Another approach to the equivalence of measure-many one-way quantum finite automata and its application

  • Authors:

  • Tianrong Lin

  • Affiliations:

  • Department of Computer Science, Fujian Normal University, Fuzhou 350007, China

  • Venue:
  • Journal of Computer and System Sciences
  • Year:
  • 2012

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Abstract

In this paper, we present a much simpler, direct and elegant approach to the equivalence problem of measure many one-way quantum finite automata (MM-1QFAs). The approach is essentially generalized from the work of Carlyle [J.W. Carlyle, Reduced forms for stochastic sequential machines, J. Math. Anal. Appl. 7 (1963) 167-175]. Namely, we reduce the equivalence problem of MM-1QFAs to that of two (initial) vectors. As an application of the approach, we utilize it to address the equivalence problem of enhanced one-way quantum finite automata (E-1QFAs) introduced by Nayak [A. Nayak, Optimal lower bounds for quantum automata and random access codes, in: Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), 1999, pp. 369-376]. We prove that two E-1QFAs A"1 and A"2 over @S are equivalence if and only if they are n"1^2+n"2^2-1-equivalent where n"1 and n"2 are the numbers of states in A"1 and A"2, respectively. As an important consequence, we obtain that it is decidable whether or not L(A"1)=L(A"2) where L(A)@?@S^@? denotes the set recognizable by MM-1QFA A (or by E-1QFA A) with cutpoint (or with non-strict cutpoint). This also extends a theorem of Eilenberg.