Qiu Daowen; Li Lvzhou; Zou Xiangfu Sun Yat Sen Univ Dept Comp Sci Guangzhou 510006 Guangdong Peoples R China. Qiu Daowen; Mateus Paulo TULisbon Inst Super Tecn Dept Matemat SQIG Inst Telecomunicacoes P-1049001 Lisbon Portugal. Qiu Daowen Chinese Acad Sci Inst Software State Key Lab Comp Sci Beijing 100080 Peoples R China. Gruska Jozef Masaryk Univ Fac Informat Brno Czech Republic.

Multi-letter quantum finite automata (QFAs) can be thought of quantum variants of the one-way multi-head finite automata (Hromkovic, Acta Informatica 19:377-384, 1983). 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 (Theor Comput Sci 237:275-306, 2000) as well as Kondacs and Watrous (66-75, 1997; Observe that 1-letter QFAs are exactly measure-once QFAs (MO-1QFAs) of Moore and Crutchfield (Theor Comput Sci 237: 275-306, 2000)). In this paper, we study the decidability of the equivalence and minimization problems of multi-letter QFAs. Three 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 Sigma, they are equivalent if and only if they are (n(2)m(k-1) - m(k-1) + k)-equivalent, where m = vertical bar Sigma vertical bar is the cardinality of Sigma, 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 (Moore and Crutchfield in Theor Comput Sci 237:275-306, 2000). (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 that if a brute force algorithm to determine equivalence would be used, as suggested by the decidability outcome of the point (1), the worst case time complexity would be exponential). 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 solvable in EXPSPACE. This implies also that the state minimization problem of MO-1QFAs (see Moore and Crutchfield in Theor Comput Sci 237: 275-306, 2000, page 304, Problem 5), an open problem stated in that paper, is also solvable in EXPSPACE.

English Abstract:

Multi-letter quantum finite automata (QFAs) can be thought of quantum variants of the one-way multi-head finite automata (Hromkovic, Acta Informatica 19:377-384, 1983). 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 (Theor Comput Sci 237:275-306, 2000) as well as Kondacs and Watrous (66-75, 1997; Observe that 1-letter QFAs are exactly measure-once QFAs (MO-1QFAs) of Moore and Crutchfield (Theor Comput Sci 237: 275-306, 2000)). In this paper, we study the decidability of the equivalence and minimization problems of multi-letter QFAs. Three 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 Sigma, they are equivalent if and only if they are (n(2)m(k-1) - m(k-1) + k)-equivalent, where m = vertical bar Sigma vertical bar is the cardinality of Sigma, 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 (Moore and Crutchfield in Theor Comput Sci 237:275-306, 2000). (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 that if a brute force algorithm to determine equivalence would be used, as suggested by the decidability outcome of the point (1), the worst case time complexity would be exponential). 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 solvable in EXPSPACE. This implies also that the state minimization problem of MO-1QFAs (see Moore and Crutchfield in Theor Comput Sci 237: 275-306, 2000, page 304, Problem 5), an open problem stated in that paper, is also solvable in EXPSPACE.

Qiu Daowen,Li Lvzhou,Zou Xiangfu,et al. multi-letter quantum finite automata: decidability of the equivalence and minimization of states[J]. ACTA INFORMATICA,2011-01-01,48(5-6):271-290.