Multi-Letter Quantum Finite Automata: Decidability of the Equivalence and Minimization of States D. Qiu 1,2,3 and L. Li 1 and X. Zou 1 and P. Mateus 2 and J. Gruska 4 1 Department of Computer Science, Sun Yat-sen University, Guangzhou, 510006, China 2 SQIG–Instituto de Telecomunica¸ c˜ oes, Departamento de Matem´ atica, Instituto Superior T´ ecnico, TULisbon, Av. Rovisco Pais 1049-001, Lisbon, Portugal 3 The State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100080, China 4 Faculty of Informatics, Masaryk University, Brno, Czech Republik February 2011 Abstract Multi-letter quantum finite automata (QFAs) are quantum variants of the one-way multi-head finite automata (J. Hromkoviˇ c, Acta Informatica 19 (1983) 377-384). It has been shown that this new one-way QFAs (multi- letter QFAs) can accept with no error some regular languages, for example (a + b) ∗ b, that are not acceptable by QFAs of Moore and Crutchfield [20] as well as Kondacs and Watrous [16]. In this paper, we study the decidability of the equivalence and minimization problems of multi-letter QFAs. Two new results presented in this paper are the following ones: (1) Given a k 1 -letter QFA A 1 and a k 2 -letter QFA A 2 over the same input alphabet Σ, they are equivalent if and only if they are (n 2 m k−1 −m k−1 +k)- equivalent, where m = |Σ| is the cardinality of Σ, k = max(k 1 ,k 2 ), and n = n 1 +n 2 , with n 1 and n 2 being numbers of states of A 1 and A 2 , respectively. When k = 1, this result implies the decidability of equivalence of measure- once QFAs [20]. (It is worth mentioning that our technical method is essentially different from the previous ones used in the literature.) (2) A polynomial-time O(m 2k−1 n 8 + km k n 6 ) algorithm is designed to determine the equivalence of any two multi-letter QFAs (see Theorems 2 and 3). Observe also that time complexity is expressed here in terms of the number of states of the multi-letter QFAs and k can be seen as a constant. (3) It is shown that the states minimization problem of multi-letter QFAs is decidable in EXPSPACE. 1