Chomsky-Schotzenberger representations for families of languages and grammatical types

  • Sorin Istrail
Session 10: M. Nivat, Chairman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 115)


The paper has two parts. In Part I, we shall present Chomsky-Schützenberger theorems for the families of context-sensitive (CS)and recursive-enumerable (RE) languages.

The results are obtained by generalizing the construction of the Dyck set from a "content-free" one to a "content-sensitive" one.

Also presented are fixed-point characterization theorems for CS and RE, which generalize the Algol-like theorem. While { xi=Fi (x1,...,xt), 1≤ i ≤ t, is the system used in the Algol-like theorem, our theorems use { xi=hi (Ri ∩ Fi (x1,...,xt)), 1 ≤ i ≤ t, with Fi as above, hi a finite substitution and Ri a regular set. The pair πi=(hi,Ri) is called a π-function, defined as πi(L)=hi(Ri ∩ L).

Part II contains the study of systems of equations with right sides polynomials in π-functions, which turn out to be regular expressions over { ·, ∪, ∩, *, finite substitution }.

This is of interest not only because they realize the CS- and RE-steps, but also because they seem to provide with a "language" in which a variety of generative mechanisms from the literature can be expressed. This gives the base to an abstract, equational-based theory for presenting generative mechanisms: Grammatical types.

Within the theory we present general techniques for deriving Chomsky-Schützenberger representations for families of languages possessing a grammatical type definition. Among such families of languages we mention: CS, RE, programmed, Turing machines, Petri-nets, regular-control, scattered-context, L-systems, N(D)TIME(f), N(D)SPACE(f) (for f(n)=nk or f(n)=kn), NP, P, EXPTIME.


Turing Machine Regular Expression Empty Word Schematic System Type Poly 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. B.
    BERSTEL, J. "Transductions and context-free languages" Teubner (1979)Google Scholar
  2. CS.
    CHOMSKY, N., SCHUTZENB RGER M.P. "The algebraic theory of context-free languages", in P.Braffort and D.Hirschberg (eds) Computer Programming and Formal Systems, pp.118–161, North Holland, (1963).Google Scholar
  3. CN.
    COCHET, Y., NIVAT,M. "Une generalisation des ensembles de Dycl Israel J.of Math. 9 (1971)Google Scholar
  4. CM.
    CULIK, K., MAURER, H.A. "On simple representations of language families" RAIRO (to appear)Google Scholar
  5. ER.
    ENGELFRIET, J., ROZENBERG, G. "Fixed-point languages, equality languages and reprezentations of recursive enumerable languages" 19th FOCS, (1978)Google Scholar
  6. G.
    GINSBURG, S. "Algebraic and automata-theoretic properties of formal languages" North Holland, Amsterdam (1975)Google Scholar
  7. H.
    HARRISON, M.A. "Introduction to formal language theory" Addison-Wesley, Reading (1978)Google Scholar
  8. HU.
    HOPCROFT,J., ULLMAN, J.D. "Introduction to Automata theory, Languages and Computation "Addison-Wesley, (1979).Google Scholar
  9. I1.
    ISTRAIL, S. "A fixed-point theorem for recursive-enumerable languages and some considerations about fixed-point semantics of monadic programs" 6th Colloq.Automata, Languages and Programming, (1979) Lect.Notes Comput. Sci 71Google Scholar
  10. I2.
    ISTRAIL, S. "Generalizations of Schützenberger-Ginsburg-Rice fixed-point theorem for context-sensitive and recursive-enumerable languages" Theor.Comput.Sci. (to appear)Google Scholar
  11. I3.
    ISTRAIL, S. "Elementary bounded languages" Information and Contr. vol. 39, 2(1978), p. 177–191Google Scholar
  12. I4.
    ISTRAIL, S. "Grammatical types of grammar-and L-forms" (in preparation).Google Scholar
  13. IS.
    ISTRAIL, S., SOIL, A. "Fixed-point theorems for Petri nets language" (submitted for publication)Google Scholar
  14. M.
    MONIEN, B. "A recursive and a grammatical characterization of the exponential-time languages" Theor.Comp.Sci. 3 (1977)Google Scholar
  15. N.
    NIVAT, M. "Transductions des languages de Chomschy" Anall.Inst. Fourier (1968)Google Scholar
  16. P.
    PETERSON, J.L. "Petri nets", Computing Surveys, vol.9,no.3, (1977)Google Scholar
  17. R.
    ROZENBERG, G. "Selective substitution grammars (towards a framework for rewriting systems) Part I: Definitions and examples" EIK (1977)Google Scholar
  18. S.
    SALOMAA, A. "Formal languages" Academic Press (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • Sorin Istrail
    • 1
  1. 1.Department of Mathematics and Computer CenterUniversity "Al.I.Cuza"IASIROMANIA

Personalised recommendations