P. Flajolet. Analytic models and ambiguity of context--free languages. Theoretical Computer Science, 49:283--309, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... = d Gamma1 (dn) Q d Gamma1 i=0 (n i) 2) dn (d Gamma1) 2 Algebraicity would imply asymptotics of the type C:A = Gamma(1 Gamma r)n r ) with C and A algebraic numbers and r a rational number 62 f1; 2; 3; g (classical result from singularity analysis [5]) In our case, for odd d 1, r is an integer and for even d 2, Gamma(1 Gamma (d Gamma 1) 2) 2 Q ( but 62 Q . Thus, the generating function of Young tableaux of height at most d (d 3) is transcendental and D finite. Due to well known one to one correspondences, this result ....

Flajolet (Philippe). -- Analytic models and ambiguity of context-free languages. Theoretical Computer Science, vol. 49, n 2-3, 1987, pp. 283--309.

....ones. The algebraic series are those satisfying an algebraic equation. More generally, the hypergeometric series are those such that the quotient of two successive terms is given by a rational fraction (see [21] The class of algebraic series is linked with the class of context free sets (see [19]) A typical example of a context free set is the set of words on the binary alphabet fa; bg having as many a s as b s. We compute below its length distribution which is an algebraic series. Example 9. The set of words on A = fa; bg having an equal number of occurrences of a and b is a submonoid ....

Flajolet, P. Analytic models and ambiguity of context-free languages. Theoret. Comput. Sci. 49 (1987), 283-309.

....semilinear languages but undecidable for DT0L languages. As a consequence all problems are decidable for context free languages. TUCS Research Group Mathematical Structures of Computer Science 1 Introduction Length considerations are often useful in language theory. For example, Flajolet [5] has shown that the inherent ambiguity of many context free languages can be deduced from the transcendentality of their generating functions. Other deep results based on length considerations are well known, e.g. in the theory of Lindenmayer systems (see Rozenberg, Salomaa [18] Andrasiu, ....

P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci. 49 (1987), 283-309.

....Dn =n =2 . Remark. The referee raised the interesting problem of whether or not 0 (z) is algebraic. If 0 (z) is algebraic then b n = P k i=1 c i n p n i n O(n q n ) where j 1 j = Delta Delta Delta = j k j = 1, p 2 Q Gamma f Gamma1; Gamma2; Gamma3; g and q p [6]. Thus our result implies that 0 (z) is not algebraic if is even. The author would like to thank Marc Bourdon and Mark Pollicott for helpful discussions about hyperbolic groups and growth series. 1. Growth series We begin by recalling the notion of growth series for finitely generated groups. ....

P. Flajolet, Analytic models and ambiguity of context-free languages, Theoretical Computer Science 49 (1987), 283-309.

....ones. The algebraic series are those satisfying an algebraic equation. More generally, 8 the hypergeometric series are those such that the quotient of two successive terms is given by a rational fraction (see [26] The class of algebraic series is linked with the class of context free sets (see [23]) A typical example of a context free set is the set of words on the binary alphabet fa; bg having as many a s as b s. We compute below its length distribution which is an algebraic series. Example 4 The set of words on A = fa; bg having an equal number of occurrences of a and b is a submonoid ....

Philippe Flajolet. Analytic models and ambiguity of context-free languages. Theoret. Comput. Sci., 49:283-309, 1987.

....languages. In this paper Parikh slender context free languages are characterized. The characterization has diverse applications. TUCS Research Group Mathematical Structures of Computer Science 1 Introduction Length considerations are often useful in language theory. For example, Flajolet [5] has shown that the inherent ambiguity of many context free languages can be deduced from the transcendentality of their generating functions. Other deep results based on length considerations are well known, e.g. in the theory of Lindenmayer systems (see Rozenberg, Salomaa [18] Andrasiu, ....

P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci. 49 (1987), 283-309.

.... of x i y j in this series is X m0 2m i j m 2m i j m i t 2m i j ; which is transcendental: the coeOEcient of t n grows like 4 n =n, up to a multiplicative constant, revealing a logarithmic singularity in the generating function that implies its transcendence (see [9] for a discussion on the possible singularities of an algebraic series) In constrast, we shall prove that for any i and j, the series S i;j (t) is algebraic. Our rst step will be to extract from S(x; y; t) the coeOEcient of y j . Before we state our result, let us introduce a few notations. ....

P. Flajolet, Analytic models and ambiguity of contextfree languages, Theoret. Comput. Sci. 49 (1987) 283309.

....n i j 2 t n ; where the sum is restricted to integers n of the same parity as i j. This series is transcendental: the coecient of t n grows like 4 n =n, up to a multiplicative constant, revealing a logarithmic singularity in the generating function that implies its transcendence (see [10] for a discussion on the possible singularities of an algebraic series) In contrast, we shall prove that for all models covered by Theorem 4, the series S i;j (t) is algebraic for all i and j. We shall describe, in terms of the set A of steps, an algebraic extension of Q(t) that contains these ....

P. Flajolet, Analytic models and ambiguity of context{free languages, Theoret. Comput. Sci. 49 (1987) 283-309.

....C 1 and C 2 be two DFA cycles such that neither is a subgraph of the other. We say that C 1 and C 2 interlace if there is an accepting computation path in the DFA containing the sequence C 1 C 2 C 1 or the sequence C 2 C 1 C 2 . The following theorem was proved by Flajolet [2]. Our proof uses a constructive argument needed for Theorem 3.3. Theorem 3.2. Every regular language is either sparse or exponentially dense. Proof. Consider L recognized by a DFA A. If L is nite, then it is trivially sparse; otherwise, L is in nite and contains strings of arbitrary ....

P. Flajolet. Analytic models and ambiguity of contextfree languages. TCS, 49:283-309, 1987.

....S, of this set is a context free language, and they conjectured that S is inherently ambiguous. To show this, they in fact conjectured Theorem 1 below. Since the generating series of these numbers is transcendental, the Chomsky Schutzenberger theorem would prove inherent ambiguity (see Flajolet [5] for a systematic exposition) The set S admits a nice combinatorial description due to Coven and Hedlund [3] and Luc Boasson (personal communication) uses this description to prove directly that S is inherently ambiguous by applying Ogden s Lemma. Mignosi [11] proved the following result. ....

P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci. (1987), 283--309.

....transcendental. Then if C = A B) 2 and D = A B) 2, we have A B = C C) D D) and at least one of C C and D D must be transcendental. As a concrete example, g(z) P 1 z=0 2n n z n is algebraic, but g g can be shown to be transcendental using the asymptotic techniques in [13]. Ideally, this result could be used to show that certain inherently ambiguous context free languages, whose generating functions aren t the Hadamard square of an algebraic function, are not QCFLs. Unfortunately, it is not obvious how to prove this, even in the case where all the f(w) are 0 or 1. ....

P. Flajolet, \Analytic models and ambiguity of context-free languages." Theo. Comp. Sci. 49 (1987) 283-309.

.... i) d Gamma1 Y i=1 i d Gamma1 p d (2) d Gamma1) 2 d dn n (d 2 Gamma1) 2 : Algebraicity would imply asymptotics of the type C:A n = Gamma(1 Gamma r)n r ) with C and A algebraic numbers and r a rational number 62 f1; 2; 3; g (classical result from singularity analysis [5]) In our case, for odd d 1, r is an integer and for even d 2, Gamma(1 Gamma (d 2 Gamma 1) 2) 2 Q ( p ) but 62 Q Gamma (d Gamma1) 2 Delta . Thus, the generating function of Young tableaux of height at most d (d 3) is transcendental and D finite. Due to well known one to one ....

Flajolet (Philippe). -- Analytic models and ambiguity of context-free languages. Theoretical Computer Science, vol. 49, n 2-3, 1987, pp. 283--309.

....the LR grammars. See e.g. 4] The Monte Carlo method assumes only that two given grammars are UCFGs. The key point that we need about a UCFG is Theorem 5 The generating series of a UCFG coincides with the generating series of its language. Details about this type of result can be found in [3, 7]. In particular, this theorem implies that the generating series of a language that can be generated by a UCFG is algebraic. The generating series of a language L is P w (w) Delta t jwj , where (w) 0 if w 62 L and (w) 1 if w 2 L. The generating series of a context free grammars is P w ....

P. Flajolet. Analytic models and ambiguity of contextfree languages. TCS, 49,(2,3):282--309, 1987.

....them as complex functions, and use analytical tools: asymptotics of the coefficients and behaviour near the singularities for algebraic functions are very specific, and proving that one of these 1 conditions is violated gives transcendence. A nice and useful survey is a 1987 paper by Flajolet [26], where such methods are in particular applied to generating series of formal languages. Another simple and already used idea that we will try to develop here is to note that: if a series with integral coefficients is algebraic of degree d over Q(X) then, its projection modulo a prime number p ....

....Petersen in [45] that the property that the language of primitive words is not unambiguous context free, implies a result of [23] the language of primitive words over an alphabet with 2 or more letters is not deterministic context free. 4 Generating series for formal languages In his 1987 paper [26] Flajolet used analytic tools to prove that some languages are not unambiguous context free. In this section we give (almost) purely algebraic proofs of most of the examples given by Flajolet in [26] in its Theorems 1 to 7. Let us consider the following languages: 4 ffl let O 3 be the language O ....

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P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci. 49 (1987) 283--309.

....ones. The algebraic series are those satisfying an algebraic equation. More generally, the hypergeometric series are those such that the quotient of two successive terms is given by a rational fraction (see [23] The class of algebraic series is linked with the class of context free sets (see [21]) A typical example of a context free set is the set of words on the binary alphabet fa; bg having as many a s as b s. We compute below its length distribution which is an algebraic series. Example 9. The set of words on A = fa; bg having an equal number of occurrences of a and b is a submonoid ....

P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci., 49 (1987), pp. 283-309.

No context found.

P. Flajolet. Analytic models and ambiguity of context--free languages. Theoretical Computer Science, 49:283--309, 1987.

....local asymptotic expansions that involve only rational exponents. A contrario, a generating function that has in nitely many singularities (e.g. a natural boundary) or that involves a transcendental element (e.g. a logarithm) in a local asymptotic expansion is by necessity transcendental; see [16] for a discussion of such transcendence criteria. In the case of generating trees, this means that the presence of a condition involving a transcendental element is expected to lead to a transcendental generating function. This is the case in the following example. Example 15. A Fredholm system ....

P. Flajolet. Analytic models and ambiguity of context{free languages. Theoretical Computer Science, 49:283-309, 1987.

....local asymptotic expansions that involve only rational exponents. A contrario, a generating function that has infinitely many singularities (e.g. a natural boundary) or that involves a transcendental element (e.g. a logarithm) in a local asymptotic expansion is by necessity transcendental; see [16] for a discussion of such transcendence criteria. In the case of generating trees, this means that the presence of a condition involving a transcendental element is expected to lead to a transcendental generating function. This is the case in the following example. Example 15. A Fredholm system ....

P. Flajolet. Analytic models and ambiguity of context--free languages. Theoretical Computer Science, 49:283--309, 1987. 25

....local asymptotic expansions that involve only rational exponents. A contrario, a generating function that has in nitely many singularities (e.g. a natural boundary) or that involves a transcendental element (e.g. a logarithm) in a local asymptotic expansion is by necessity transcendental; see [14] for a discussion of such transcendence criteria. In the case of generating trees, this means that the presence of a condition involving a transcendental element is expected to lead to a transcendental generating function. An instance that we examine now is System (2:e) in which the rules are ....

P. Flajolet. Analytic models and ambiguity of context{free languages. Theoretical Computer Science, 49:283-309, 1987.

....of a Q(z) algebraic function are asymptotic to a sum of algebraic elements of the form Gamma(r=s 1) n n r=s ; 12) where ; are algebraic numbers, and the exponent r=s is a rational number. This furnishes a generalized density theorem for context free languages and was used in [23] in order to establish the inherent ambiguity of several context free languages. Similar singular expansions involving fractional powers also hold for functions implicitly defined by equations of the form Phi(z; f(z) 0, where Phi, analytic function of two complex variables, need no longer be ....

Flajolet, P. Analytic models and ambiguity of context--free languages. Theoretical Computer Science 49 (1987), 283--309.

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Flajolet, P. Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci. 49 (1987), 283-309.

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P. Flajolet. Analytic models and ambiguity of contextfree languages. TCS, 49,(2,3):282--309, 1987.

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P. Flajolet, Analytic models and ambiguity of context-free languages, Theoretical Computer Science 49 (1987), 283-309.

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P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci. 49 (1987) 283-309.

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P. Flajolet, Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci. 49 (1987) 283--309.

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