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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)

Abstract

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.

Keywords

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.

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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

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