Skip to main content
SpringerLink
Log in

Formes de langages et de grammaires

Grammars and languages forms

  • Published:
Acta Informatica Aims and scope Submit manuscript

Summary

This paper is devoted to the study of context-free languages over infinite alphabets. This work can be viewed as a new attempt to study families of grammars, replacing the usual “grammar forms” and giving a new point of view on these questions. A language over an infinite alphabet or I-language appears as being a model for a family of usual languages; an interpretation is an homomorphism from the infinite alphabet to any finite alphabet. Using this notion of interpretation we can associate to each family of I-languages an image, called its shadow, which is a family of usual languages.

The closure properties of families, generalizing to infinite alphabets the family of context-free languages, lead to define rational transductions between infinite alphabets or I-transductions, and then, families of I-languages closed under I-transductions, or I-cones. We study here relations between the closure properties of a family of I-languages and these of its shadow. As a result, we obtain that any union closed rational cone of context-free languages, principal or not, is the shadow of a principal I-cone.

This work leads to new results about the classical theory of context-free languages. For instance, we prove that any principal rational cone of context-free languages can be generated by a context-free language, whose grammar has only 6 variables. This work also leads to more general considerations about the adequacy of some generating devices to the generated languages. It appears that the context-free grammars are fair, in a sense that we define, for generating context-free languages but that non-expansive context-free grammars are not for generating non-expansive context-free languages. This point of view raises a number of questions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Bibliographie

  1. Autebert, J.M.: Non-principalité du cylindre des langages à compteur. Math. System Theory 11, 157–167 (1977)

    Google Scholar 

  2. Autebert, J.M.: Cylindres de langages algébriques. Thèse de Doctorat d'Etat de l'Université de Paris VII, Paris 1978

  3. Autebert, J.M., Beauquier, J., Boasson, L.: Langages sur des alphabets infinis. Discrete Math. 2, 1–20 (1980)

    Google Scholar 

  4. Autebert, J.M., Beauquier, J., Boasson, L., Nivat, M.: Quelques problèmes ouverts en théorie des langages algébriques. R.A.I.R.O. Informatique Théorique 13, 363–379 (1979)

    Google Scholar 

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

    Google Scholar 

  6. Boasson, L.: The inclusion of the substitution closure of linear and one counter languages in the largest sub-AFL of the family of algebraic languages is proper. Information Processing Lett. 2, 135–140 (1974)

    Google Scholar 

  7. Cremers, A.B., Ginsburg, S.: Context-free grammar forms. Comput. System Sci. 11, 86–116 (1975)

    Google Scholar 

  8. Culik, K., Maurer, H.A.: Propagating chain-free normal forms for EOL systems. Information and Control 36, 309–319 (1978)

    Google Scholar 

  9. Elgot, C.C., Mezei, J.E.: On relations defined by generalized finite automata. I.B.M. Journal of Research and Development 9, 47–68 (1965)

    Google Scholar 

  10. Ginsburg, S.: A survey of context-free grammar forms. In: Formal Languages and Programming (R. Aguilar, ed.), pp. 97–110. Amsterdam: North-Holland 1976

    Google Scholar 

  11. Greibach, S.A.: Chains of full AFL's. Math. System Theory 4, 231–242 (1970)

    Google Scholar 

  12. Greibach, S.A.: The hardest context-free language. SIAM J. Comput. 2, 304–311 (1973)

    Google Scholar 

  13. Greibach, S.A.: Control sets and context-free grammar forms. J. Comput. System Sci. 15, 35–98 (1977)

    Google Scholar 

  14. Maurer, H.A., Salomaa, A., Wood, D.: Good EOL forms. SIAM J. Comput. 7 (1978)

  15. Nivat, M.: Transductions des langages de Chomsky. Ann. Inst. Fourier 18, 339–456 (1968)

    Google Scholar 

  16. Rozenberg, G., Herman, G.T.: in Developmental Systems and Languages: North-Holland, Amsterdam 1975

    Google Scholar 

  17. Salomaa, A.: Formal languages. Academic Press 1973

Download references

Author information

Authors and Affiliations

  1. Université de Paris 7, U.E.R. de Mathématiques, 2 Place Jussieu, 75221, Paris Cedex 05, France

    Jean -Michel Autebert

  2. Université de Picardie, U.E.R. de Mathématiques, 2 Place Jussieu, 75221, Paris Cedex 05, France

    Joffroy Beauquier

  3. Université Paris 7, U.E.R. de Mathématiques, 2 Place Jussieu, 75221, Paris Cedex 05, France

    Luc Boasson

Authors
  1. Jean -Michel Autebert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joffroy Beauquier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luc Boasson
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Autebert, J.M., Beauquier, J. & Boasson, L. Formes de langages et de grammaires. Acta Informatica 17, 193–213 (1982). https://doi.org/10.1007/BF00288970

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00288970

Search

Navigation