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**1 - 7**of**7**### The average state complexity of the star of a ﬁnite set of words is linear

"... We prove that, for the uniform distribution over all sets X
of m (that is a ﬁxed integer) non-empty words whose sum of lengths is
n, D_X , one of the usual deterministic automata recognizing X^∗ , has on
average O(n) states and that the average state complexity of X^∗ is Θ(n).
We also show that the ..."

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We prove that, for the uniform distribution over all sets X
of m (that is a ﬁxed integer) non-empty words whose sum of lengths is
n, D_X , one of the usual deterministic automata recognizing X^∗ , has on
average O(n) states and that the average state complexity of X^∗ is Θ(n).
We also show that the average time complexity of the computation of
the automaton D_X is O(n log n), when the alphabet is of size at least
three.

### July 9, 2012 12:27 WSPC/INSTRUCTION FILE muldebru˙new De Bruijn Sequences Revisited

"... A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings: the multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of or ..."

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A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings: the multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of order n is a word such that every word of length n appears exactly once in w as a factor that starts at a position im + 1 for some integer i ≥ 0. A pseudo de Bruijn sequence of order n with respect to an antimorphic involution θ is a word such that for every word u of length n the total number of appearances of u and θ(u) as a factor is one. We show that the number of m-shift de Bruijn sequences of order n is an!a(m−n)(an−1) for 1 ≤ n ≤ m and is (am!) a n−m for 1 ≤ m ≤ n, where a is the size of the alphabet. We provide two algorithms for generating a multi-shift de Bruijn sequence. The multi-shift de Bruijn sequence is important for solving the Frobenius problem in a free monoid. We show that the existence of pseudo de Bruijn sequences depends on the given alphabet and antimorphic involution, and obtain formulas for the number of such sequences in some particular settings. 1.

### De Bruijn Sequences Revisited

"... A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings: the multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of or ..."

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A (non-circular) de Bruijn sequence w of order n is a word such that every word of length n appears exactly once in w as a factor. In this paper, we generalize the concept to different settings: the multi-shift de Bruijn sequence and the pseudo de Bruijn sequence. An m-shift de Bruijn sequence of ordern is a word such that every word of length n appears exactly once in w as a factor that starts at a position im + 1 for some integer i ≥ 0. A pseudo de Bruijn sequence of order n with respect to an antimorphic involutionθ is a word such that for every worduof lengthnthe total number of appearances of u and θ(u) as a factor is one. We show that the number of m-shift de Bruijn sequences of ordernis a n!a (m−n)(an −1) for1 ≤ n ≤ m and is (am!) an−m for 1 ≤ m ≤ n, where a is the size of the alphabet. We provide two algorithms for generating a multi-shift de Bruijn sequence. The multi-shift de Bruijn sequence is important for solving the Frobenius problem in a free monoid. We show that the existence of pseudo de Bruijn sequences depends on the given alphabet and antimorphic involution, and obtain formulas for the number of such sequences in some particular settings. 1

### and

, 2009

"... The computational complexity of universality problems for prefixes, suffixes, factors, and subwords of regular languages ..."

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The computational complexity of universality problems for prefixes, suffixes, factors, and subwords of regular languages

### The Average State Complexity of Rational Operations on Finite Languages ∗

"... Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear. ..."

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Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear.

### BGN The Average State Complexity of Rational Operations on Finite Languages ∗

"... Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear. 1. ..."

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Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear. 1.