = 1, we obtain the decidability of equivalence of measure-once QFAs in the literature. (2) However, if we determine the equivalence of multi-letter QFAs by checking all strings of length not more than nm −m + k, then the worst time complexity is exponential, i.e., O(nm 2 m k−1 −m k−1+2k−1). Therefore, we design a polynomial-time O(mn + kmn) algorithm for determining the equivalence of any two multi-letter QFAs. Here, the time complexity is concerning the number of states in the multi-letter QFAs, and k is thought of as a constant.