Another approach to the equivalence of measure-many one-way quantum finite automata and its application
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-1QFA5). 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-1QFA5 A(1) and A(2) over Sigma 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) subset of Sigma* 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. (C) 2012 Elsevier Inc. All rights reserved.
equivalence measure-many one-way quantum finite automata quantum finite automata enhanced one-way quantum finite automata