## Context-Free Languages and Push-Down Automata (1997)

Venue: | Handbook of Formal Languages |

Citations: | 67 - 0 self |

### BibTeX

@INPROCEEDINGS{Autebert97context-freelanguages,

author = {Jean-michel Autebert and Jean Berstel and Luc Boasson},

title = {Context-Free Languages and Push-Down Automata},

booktitle = {Handbook of Formal Languages},

year = {1997},

pages = {111--174},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.1 Grammars : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2. Systems of equations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.1 Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.2 Resolution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.3 Linear systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2.4 Parikh's theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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Citation Context ... parallel Turing machines nor of "succinctness" (see e.g. [52]), that is a measure of the size of the description of a language. We have chosen to present material which is not available in =-=textbooks [17, 29, 1, 47, 28, 4, 30, 32, 2]-=- (more precisely not available in more than one textbook) because it is on the borderline between classical stuff and advanced topics. However, we feel that a succinct exposition of these results may ... |

511 | Formal Languages
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Citation Context ... parallel Turing machines nor of "succinctness" (see e.g. [52]), that is a measure of the size of the description of a language. We have chosen to present material which is not available in =-=textbooks [17, 29, 1, 47, 28, 4, 30, 32, 2]-=- (more precisely not available in more than one textbook) because it is on the borderline between classical stuff and advanced topics. However, we feel that a succinct exposition of these results may ... |

336 |
Introduction to Formal Language Theory
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Citation Context ... parallel Turing machines nor of "succinctness" (see e.g. [52]), that is a measure of the size of the description of a language. We have chosen to present material which is not available in =-=textbooks [17, 29, 1, 47, 28, 4, 30, 32, 2]-=- (more precisely not available in more than one textbook) because it is on the borderline between classical stuff and advanced topics. However, we feel that a succinct exposition of these results may ... |

303 | Formal languages and their relation to automata - Hopcroft, Ullman - 1969 |

273 |
On certain formal properties of grammars
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Citation Context ...and Luc Boasson 1. Introduction This chapter is devoted to context-free languages. Context-free languages and grammars were designed initially to formalize grammatical properties of natural languages =-=[9]-=-. They subsequently appeared to be well adapted to the formal description of the syntax of programming languages. This led to a considerable development of the theory. The presentation focuses on two ... |

246 | Transductions and context-free languages
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210 |
The Theory of Parsing
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199 |
The Use of Language
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- 2001
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Citation Context ...oposition 2.2. A cycle-free (right or left) linear system has a unique solution. 14 Jean-Michel Autebert, Jean Berstel, and Luc Boasson 2.4 Parikh's theorem In this section, we prove Parikh's theorem =-=[43]-=-. Our presentation follow [44]. A more general result is given in Kuich's chapter [37]. As already mentioned, all results concerning systems of equations, provided they make sense (e.g. Greibach norma... |

163 |
On the translation of languages from left to right
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(Show Context)
Citation Context ...most derivation, because it depends on whether the word contains a letter c or a letter d. Using rightmost derivations instead of leftmost derivations leads to define the LR-grammars: Definition 6.4. =-=[28, 34]-=- A context-free grammar G = (V; P ) over the terminal alphabet A is a LR(k)-grammar if, S ? \Gamma! r mXu \Gamma! mffu = pv S ? \Gamma! r m 0 X 0 u 0 \Gamma! m 0 ff 0 u 0 = pv 0 (with u; u 0 2 A ? ; p... |

127 |
The algebraic theory of context-free languages
- Chomsky, Schutzenberger
- 1963
(Show Context)
Citation Context ...Context-free languages may indeed be defined as the components of the least solution of systems of polynomial equations, whence the term "algebraic" languages introduced by Chomsky and Schut=-=zenberger [10]. The same-=- construction was used by Ginsburg and Rice [20]. They preferred to call them ALGOL-like languages because they are "a model for the syntactic classes of the programming language ALGOL". Ind... |

113 |
The Mathematical Theory of Context-free Languages
- Ginsburg
- 1966
(Show Context)
Citation Context |

77 |
Syntax-directed transduction
- Lewis, Stearns
- 1968
(Show Context)
Citation Context ... in syntactical analysis. Given a word w over the alphabet A, define F irst k (w) as the prefix of length k of w; if jwj ! k, F irst k (w) is equal to w. We may now define LL-grammars Definition 6.3. =-=[1, 38]-=- A context-free grammar G = (V; P ) over the terminal alphabet A is a LL(k)-grammar if S ? \Gamma! ` uXm \Gamma! uffm ? \Gamma! ` uv S ? \Gamma! ` uXm 0 \Gamma! uff 0 m 0 ? \Gamma! ` uv 0 62 Jean-Mich... |

42 |
Two families of languages related to ALGOL
- Ginsburg, Rice
- 1962
(Show Context)
Citation Context ...ents of the least solution of systems of polynomial equations, whence the term "algebraic" languages introduced by Chomsky and Schutzenberger [10]. The same construction was used by Ginsburg=-= and Rice [20]. They preferred to call -=-them ALGOL-like languages because they are "a model for the syntactic classes of the programming language ALGOL". Indeed, one says "an instruction is: : : " rather than "the s... |

39 |
Parenthesis grammars
- McNaughton
- 1967
(Show Context)
Citation Context ...bfamilies of Det. We only quote here a few among the results published in the literature, in the positive case. The equivalence problem is decidable for parenthesis languages (see paragraph 6.6 below)=-=[39]-=-, for simple languages (see paragraph 6.7 below)[36], for finite-turn languages (see paragraph 6.4 below)[53], realtime languages[42] ,(pre-)NTS languages[49]. A result of S'enizergues[50] shows that ... |

38 |
Simple deterministic languages
- Korenjak, Hopcroft
- 1966
(Show Context)
Citation Context ... results published in the literature, in the positive case. The equivalence problem is decidable for parenthesis languages (see paragraph 6.6 below)[39], for simple languages (see paragraph 6.7 below)=-=[36]-=-, for finite-turn languages (see paragraph 6.4 below)[53], realtime languages[42] ,(pre-)NTS languages[49]. A result of S'enizergues[50] shows that if C is an effective cylinder (i. e. a family of lan... |

31 |
The hardest context-free language
- Greibach
- 1973
(Show Context)
Citation Context ...s ns3 letters,swill be the usual encoding : (a i ) = ab i a and (a i ) = ab i a for 1sisn. For the three other letters, we define ([) = [, (]) =] and (+) = +. We now state the Theorem 4.3 (Greibach). =-=[23] A languag-=-e L over the alphabet A is context -free iff there exists a morphism ' such that $L = ' \Gamma1 (H), where $ is a new letter. Proof. The "if" part follows directly from the closure propertie... |

26 |
Syntactic analysis and operator precedence
- Floyd
- 1963
(Show Context)
Citation Context ...grammars, an operator precedence can be defined which is inspired of the classical precedence relations of usual arithmetic operators. From a general point of view, the following holds : Theorem 3.5. =-=[28, 16]-=- Given a context-free language, an equivalent contextfree grammar in operator normal form can effectively be constructed. Proof. It is very easy. Given a grammar G in Chomsky normal form, to each pair... |

25 |
A new normal-form theorem for context-free phrase structure grammars
- Greibach
- 1965
(Show Context)
Citation Context ...oper context-free grammar G, an equivalent context free grammar in quadratic Greibach normal form can effectively be constructed from G. A weaker similar result has originally been proved by Greibach =-=[24]-=-: she showed that, given a proper context-free grammar, an equivalent contextfree grammar in Greibach normal form can effectively be constructed. The additional statement stating that this grammar can... |

23 |
A characterization of parenthesis languages
- Knuth
- 1967
(Show Context)
Citation Context ...the grammar G given by S \Gamma! aSSa + axa is parenthetic. Clearly, any pure parenthetic language over A = fa; ag is included in the Dyck language D ? 1 . The following characterization due to Knuth =-=[35]-=- shows, in particular, that D ? 1 is not (purely) parenthetic. A word u over the alphabet A = fa; ag [B is balanced if it satisfies juj a = juj a and, for any prefix v of u, jvj asjvj a . It should be... |

21 |
Bracketed context-free languages
- Ginsburg, Harrison
- 1967
(Show Context)
Citation Context ...oasson S \Gamma! SS \Gamma! xS \Gamma! xSS \Gamma! xxS \Gamma! xxx and S \Gamma! SS \Gamma! SSS \Gamma! xSS \Gamma! xxS \Gamma! xxx: A very similar family of languages has been introduced by Ginsburg =-=[19]-=-. Let ks1. Given an alphabet A = fa 1 ; : : : ; a k g, we associate to it the copy A = fa 1 ; : : : ; a k g and the alphabet Z = A[A. A grammar over Z [B with k rules is completely parenthetic if the ... |

15 |
Context-free language and push-down automata,” Information and Control
- Schützenberger
- 1963
(Show Context)
Citation Context ...unded number of topmost stack symbols in one step. Several such mechanisms have been introduced and studied in the literature (see e.g. Cole[11], Courcelle[12], Greibach[26], Nivat[41], Schutzenberger=-=[48]-=-). We present now one such accepting device. A jump pdm over A is a 4-tuple A = (Q; Z; J; T ), where Q and Z have the same meaning as in a pdm, and J is a new alphabet in bijection with Z, the element... |

14 |
Finite-turn pushdown automata
- Ginsburg, Spanier
- 1966
(Show Context)
Citation Context ...ong to the family F turn(k) if it is recognized by a k-turn pda. Then a finite-turn pda is a pda which is k-turn for some integer k. A language is finite-turn if it is recongized by a finite-turn pda =-=[21]-=-. It is easy to prove that 0-turn languages are rational. The family of finite-turn languages can be described using the bracket operation too. This definition is similar to the one of strong quasi-ra... |

11 |
The theory and application of pushdown store machines
- Evey
- 1963
(Show Context)
Citation Context ...= fa i b i c j j i; j ? 0g, then \Theta(L) = fa i b i c i+1 j i ? 0g. Hence Fact 5.3. The image through \Theta of a context-free language is not always context-free. Nevertheless, Theorem 5.4 (Evey). =-=[14]-=- Given a pushdown transducer T , if L is the context-free language recognized by the associated pda, the image \Theta(L) is a context-free language. Proof. Let T = (Q; Z; B; i; K; fl) be a pdt over A.... |

10 |
The equivalence and inclusion problems for NTS languages
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Citation Context ...thesis languages (see paragraph 6.6 below)[39], for simple languages (see paragraph 6.7 below)[36], for finite-turn languages (see paragraph 6.4 below)[53], realtime languages[42] ,(pre-)NTS languages=-=[49]-=-. A result of S'enizergues[50] shows that if C is an effective cylinder (i. e. a family of languages effectively closed under inverse homomorphism and intersection with rational sets) containing the f... |

9 |
Derivation-bounded languages
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(Show Context)
Citation Context ... it is generated by a grammar of finite index. This result can be made even more precise: the family Qrt(k) is exactly the family of languages generated by grammars of index k. We refer the reader to =-=[22, 27, 47, 4]-=- for a proof of this proposition. 6.3 Strong quasi-rational languages We present now a less usual family of languages. It is derived from the bracket operation defined above. Recall that Lin is the sm... |

9 |
Transductions des Langages de Chomsky. Annales de l’Institut Fourier 18
- Nivat
- 1968
(Show Context)
Citation Context ... that erases an unbounded number of topmost stack symbols in one step. Several such mechanisms have been introduced and studied in the literature (see e.g. Cole[11], Courcelle[12], Greibach[26], Nivat=-=[41]-=-, Schutzenberger[48]). We present now one such accepting device. A jump pdm over A is a 4-tuple A = (Q; Z; J; T ), where Q and Z have the same meaning as in a pdm, and J is a new alphabet in bijection... |

9 |
Commutative regular equations and Parikh’s theorem
- Pilling
- 1973
(Show Context)
Citation Context ...ight or left) linear system has a unique solution. 14 Jean-Michel Autebert, Jean Berstel, and Luc Boasson 2.4 Parikh's theorem In this section, we prove Parikh's theorem [43]. Our presentation follow =-=[44]-=-. A more general result is given in Kuich's chapter [37]. As already mentioned, all results concerning systems of equations, provided they make sense (e.g. Greibach normal form makes no sense in free ... |

8 |
Matrix equations and normal forms for context-free grammars
- ROSENKRANTZ
- 1967
(Show Context)
Citation Context ...contextfree grammar in Greibach normal form can effectively be constructed. The additional statement stating that this grammar can be in quadratic Greibach normal form was proved later by Rosenkrantz =-=[45]-=-. We sketch here the proof he gave; we will see below an alternative proof. Sketch of the construction: We may assume that the grammar is proper and in Chomsky normal form, that is that each right-han... |

6 |
Context-free grammar forms
- AB, Ginsburg
- 1975
(Show Context)
Citation Context ... S \Gamma! aS b bS b b + b S b \Gamma! aS b bS b + ": If we need a proper grammar in operator normal form, we just apply the usual algorithm to make it proper. Remark 3.2. The theory of grammar f=-=orms [13]-=- develops a general framework for defining various similar normal forms. These are defined through patterns like VAV + A indicating that the right members have to lie in VAV [ A. From this point of vi... |

6 |
Normal-form transformations of context-free grammars
- HOTZ
- 1978
(Show Context)
Citation Context ...orem 3.3. Given a proper context-free grammar G, an equivalent context free grammar in quadratic double Greibach normal form can effectively be constructed from G. This result has been proved by Hotz =-=[31]-=-. We follow his proof. It should be noted that the same technique allows to give an alternative proof of the previous theorem 3.2. The proof of theorem 3.3 turns out to be a complement to the proof of... |

5 |
On the intersection of stacks and queues
- Brandenburg
- 1988
(Show Context)
Citation Context ...aracterize the languages in Lin " Ocl on one hand, and, to describe any context-free language by substituting linear and one-counter languages on the other hand. The first question is still open =-=(see [8]-=- for results on this theme). The second question has a negative answer: defining the family of Greibach languages as the substitution closure of linear and one-counter languages, we get a large strict... |

5 |
Transductions de séries formelles
- Fliess
- 1974
(Show Context)
Citation Context ...ired. Even finiteness of the number of equations is not necessary. Next, the reader should check that all results also hold for partially commutative free monoids (this was observed already by Fliess =-=[15]-=-). Indeed, the argument used in most proofs is just an induction on length, and thus carries over to such monoids. 6 Jean-Michel Autebert, Jean Berstel, and Luc Boasson 2.1 Systems For the definition ... |

4 |
Two iteration theorems for some families of languages
- Boasson
- 1973
(Show Context)
Citation Context .... The proof that this holds is more technical than in the case of linear languages. The idea is to prove an iteration lemma for one-counter languages and to use it to get the desired strict inclusion =-=[6]-=-. We will give later on such counter-examples, but we will not state this lemma which is too technical and beyond the scope of this presentation. As in the case of linear languages, the definition of ... |

4 |
A few remarks on the index of context-free grammars and languages
- Gruska
- 1971
(Show Context)
Citation Context ... it is generated by a grammar of finite index. This result can be made even more precise: the family Qrt(k) is exactly the family of languages generated by grammars of index k. We refer the reader to =-=[22, 27, 47, 4]-=- for a proof of this proposition. 6.3 Strong quasi-rational languages We present now a less usual family of languages. It is derived from the bracket operation defined above. Recall that Lin is the sm... |

4 |
The equivalence problem for real-time d.p.d.a's
- Oyamaguchi
- 1987
(Show Context)
Citation Context ...em is decidable for parenthesis languages (see paragraph 6.6 below)[39], for simple languages (see paragraph 6.7 below)[36], for finite-turn languages (see paragraph 6.4 below)[53], realtime languages=-=[42]-=- ,(pre-)NTS languages[49]. A result of S'enizergues[50] shows that if C is an effective cylinder (i. e. a family of languages effectively closed under inverse homomorphism and intersection with ration... |

3 |
Theorie des langages et des automates
- Autebert
(Show Context)
Citation Context |

3 |
Deterministic pushdown store machines and realtime computation
- Cole
- 1971
(Show Context)
Citation Context ...device, this leads to use some mechanism that erases an unbounded number of topmost stack symbols in one step. Several such mechanisms have been introduced and studied in the literature (see e.g. Cole=-=[11]-=-, Courcelle[12], Greibach[26], Nivat[41], Schutzenberger[48]). We present now one such accepting device. A jump pdm over A is a 4-tuple A = (Q; Z; J; T ), where Q and Z have the same meaning as in a p... |

2 |
On jump deterministic pushdown automata
- Courcelle
- 1977
(Show Context)
Citation Context ...ads to use some mechanism that erases an unbounded number of topmost stack symbols in one step. Several such mechanisms have been introduced and studied in the literature (see e.g. Cole[11], Courcelle=-=[12]-=-, Greibach[26], Nivat[41], Schutzenberger[48]). We present now one such accepting device. A jump pdm over A is a 4-tuple A = (Q; Z; J; T ), where Q and Z have the same meaning as in a pdm, and J is a ... |

2 |
Church-Rosser controlled rewriting systems and equivalence problems for deterministic context-free languages
- S'enizergues
- 1989
(Show Context)
Citation Context ...h 6.6 below)[39], for simple languages (see paragraph 6.7 below)[36], for finite-turn languages (see paragraph 6.4 below)[53], realtime languages[42] ,(pre-)NTS languages[49]. A result of S'enizergues=-=[50]-=- shows that if C is an effective cylinder (i. e. a family of languages effectively closed under inverse homomorphism and intersection with rational sets) containing the family Rat of rational sets for... |

1 |
Some observations on hardest context-free languages
- Autebert, Boasson, et al.
- 1981
(Show Context)
Citation Context ...istic version of L comes from the following Proposition 4.1. The language ND(L) is deterministic context-free iff L is regular; in this case, ND(L) is regular too. For a proof, we refer the reader to =-=[3]-=-. We now end this short preparation by the Lemma 4.1. Given an alphabet A, there exists a morphismssuch that HA =s\Gamma1 (H). 28 Jean-Michel Autebert, Jean Berstel, and Luc Boasson Proof. Let H = HB ... |

1 |
Canonical forms of context-free grammars and position restricted grammar forms
- Blattner, Ginsburg
- 1977
(Show Context)
Citation Context ...L 1 ; : : : ; Ln ) be a vector of languages over V [A. This defines a substitution as follows. (1) "(L) = f"g (2) a(L) = a a 2 A (3) X i (L) = L i i = 1; : : : ; n (4) uv(L) = u(L)v(L) u; v =-=2 (V [ A) (5)-=- Q(L) = S w2Q w(L) Q ae (V [ A) Observe that the last equation implies that Q(L[M ) = Q(L)[Q(M ), where L[M is componentwise union. A vector L = (L 1 ; : : : ; Ln ) of languages over A is a solution i... |

1 |
Familles de langages translatables et ferm'ees par crochet
- Boasson, Crestin, et al.
- 1973
(Show Context)
Citation Context ...rstel, and Luc Boasson where [L] denotes the bracket closure of the family L. Finally, we denote by Sqrt, the infinite union of the families Sqrt(k). This is the family strong quasirationalslanguages =-=[7]-=-. Clearly, for each k, Sqrt(k) ` j Qrt(k). The following more precise fact holds Fact 6.2. There exists a language in Qrt(2) which is not in Sqrt. Such a language is the language M of the example abov... |

1 |
Some uniformely erasable families of languages. Theoret
- Ginsburg, Goldstine, et al.
- 1976
(Show Context)
Citation Context ...ent of the above proof can be also used to show that the linear language fw#ew j w 2 fa; bg ? g is not one-counter; it needs to prove that a one-counter pda may always be supposed to be realtime (see =-=[18]-=-). This shows, in particular, that the family Ocl is a strict subfamily of the family of context-free languages. As in the case of linear languages, it can be seen that OclsRat = Ocl whence the inclus... |

1 |
pda's, deterministic context-free languages, principal afdl's and polynomial time recognition
- Jump
- 1973
(Show Context)
Citation Context ...terministic case, it only remains to prove that a deterministic language can be recognized by a jump dpda. The proof is very technical and lengthy, so we refer the interested reader either to Greibach=-=[25]-=-, or to Cole[11]. An other model considered allows to erase rational segments of the stack word. This is clearly a generalization of jump pdm, since in a jump pdm, the erased factors have the form zh ... |

1 |
pda's and hierarchies of deterministic cf languages
- Jump
- 1974
(Show Context)
Citation Context ...e mechanism that erases an unbounded number of topmost stack symbols in one step. Several such mechanisms have been introduced and studied in the literature (see e.g. Cole[11], Courcelle[12], Greibach=-=[26]-=-, Nivat[41], Schutzenberger[48]). We present now one such accepting device. A jump pdm over A is a 4-tuple A = (Q; Z; J; T ), where Q and Z have the same meaning as in a pdm, and J is a new alphabet i... |

1 |
Formale Sprachen. B.I.-Wissenschaftsverlag
- Hotz, Estenfeld
- 1981
(Show Context)
Citation Context |

1 |
The power of the Greibach normal form
- Hotz, Kretschmer
- 1989
(Show Context)
Citation Context ...ces of a very general situation (see [5]). 4. Applications of the Greibach normal form 4.1 Shamir's theorem We present a first application of Greibach normal form. The presentation given here follows =-=[33]-=-. Recall that, given an alphabet V containing n letters, we denote by D ? V the Dyck set over the alphabet (V [ V ). Given a word m 2 (V [ V ) ? , we denote e m the reversal of m. We denote P((V [ V )... |

1 |
Formal power series, chapter 9
- Kuich
(Show Context)
Citation Context ...fact, most results can be treated differently, in at least two ways: first, they hold in a much more general framework, namely for formal power series over suitable semirings (see the chapter of Kuich=-=[37]-=-); next, there are general results, such as fixed-point theorems in conveniently ordered sets, that imply easily the present results. The present style of exposition was chosen to show what the minima... |

1 |
Transductions des langages de Chomsky, Ch. VI, mim'eographi'e
- Nivat
- 1967
(Show Context)
Citation Context ...ng to each letter a the language L a generated by the grammar G a . We then may repeat such an operation giving raise to an iterated sum of grammars. It then follows immediately that Proposition 6.4. =-=[40]-=- A language L is quasi-rational iff there exists a grammar generating L that is an iterated sum of linear grammars. Example 6.2. (continued) Consider the language L generated by the linear grammar S \... |

1 |
Pushdown automata with terminal languages
- Sakarovitch
- 1981
(Show Context)
Citation Context ...ode of acceptance is generally chosen to give the simplest proofs for one's purpose. Other modes of acceptance than the ones quoted above have been investigated. For instance, a result of Sakarovitch =-=[46]-=- shows that if K = [ q2Q fqg \Theta L q with L q context-free, then the language recognized remains context-free. The characterization of context-free languages in terms of languages recognized by pda... |