Jean Berstel. Transductions and Context-Free Languages. Teubner Verlag, Stuttgart, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....w if and only if block has neither a forward nor a backward correlation. Example 1. We illustrate these de nitions by the following diagram. The relevant relations between blocks are on a spiral from the leftmost block to the block in the middle, indicated by solid and dotted arrows. 3] 5] [1] [6] 10] 8] 4] 5] 2] 10] 6] 6 6 6 6 6 The blocks 1, 2, and 3 have a forward correlation. Block 4 and 5 have a forward crack. Forward correlations and cracks are indicated by solid and dotted arrows from left to right, respectively. Blocks 2, 3, and 4 have a backward crack, ....

....again indicated by dotted and solid arrows, this time from right to left. Block 4 is isolated, since it has neither a forward nor a backward correlation. De nition 4. Let m; r 2 N, i 1 ; i 2m 1 2 N, and w = i 1 ] i 2m 1 ] r w) ri 1 ] ri 2m 1 ] z m = [1] for m = 1 [1] 4 z m 1 ) 2] for m 1 For example z 4 = 1] 4] 16] 64] 32] 8] 2] L even;r : f(r z m ) j m 2 Ng L even : S r2N L even;r Lmin : L even;1 . Note that Lmin = fzm j m 2 Ng, and that a word is in L even if and only if it has no cracks. For a word with an isolated ....

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J. Berstel. Transductions and Context-Free Languages. Teubner, 1979.

....a letter (section 7) and the class UC(F ) is equal to the class of deterministic context sensitive transductions (section 8) A part of this work has been already published as extended abstract in [5] 2. Preliminaries We assume the reader to be familiar with basic formal language theory (see [2,3] for more precisions) The goal of this section is to x notations and terminology. 2.1. Words and languages For a nite alphabet Sigma, we denote by Sigma the free monoid generated by Sigma. The neutral element of this monoid is the empty word, which is denoted by . The size of the alphabet ....

....state q i 2 I and a nal state q f 2 F such that (q f ; v) 2 (q i ; u)ffi . The transduction associated with T is dened by: q i 2I; qf 2F f(u; v) 2 X j (q f ; v) 2 (q i ; u)ffi g: The next theorem presents well known closure properties of the class of rational transductions (see [2] for detailed proofs of these properties) Theorem 2.2. The class of rational transductions is closed under union, composition and inverse. A transduction is length preserving if and only if for each couple (u; v) 2 we have juj = jvj. In the remainder of the paper, we consider only ....

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Berstel, J. Transductions and Context-Free Languages. Teubner Studienb#cher, Stuttgard, 1979.

....will elaborate on this and show that, if the homomorphism is simple in the sense of [21] one can conclude that relative liveness properties that hold on the abstract system also hold on the concrete system. 3 Preliminaries For de ning our concepts, we need several notions from language theory [5, 7, 11, 24]. Let L be a language and let L be an language. De nition 3.1 The left quotient of L by a word w 2 is de ned by cont(w; L) fv 2 j wv 2 Lg. The left quotient of L by w 2 is similarly de ned by cont(w; L ) fx 2 j wx 2 L g. The left quotient describes ....

Berstel, J. Transductions and Context-Free Languages, rst ed. Studienbucher Informatik. Teubner Verlag, Stuttgart, 1979.

....if (L) normalized rational if (L) rational if L = L ) for some regular language L . The corresponding classes are denoted by REC(M ) NRAT(M ) and RAT(M ) respectively. We have REC(M) NRAT(M) RAT(M ) The classes REC(M) and RAT(M) are classical, see e.g. [4], their de nitions do neither depend on nor on as can be seen easily. The de nition of NRAT(M) is less robust, it depends on the normal form mapping . The classes REC(M) and NRAT(M) are Boolean algebras, whereas RAT(M) is not a Boolean algebra in general. For free monoids we have REC(M) ....

....Z=2Z, then there must exist an a 2 n such that a 6= a in G P. Next assume that the graph (V; E) consists of two non empty disjoint components (V 1 ; E 1 ) and (V 2 ; E 2 ) which de ne graph products G P 1 and G P 2 , respectively. Then G P = G P 1 GP 2 . Furthermore by Mezei s Theorem, see e.g. [4], every L 2 REC(GP) is a nite union of sets of the form L 1 L 2 with L i 2 REC(G P i ) Thus, we may apply Proposition 2 and proceed with the two graphs (V 1 ; E 1 ) and (V 2 ; E 2 ) Hence, for the rest of the proof we may assume that the graph (V; E) is connected. Furthermore since by ....

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J. Berstel. Transductions and context{free languages. Teubner Studienbucher, Stuttgart, 1979.

....has moreover a deterministic input automaton. It is then called sequential. Used as an encoder, this means that the output codeword is obtained sequentially from the input data. Transducers which are not sequential, but which realize sequential functions, can be rst determinized (see for instance [4] Institut Gaspard Monge, Universit e de Marne la Vall ee, 5 boulevard Descartes, 77454 Marne la Vall ee Cedex 2, France. http: www igm.univ mlv.fr fbeal,cartong or [3] In the case of sequential transducers, there exists a minimal equivalent sequential transducer, even if the output labels ....

....in the coming algo2 rithm. The time complexity is independent of the size of the alphabet. The algorithm consists in decreasing by 1 the value P at each step. We present our algorithm for sequential transducers but it can be directly extended to the case of subsequential transducers (see [7] or [4] for the de nition of a subsequential transducer) In Section 2, we recall some basic de nitions from automata theory and we de ne the pre x automaton of an automaton. The computation algorithm of the pre x of an automaton is presented in Section 3. The complexity is analyzed in Section 4. ....

Berstel, J. Transductions and Context-Free Languages. B.G. Teubner, 1979.

.... on the concepts of recognizable (i.e. recognized by finite automata) and rational (i.e. defined by means of rational expressions) trace languages, which led to the development of a common generalization of the classical theories of regular languages, semi )linear sets and rational relations [4]. Similarly to the theory of regular languages, the relationships between descriptions of languages using finite monoids, rational operations, standard automata, asynchronous automata and logic were established (see [15] It is well known that for regular word languages all basic problems are ....

J. Berstel, Transductions and Context-Free Languages, Teubner, Stuttgart, 1979.

....belongs to R, b) if S and T are in R, then so are ST and S [ T , c) if S is in R, then so is S , the submonoid of R generated by S. It is easy to show that if S is rational, then hSi is also rational. In fact, the rational subgroups of the free group can be characterized as follows (see [5] for a proof) Proposition 3.1 A subgroup of FG(A) is rational if and only if it is nitely generated. We can now state our rst characterization. Theorem 3.2 A subset S of the free group FG(A) is accepted by a reversible automaton if and only if S is a nite union of left cosets of nitely ....

J. Berstel, Transductions and Context Free Languages, Teubner Verlag, 1979.

....the free group are related as follows. Let A = A[A , where A A = and let K ae A be the set of group reduced words. We have the canonical injection : FG(A) A and the canonical maps : A K, where wffi is the unique reduced word v with v = w. Then a theorem of Benois [7] states a subset S of FG(A) is rational if and only if the subset S ffi of A is rational. The profinite group topology on A (FG(A) is the smallest topology such that every monoid (group) morphism from A (FG(A) onto a finite group G is continuous. This topology was first considered ....

....next theorem follows from well known results about rational sets in free groups. Lemma 5.11. Let L; L 2 Rat(FG(A) be given by rational expressions. Then it is decidable whether or not L = L . Proof. Let L 2 Rat(FG(A) be given by a rational expression. Then by using the theorem of Benois [7], one can effectively construct a finite state automaton B(L) over A that jB(L)j = Lfi , the set of reduced words representing L. But L = L Lfi = L fi and the result follows since equality is decidable for languages specified by finite state automata. Corollary 5.12. Let L 2 ....

J. Berstel, Transductions and Context Free Languages, Teubner Verlag, 1979.

....is ffflg [ X [ X : is the family of rational subsets and is denoted by Rat(M ) As a particular case, given two monoids M and N , a function of M into N is rational if its graph is a rational subset of the product monoid M Theta N . We refer the reader to the two handbooks [5] and [2] for basic results in this theory. It is well known that in the case of a direct product of free monoids n , the family of rational subsets, also called rational relations, is precisely the family of relations recognized by finite automata. Indeed, the notion of finite automaton designed to ....

J. Berstel. Transductions and context-free languages. B. G. Teubner, 1979.

....algorithm when used with an acyclic graph weighted with real valued numbers. The Generic Topological Single SourceShortest Distance algorithm is useful among other things for computing the coefficients of a rational power series represented by a weighted automaton or a weighted transducer [5, 6, 10, 36, 23]. An e#cient implementation of the Generic Single Source Shortest Distance algorithm with various queue disciplines including the generic topological order can be found in the latest version of the FSM library [30] The complexity of the Generic Topological Single Source Shortest Distance ....

J. Berstel, Transductions and Context-Free Languages. Teubner Studienbucher, Stuttgart, 1979.

....language L, there is a minimal one for this partial order. This automaton is called the minimal automaton of L. Again, there are standard algorithms for minimizing a given finite automaton [21] 5. 2 Transducers The modelling power of finite automata can be enriched by adding an output function [1, 11]. Let k be a semiring. The definition of a k transducer (or automaton with output in k) is quite similar to that of a finite automaton. It is also a quintuple A = Q; A; E; I; F ) where Q (resp. I, F ) is the set of states (resp. initial and final states) and A is the alphabet. But the set of ....

J. Berstel, 1979, Transductions and Context-Free Languages , Teubner, Stuttgart.

....parsing time. 2 Preliminaries Let be a nite alphabet. Let u = a 1 an 2 be a word over , where a i 2 for 1 i n. The length of u is juj : n. For 1 i n we de ne u[i] a i . The empty word is denoted by . Formal languages and their concatenation are de ned as usual [1]. Let and be two alphabets. A substitution is the homomorphic extension of a mapping : 2 . For a 2 , L , and L , the single symbol substitution de ned by (a) L and (b) fbg for each b 2 n fag is denoted [a= L] We write L[a= L] for [a= L] L) For L ....

....only if it is not Parikh bounded. Proof. If X is pumpable then by de nition we have a pumping tree, which pumps occurrences of X . Hence X is not Parikh bounded. If X is not Parikh bounded, we choose a word with suciently many occurrences of X . We mark them according to Ogden s iteration Lemma [1] for context free grammars, and obtain a pumping tree which is appropriate to show the pumpability of X . ut In the sequel we will use the notion pumpable symbol synonymous for non Parikh bounded symbols, without explicitly referencing the previous lemma. 5 Computation of Parikh Suprema There is ....

J. Berstel. Transductions and context-free languages. Teubner, Stuttgart, 1979.

....; ng is abbreviated as [n] with [0] We write Dom(R) resp. Ran(R) for the domain (resp. range) of a relation R. We assume that the reader is familiar with the notions of monoid, word, rational and recognizable subsets of a monoid, regular expression, nite automaton and transducer (see e.g. [3]) as well as pushdown automaton (see e.g. 1] and Turing machine (see e.g. 12] The family of nite (resp. recognizable, rational) subsets of a monoid M is written Fin(M) resp. Rec(M) Rat(M) The free monoid over is written , stands for the empty word and the length of a word w 2 ....

J. Berstel. Transductions and ContextFree Languages. B. G. Teubner, Stuttgart, 1979.

....section a new proof that the recognizable power series in commuting variables are strictly included in the rational power series. To this aim, we extend to formal power series a theorem due to Mezei, characterizing the recognizable languages over a product monoid. Theorem 3.4. 2 (Mezei s Theorem, [6], 21] A language L over a product monoid M x M2 is recognizable if and only if it is a finite union of languages L x L2, with Li recognizable language over Mi, i 1, 2. This result has been extended in [71] to recognizable series over X x X, with X, X2 alphabets, and we prove it here in ....

....To this aim, we use the so called multiplicity approach, as detailed in the following. 6.6. 1 The multiplicity approach The commutation of two polynomials, or of two formal power series, in noncommuting variables, with coefficients in a field, is characterized in two results due to Bergman, 1969, [6], for polynomials, and to Cohn, 1962, 16] for formal power series. These two results, recalled in Section 6.1, characterize the commutation of two multisets of words similarly as in free monoids. In this sense, the BTC property and Conjecture 3 are the equivalent of Bergman s and Cohn s results ....

J. Berstel, Transductions and Context-free Languages, B.G. Teubner, Stuttgart, 1979.

....A and B, respectively. Then, L is SNT reducible to L , denoted by L L , iff there exists an SNT T such that for each string W 2 A , W 2 L iff T (W ) L 6= The following proposition is a simple special case of more general results on transductions which can be found e.g. in [12, 5, 23] (in particular, see [23, Theorem 11.2, p. 276] Proposition 8.1 The class of regular languages is closed under SNT reductions, i.e. if L L and L regular, then also L is regular. Theorem 8.2 The class ESO(9 ) is regular. Proof. Let Phi be an ESO(9 ) formula of the form 9P9z 1 : ....

J. Berstel. Transductions and Context-Free Languages. Teubner, Stuttgart, 1979.

....the study of these tools has led to efficient parallel algorithms (see [2, 15] for instance) The theory of rational transductions was mainly developed by M. P. Schutzenberger, S. Eilenberg and M. Nivat (see [4, 16, 17] This theory is now well established and its basic results can be found in [3, 4]. More recently, some representation theorems were achieved in terms of compositions of morphisms and inverse morphisms [19, 8] At the contrary, there is only a few papers dealing with iteration of rational transductions (see [7, 20] Since several mechanisms of computation are actually ....

....languages by transductions of Fin(F ) Proposition 4.13) and in the last subsection we show that this class corresponds to the class of deterministic context sensitive transductions (Proposition 4. 18) 2 Preliminaries We assume the reader to be familiar with basic formal language theory (see [3, 4] for more precisions) The goal of this section is to fix notations and terminology. 2.1 Words and languages For a finite alphabet Sigma, we denote by Sigma the free monoid generated by Sigma. The neutral element of this monoid is the empty word, which is denoted by . The size of the ....

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Berstel, J. Transductions and Context-Free Languages. Teubner Studienbucher, Stuttgard, 1979.

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J. Berstel, Transductions and context-free languages, Teubner, Stuttgart, (1979).

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J. Berstel, Transductions and context-free languages, Teubner, Stuttgart, (1979).

....not consider complexity results at all, neither of recognition by various classes of sequential or 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 give some insight in the theory of context free languages for advanced beginners, and also provide ....

....of LL and LR languages. 6.1 Linear languages The simplest way to define the family of linear languages is by grammars: a context free grammar is linear if each right member of the rules contain at most one variable. A context free language is linear if there exists a linear grammar generating it [10, 4]. We denote by Lin the family of linear languages. Naturally, the first question that arises is whether Lin is a proper subfamily of the family of context free languages. This is easily seen to be true. Many proofs are possible. Here is an example of a context free language which is not linear: ....

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J. Berstel. Transductions and Context-Free Languages. Teubner Verlag, 1979.

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Jean Berstel. Transductions and Context-Free Languages. Teubner Verlag, Stuttgart, 1979.

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J. Berstel. Transductions and Context-Free Languages. Teubner, Stuttgart, 1979.

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J. Berstel. Transductions and Context-Free Languages. Teubner, 1979.

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Berstel, J.: Transductions and Context-Free Languages. Teubner (1979)

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Berstel, Jean. 1979. Transductions and Context-Free Languages. Teubner, Stuttgart.

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Jean Berstel. 1979. Transductions and Context-Free Languages. Teubner, Stuttgart.

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Jean Berstel. Transductions and context-free languages. Teubner Studienb ucher, Stuttgart, 1979.

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J. Berstel. Transductions and context-free languages. Teubner Studienbucher, Stuttgart, 1979.

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