东华大学学报：英文版

Closure of q-Quantum Languages

Zhiwen Mo

Published Jan 1, 2007• 0 citations

Abstract

η-quantum languages are discussed and some of their properties are derived. Furthermore the q-quantum language is defined. It is shown that L(A1A2)=L(A1)∩L(A2), L(A)=L(A1)∪L(A2). So over the same alphabet the intersection and union of two different q-quantum languages are also q-quantum languages.

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东华大学学报：英文版

Closure of q-Quantum Languages

Zhiwen Mo

Published Jan 1, 2007• 0 citations

OpenAlex

Abstract

Related papers

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Jan 1, 20030 citations

Two quantum finite automata are equivalent if for all input string

ω

over the input alphabet the two automata accept

ω

with equal probability. In [Theoret. Comput. Sci. 410 (2009) 3006-3017], it was shown that a

k_1

-letter QFA

\mathcal{A}_1

and a

k_2

-letter QFA

\mathcal{A}_2

over

Σ=\{σ\}

, are equivalent if and only if they are

(n_1+n_2)^4+k-1

-equivalent where

n_i

is the number of states of

\mathcal{A}_i

i=1,2

, and

k=\max\{k_1,k_2\}

. In this letter, we improve the above upper-bound to

(n_1^2+n_2^2-1)+k

. This also answers an open problem of Qiu et al. [Acta Informatica 48 (2011) 271-290]. Further, we show that, in the case of

Σ=\{σ_1,...,σ_t\}

with

2\leq t<\infty

, there exists an integer

z

such that

\mathcal{A}_1

and

\mathcal{A}_2

are equivalent if and only if they satisfy

z

-equivalent.

{\cal A}_1

and

{\cal A}_2

with control languages (regular languages)

{\cal L}_1

and

{\cal L}_2

, respectively, are equivalent if and only if they are

(c_1n_1^2+c_2n_2^2-1)

-equivalent. Furthermore, if

{\cal L}_1

and

{\cal L}_2

are given in the form of DFAs, with

m_1

and

m_2

states, respectively, then there exists a polynomial-time algorithm running in time

O ((m_1n_1^2+m_2n_2^2)^4)

that takes as input

{\cal A}_1

and

{\cal A}_2

and determines whether they are equivalent. \item[(ii)] Two MM-1QFAs

{\cal A}_1

and

{\cal A}_2

with

n_1

and

n_2

states, respectively, are equivalent if and only if they are

(3n_1^2+3n_2^2-1)

-equivalent. Furthermore, there is a polynomial-time algorithm running in time

O ((3n_1^2+3n_2^2)^4)

that takes as input

{\cal A}_1

and

{\cal A}_2

and determines whether

{\cal A}_1

and

{\cal A}_2

are equivalent.