By A. de Luca, F. Mignosi, D. Perrin, G. Rozenberg (eds.)
Read or Download Theoretical Computer Science, Volume 340, Issue 2, Pages 179-456 (27 June 2005), The Art of Theory PDF
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Additional resources for Theoretical Computer Science, Volume 340, Issue 2, Pages 179-456 (27 June 2005), The Art of Theory
Sample text
Therefore, the only return word that appears before position (an+1 − i)qn is tn , the shortest return word. We conclude that the first occurrence of tni tn−1 as a return word of tni tn−1 in f is (an+1 − i)qn . Propositions 3 and 4 are actually valid for n 0, though we have proved them only for n 2. The proofs for smaller values of n have to be made separately and are rather technical; they appear in an appendix at the end of this paper. Example 8. Let f and w = 001 be as in Example 2. Thus |w| = 3 and hence |w| ∈ ]q1 + q0 − 2, 2q1 + q0 − 2] = [3, 5].
The length of is 0. For any w ∈ A∗ and a ∈ A, |w|a denotes the number of occurrences of the letter a in w. Let w ∈ A∗ . The word u is a factor (or subword) of w if there exist words p, q such that w = puq. A factor u of w is called proper if u = w. , a suffix) of w. For any w ∈ A∗ , we denote by Fact w, the sets of its factors. For any X ⊆ A∗ , we set Fact X = u∈X Fact u. An element of Fact X will be also called a factor of X. A. Carpi, A. de Luca / Theoretical Computer Science 340 (2005) 220 – 239 223 A set X is called dense if any word of A∗ is a factor of X.
If aba 3 u ∈ X∗ , there exist v ∈ X∗ and x ∈ X such that aba 3 u = xv . , x = aba or a. In the first case, v = aau ∈ X ∗ . , X = A. A. Carpi, A. de Luca / Theoretical Computer Science 340 (2005) 220 – 239 229 Lemma 14. Let X be a finite PER-complete central code. Then a ∈ X or b ∈ X. Proof. Consider the word w = (aab)n aaa(baa)n with 3n (X). As one easily verifies, w = (a 2 bn a), so that w ∈ PER. Since X is PERcomplete, there exist words , ∈ A∗ such that w ∈ X ∗ . We have to distinguish three cases: (1) (aab)n a, aa(baa)n ∈ X∗ , (2) (aab)n aa, a(baa)n ∈ X∗ , (3) (aab)n = u, (baa)n = v with x = uaaav ∈ X, , ∈ X∗ , u, v ∈ A∗ .
Theoretical Computer Science, Volume 340, Issue 2, Pages 179-456 (27 June 2005), The Art of Theory by A. de Luca, F. Mignosi, D. Perrin, G. Rozenberg (eds.)
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