Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Formal languages of labelled graphs

Formale Sprachen von markierten Graphen

  • Published:
Computing Aims and scope Submit manuscript

Abstract

The definition of the direct derivation of strings in a formal grammar can be generalized to graphs with labelled nodes and edges to get a sequential rewriting system for graphs. A graph production consists of three parts: left hand side, right hand side, and a description how the embedding of the left hand side in a host graph is transformed if the right hand side is substituted. In the following a production mechanism is introduced which is very general with respect to the transformation of the embedding.

We get results similar to those of formal languages' theory:

$${\text{[}}U{\text{]}} \supset {\text{[}}M{\text{] = [}}CS{\text{]}} \supseteq {\text{[}}CF{\text{] = [}}N{\text{]}} \supseteq {\text{[}}R{\text{] = [}}RN{\text{]}}$$

where [U], ..., [RN] means the class of graph languages which is unrestricted, monotone, context sensitive, context free, normal, regular, and regular in normal form respectively.

The class of regular graph languages was investigated in more detail by successively restricting the embedding transformation. Some generalizations and open problems are given.

Zusammenfassung

Als Verallgemeinerung des Ableitungsbegriffs für Zeichenketten mit Hilfe von Ersetzungsregeln einer formalen Grammatik wird im folgenden ein Ersetzungsmechanismus für Graphen mit Kanten- und Knotenmarkierung vorgestellt, der, bei Ersetzung eines Untergraphen durch einen anderen, bezüglich der Überführung der Einbettungen dieser beiden Untergraphen sehr allgemein ist.

Es ergeben sich analoge Aussagen zur Theorie der Formalen Sprachen:

$${\text{[}}U{\text{]}} \supset {\text{[}}M{\text{] = [}}CS{\text{]}} \supseteq {\text{[}}CF{\text{] = [}}N{\text{]}} \supseteq {\text{[}}R{\text{] = [}}RN{\text{]}}$$

wobei [U] bzw. ... bzw. [RN] die Klasse der uneingeschränkten, beschränkten, kontextsensitiven, kontextfreien, normalen, regulären Graph-Sprachen bzw. regulären Graph-Sprachen in Normal-form ist.

Die Klasse der regulären Graph-Sprachen wurde genauer untersucht, indem die Vorschrift eingeschränkt wurde, die die Einbettungsüberführung festlegt. Einige Verallgemeinerungen des eingeführten Ableitungsbegriffs und einige offene Fragen sind angegeben.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Abe, N., Mizumoto, M., Toyoda, J.-I., Tanaka, K.: Web Grammars and Several Graphs. Journ. Comp. Syst. Sciences7, 37–65 (1973).

    Google Scholar 

  2. Ehrig, H., Pfender, H., Schneider, H. J.: Graph-Grammars: An algebraic approach. Proc. 14th Annual Conf. on Switching and Automata Theory, 167–180 (1973).

  3. Feder, J.: Plex Languages. Inf. Sciences3, 225–241 (1971).

    Google Scholar 

  4. Lucas, P., Walk, K.: On the formal description of PL/I. Ann. Rev. Aut. Progr.6, 3, 105–182 (1969).

    Google Scholar 

  5. Milgram, D., Rosenfeld, A.: Array automata and array grammars. Proc. IFIP Congr. 1971, Booklet TA-2, 166–173.

  6. Montanari, U. G.: Separable Graphs, Planar Graphs and Web Grammars. Inf. Contr.16, 243–267 (1970).

    Google Scholar 

  7. Nagl, M.: Formale Sprachen von markierten Graphen. Arbeitsber. d. Inst. f. Math. Maschinen u. Datenverarb.7, 4, Erlangen 1974.

    Google Scholar 

  8. Pfaltz, J. L., Rosenfeld, A.: Web Grammars. Proc. Int. Joint Conf. Art. Int., 609–619, Washington 1969.

  9. Rosenfeld, A.: Progress in Picture Processing 1969–71. Computung surveys5, 2, 81-108 (1973).

    Google Scholar 

  10. Rosenfeld, A., Milgram, D. L.: Web automata and web grammars. Machine Intelligence7, 307–324 (1972).

    Google Scholar 

  11. Shaw, A. C.: Parsing of Graph-Representable Pictures. Journ. ACM17, 3, 453–381 (1970).

    Google Scholar 

  12. Schneider, H. J.: Chomsky-Systeme für partielle Ordnungen. Arbeitsber. d. Inst. f. Math. Masch. u. Datenverarb.3, 3, Erlangen 1970.

    Google Scholar 

  13. Schneider, H. J.: A necessary and sufficient conditions for Chomsky-productions over partially ordered symbol sets. Lect. Notes in Econ. and Math. Systems78, 90–98 (1973).

    Google Scholar 

  14. Schneider, H. J.: Syntax-directed description of incremental compilers. Lect. Notes in Comp. Sci.26, 192–201 (1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Inst. f. Math. Masch. u. Datenver. (II), Egerlandstraße 13, D-8520, Erlangen, Federal Republic of Germany

    M. Nagl

Authors
  1. M. Nagl
    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

Nagl, M. Formal languages of labelled graphs. Computing 16, 113–137 (1976). https://doi.org/10.1007/BF02241984

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02241984

Keywords

Search

Navigation