On differentiation functions, structure functions, and related languages of context-free grammars

Dassow, Jürgen; Mitrana, Victor; Păun, Gheorghe; Stiebe, Ralf

doi:10.1051/ita:2004013

Dassow, Jürgen ; Mitrana, Victor ; Păun, Gheorghe ; Stiebe, Ralf

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) no. 3, pp. 257-267.

Résumé

We introduce the notion of a differentiation function of a context-free grammar which gives the number of terminal words that can be derived in a certain number of steps. A grammar is called narrow (or $k$ -narrow) iff its differentiation function is bounded by a constant (by $k$ ). We present the basic properties of differentiation functions, especially we relate them to structure function of context-free languages and narrow grammars to slender languages. We discuss the decidability of the equivalence of grammars with respect to the differentiation function and structure function and prove the decidability of the $k$ -narrowness of context-free grammars. Furthermore, we introduce languages representing the graph of the differentiation and structure function and relate these languages to those of the Chomsky hierarchy.

MR Zbl

DOI : 10.1051/ita:2004013

Classification : 68Q45, 68Q50
Mots clés : differentiation function, structure function, slender languages

@article{ITA_2004__38_3_257_0,
     author = {Dassow, J\"urgen and Mitrana, Victor and P\u{a}un, Gheorghe and Stiebe, Ralf},
     title = {On differentiation functions, structure functions, and related languages of context-free grammars},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {257--267},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     doi = {10.1051/ita:2004013},
     mrnumber = {2076403},
     zbl = {1082.68049},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ita:2004013/}
}

TY  - JOUR
AU  - Dassow, Jürgen
AU  - Mitrana, Victor
AU  - Păun, Gheorghe
AU  - Stiebe, Ralf
TI  - On differentiation functions, structure functions, and related languages of context-free grammars
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2004
SP  - 257
EP  - 267
VL  - 38
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ita:2004013/
DO  - 10.1051/ita:2004013
LA  - en
ID  - ITA_2004__38_3_257_0
ER  -

%0 Journal Article
%A Dassow, Jürgen
%A Mitrana, Victor
%A Păun, Gheorghe
%A Stiebe, Ralf
%T On differentiation functions, structure functions, and related languages of context-free grammars
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2004
%P 257-267
%V 38
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ita:2004013/
%R 10.1051/ita:2004013
%G en
%F ITA_2004__38_3_257_0

Dassow, Jürgen; Mitrana, Victor; Păun, Gheorghe; Stiebe, Ralf. On differentiation functions, structure functions, and related languages of context-free grammars. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) no. 3, pp. 257-267. doi : 10.1051/ita:2004013. http://archive.numdam.org/articles/10.1051/ita:2004013/

Bibliographie
Cité par

[1] N. Chomsky and M.P. Schützenberger, The algebraic theory of context-free languages, in Computer Programming and Formal Systems, edited by P. Braffort, D. Hirschberg. North-Holland, Amsterdam (1963) 118-161. | Zbl

[2] J. Dassow, Eine neue Funktion für Lindenmayer-Systeme. EIK 12 (1976) 515-521. | Zbl

[3] J. Dassow, Numerical parameters of evolutionary grammars, in Jewels are forever, edited by J. Karhumäki, H. Maurer, Gh. Păun, G. Rozenberg. Springer-Verlag, Berlin (1999) 171-181. | Zbl

[4] J. Dassow and Gh. Păun, Regulated Rewriting in Formal Language Theory. Akademie-Verlag, Berlin and Springer-Verlag, Berlin (1989). | MR | Zbl

[5] D. Hauschildt and M. Jantzen, Petri nets algorithms in the theory of matrix grammars. Acta Inform. 31 (1994) 719-728. | Zbl

[6] S. Ginsburg, The Mathematical Theory of Context-Free Languages. McGraw Hill Book Comp., New York (1966). | MR | Zbl

[7] O. Ibarra, Restricted one-counter machines with undecidable universe problems. Math. Syst. Theory 13 (1979) 181-186. | Zbl

[8] L. Ilie, On a conjecture about slender context-free languages. Theor. Comput. Sci. 132 (1994) 427-434. | Zbl

[9] L. Ilie, On lengths of words in context-free languages. Theor. Comput. Sci. 242 (2000) 327-359. | Zbl

[10] R. Incitti, The growth function of context-free languages. Theor. Comput. Sci. 255 (2001) 601-605. | Zbl

[11] T. Katayama, M. Okamoto and H. Enomoto, Characterization of the structure-generating functions of regular sets and the D0L systems. Inform. Control 36 (1978) 85-101. | Zbl

[12] W. Kuich and R.K. Shyamasundar, The structure generating function of some families of languages. Inform. Control 32 (1976) 85-92. | Zbl

[13] M. Kunze, H.J. Shyr and G. Thierrin, $h$ -bounded and semidiscrete languages. Inform. Control 51 (1981) 147-187. | Zbl

[14] M. Latteux and G. Thierrin, Semidiscrete context-free languages. Internat. J. Comput. Math. 14 (1983) 3-18. | Zbl

[15] Gh. Păun and A. Salomaa, Thin and slender languages. Discrete Appl. Math. 61 (1995) 257-270. | Zbl

[16] D. Raz, Length considerations in context-free languages. Theor. Comput. Sci. 183 (1997) 21-32. | Zbl

[17] A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag (1978). | MR | Zbl

[18] R. Stiebe, Slender matrix languages, in Developments in Language Theory, edited by G. Rozenberg, W. Thomas. World Scientific, Singapore (2000) 375-385. | Zbl

Cité par Sources :