Hierarchy and equivalence of multi-letter quantum finite automata

Abstract

Multi-letter quantum finite automata (QFAs) are a new one-way QFA model proposed recently by Belovs, Rosmanis, and Smotrovs [A. Belovs, A. Rosmanis, J. Smotrovs, Multi-letter reversible and quantum finite automata, in: Proceedings of the 13th International Conference on Developments in Language Theory, DLT’2007, Harrachov, Czech Republic, in: Lecture Notes in Computer Science, 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 continue to study multi-letter QFAs. We mainly focus on two issues: (1) we show that ( k + 1 ) -letter QFAs are computationally more powerful than k -letter QFAs, that is, ( k + 1 ) -letter QFAs can accept some regular languages that are unacceptable by any k -letter QFA. A comparison with the one-way QFAs is made by some examples; (2) we prove that a k 1 -letter QFA A 1 and another k 2 -letter QFA A 2 are equivalent, if and only if, they are ( n 1 + n 2 ) 4 + k − 1 -equivalent, and the time complexity of determining the equivalence of two multi-letter QFAs using this method is O ( n 12 + k 2 n 4 + k n 8 ) , where n 1 and n 2 are the numbers of states of A 1 and A 2 , respectively, and k = max ( k 1 , k 2 ) . Some other issues are addressed for further consideration.