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

Published online by Cambridge University Press:  15 June 2004

Jürgen Dassow
Affiliation:
Otto-von-Guericke-Universität Magdeburg, Fakultät für Informatik, PSF 4120, 39016 Magdeburg, Germany; [email protected].; [email protected].
Victor Mitrana
Affiliation:
University of Bucharest, Institute of Mathematics, Str. Academiei 14, 70109 Bucuresti, Romania. Rovira i Virgili University, Research Group in Mathematical Linguistics, Pça. Imperial Tarraco 1, 43005, Tarragona, Spain; [email protected].
Gheorghe Păun
Affiliation:
Institute of Mathematics of the Romanian Academy, PO Box 1–764, 70700 Bucuresti, Romania; [email protected].
Ralf Stiebe
Affiliation:
Otto-von-Guericke-Universität Magdeburg, Fakultät für Informatik, PSF 4120, 39016 Magdeburg, Germany; [email protected].; [email protected].
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Abstract

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.

Type
Research Article
Copyright
© EDP Sciences, 2004

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References

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.
Dassow, J., Eine neue Funktion für Lindenmayer-Systeme. EIK 12 (1976) 515521.
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.
J. Dassow and Gh. Păun, Regulated Rewriting in Formal Language Theory. Akademie-Verlag, Berlin and Springer-Verlag, Berlin (1989).
Hauschildt, D. and Jantzen, M., Petri nets algorithms in the theory of matrix grammars. Acta Inform. 31 (1994) 719728. CrossRef
S. Ginsburg, The Mathematical Theory of Context-Free Languages. McGraw Hill Book Comp., New York (1966).
Ibarra, O., Restricted one-counter machines with undecidable universe problems. Math. Syst. Theory 13 (1979) 181186. CrossRef
Ilie, L., On a conjecture about slender context-free languages. Theor. Comput. Sci. 132 (1994) 427434. CrossRef
Ilie, L., On lengths of words in context-free languages. Theor. Comput. Sci. 242 (2000) 327359. CrossRef
Incitti, R., The growth function of context-free languages. Theor. Comput. Sci. 255 (2001) 601605. CrossRef
Katayama, T., Okamoto, M. and Enomoto, H., Characterization of the structure-generating functions of regular sets and the D0L systems. Inform. Control 36 (1978) 85101. CrossRef
Kuich, W. and Shyamasundar, R.K., The structure generating function of some families of languages. Inform. Control 32 (1976) 8592. CrossRef
Kunze, M., Shyr, H.J. and Thierrin, G., h-bounded and semidiscrete languages. Inform. Control 51 (1981) 147187. CrossRef
Latteux, M. and Thierrin, G., Semidiscrete context-free languages. Internat. J. Comput. Math. 14 (1983) 318. CrossRef
Păun, Gh. and Salomaa, A., Thin and slender languages. Discrete Appl. Math. 61 (1995) 257270. CrossRef
Raz, D., Length considerations in context-free languages. Theor. Comput. Sci. 183 (1997) 2132. CrossRef
A. Salomaa and M. Soittola, Automata-Theoretic Aspects of Formal Power Series. Springer-Verlag (1978).
R. Stiebe, Slender matrix languages, in Developments in Language Theory, edited by G. Rozenberg, W. Thomas. World Scientific, Singapore (2000) 375–385.