Hierarchy and equivalence of multi-letter quantum finite automata

  • Authors:

  • Daowen Qiu;Sheng Yu

  • Affiliations:

  • Department of Computer Science, Zhongshan University, Guangzhou 510275, China and SQIGInstituto de Telecomunicações, IST, TULisbon, Av. Rovisco Pais 1049-001, Lisbon, Portugal;Department of Computer Science, The University of Western Ontario, London, Ontario, N6A 5B7, Canada

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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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^1^2+k^2n^4+kn^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.