Sheng Yu. Handbook of Formal Languages, Regular Languages, volume 1, chapter 2, pages 41--110. Springer Verlag, 1997. 32  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in section 5. And we nally prove that three of the classes de ned here are identi able in the limit using positive data. We present some comments in the last section. 2 Preliminaries 2.1 Languages and automata We present here classical results on regular languages. The reader can refer to [Yu97] for general results and to [DLT01b] for results concerning RFSA. Let be a nite alphabet and be the set of words built on . We denote the empty word. We assume that words of are ordered the following way : u v i [juj jvj or (juj = jvj and u is before v in the lexicographic order) ....

Sheng Yu. Handbook of Formal Languages, Regular Languages, volume 1, chapter 2, pages 41-110. Springer Verlag, 1997. 12

.... a la limite par exemples positifs seuls dans la section 6. Nous pr esentons quelques commentaires dans la derni ere section. 2 Pr eliminaires 2.1 Langages et automates Nous pr esentons ici quelques notions et r esultats classiques sur les langages r eguliers. Le lecteur pourra se r ef erer a [Yu97] pour les r esultats g en eraux et a [DLT01b] pour tout ce qui concerne les AFER. Soient un alphabet ni et l ensemble des mots construits sur . On note la cha ne vide et juj la longueur d une cha ne u de . On suppose que les mots de sont ordonn es comme suit : u v ssi ....

Sheng Yu. Handbook of Formal Languages, Regular Languages, volume 1, chapter 2, pages 41-110. Springer Verlag, 1997. 12

....languages and uses them to add transitions to the current automaton, and obtain a RFSA. Section 5 presents experimental results of DeLeTe2, and shows that, when regular languages are generated using non deterministic ways, DeLeTe2 has good performances. 2 Preliminaries The reader may refer to [Yu97], HU79] for classical de nitions and proofs on formal language theory. The notions of prime and composed residuals languages and RFSA have been introduced and studied in [DLT01] 2.1 Regular languages, regular expressions and automata Let be a nite alphabet and let be the set of words ....

Sheng Yu. Handbook of Formal Languages, Regular Languages, volume 1, chapter 2, pages 41-110. Springer Verlag, 1997.

....alphabet are studied in Section 7. Section 8 is devoted to the study of the complexity of decision and construction problems on RFSAs. We conclude in Section 9. 2. Preliminaries In this section, we recall some definitions on finite automata. For more information, we invite the reader to consult [9, 14]. 2.1. Automata and languages Let be a finite alphabet, and let be the set of words on . We denote by the empty word and by juj the length of a word u. For an integer n, we define j juj = ng and j juj ng. A language is a subset of . A nondeterministic finite ....

Yu, S.: Handbook of Formal Languages, Regular Languages, vol. 1, chapter 2, Springer Verlag, 1997, 41--110.

....particular (and pathological) RFSA. Section 7 is devoted to the study of the complexity of our constructions : most of them are PSPACE complete. 2 Preliminaries In this section, we will remind some de nitions concerning nite states automata. For more informations, we invite the reader to consult [HU80, Yu97]. 2.1 Automata and languages Let be a nite alphabet, and let be the set of words on . We note the empty string and juj the length of a word u in . A language is a subset of . A non deterministic nite automaton (NFA) is a quintuplet A = h ; Q; Q 0 ; F; i where Q is a ....

S. Yu. Handbook of Formal Languages, Regular Languages, volume 1, chapter 2, pages 41110. Springer, 1997. 12

....We will give an example of a non regular beta shift which does satisfy the Weak Pumping Lemma. However, the Strong Pumping Lemma does completely characterize the regular beta shifts. There are pumping lemmas which give necessary and sucient conditions for regularity in the general case; see [35] for more details. First we prove Theorem 4.3.6, which shows that in general a beta shift satis es the Weak Pumping Lemma if and only if the lexicographically smallest element in the orbit closure of the expansion of 1 base does not appear in the orbit of the expansion of 1. We do this by rst ....

.... 1g satis es the Strong Pumping Lemma, but it is not regular as there is no nite state automaton which recognizes the language (if there were, there would be a nite state automaton recognizing the set of palindromes in f0; 1g , but that set is not regular see [16] For more examples, see [35]. Although the Strong Pumping Lemma is in general not sucient to show regularity, it is sucient within the class of beta shifts. Thus Theorems 4.3.3 and 4.3.12 together say that a beta shift is regular if and only if it satis es the Strong Pumping Lemma. Theorem 4.3.12. Let X be a beta shift ....

Sheng Yu, Regular languages, Handbook of Formal Languages (G. Rozenberg and A. Salomaa, eds.), vol. 1, Springer, 1997, pp. 41-105. 124

....Roughly speaking, in terms of state complexity these are ecient operations for NFAs. Again, this is essentially di erent when deterministic nite automata come to play. For example, for arbitrary alphabets in [28] a bound of (2m 1) 2 states has been shown for the DFA concatenation, and in [25] a bound of 2 states for the iteration. n state NFA. Then m n states are sucient and necessary in the worst case for an NFA to accept the language L(A)L(B) Proof. The upper bound is due to the observation that in C one has simply to connect the accepting states in A with the states in B ....

S. Yu, \Regular languages," in Handbook of Formal Languages 1 , eds. G. Rozenberg and A. Salomaa (Springer, Berlin, 1997) chapter 2, pp. 41-110.

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Sheng Yu. Handbook of Formal Languages, Regular Languages, volume 1, chapter 2, pages 41--110. Springer Verlag, 1997. 32

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