Skip to main content
SpringerLink
Log in

Transformations of structures: An algebraic approach

  • Published:
Mathematical systems theory Aims and scope Submit manuscript

Abstract

This paper introduces a new mathematical approach to transformations of structures, where the concept of “structure” is extremely general. Many structures and transformations that arise in biology as well as computer science are special cases of our concepts. A structure may be changed by finding an occurrence of a pattern and replacing it by another pattern as specified by a rule. To prove theorems about long sequences of applications of complicated rules, we need precise and tractable mathematical definitions of rules and how to apply them. This paper presents such definitions and three fundamental theorems, together with brief remarks on applications to control flow analysis, record handling, and evaluation of recursively defined functions. Unlike some previous efforts toward a rigorous theory of transformations of structures, this paper uses ideas and results from abstract algebra to minimize the need for elaborate constructions.

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

References

  1. M. A. Arbib and E. G. Manes,Arrows, Structures, and Functors, Academic Press, New York, 1975.

    Google Scholar 

  2. G. Berry and J.-J. Levy, Minimal and optimal computations of recursive programs,J. ACM 26 (1979), 148–175.

    Google Scholar 

  3. G. Birkhoff and J. D. Lipson, Heterogeneous algebras,J. Combinatorial Theory, 8 (1970), pp. 115–133.

    Google Scholar 

  4. V. Claus, H. Ehrig, and G. Rozenberg (Editors),Graph Grammars and their Application to Computer Science and Biology, Lecture Notes in Computer Sci., 73 (1979).

  5. E. F. Codd,A relational model of data for large shared data banks, Comm. ACM, 13 (1970), pp. 377–387.

    Google Scholar 

  6. J. A. Darringer and W. H. Joyner,A new look at logic synthesis, IBM Research Report RC 8268, Yorktown Heights NY, May 1980.

  7. H. Ehrig and H.-J. Kreowski, Contributions to the algebraic theory of graph grammars,Math. Nachr., to appear.

  8. H. Ehrig and H.-J. Kreowski, Applications of graph grammar theory to consistency, synchronization, and scheduling in data base systems,Information Sciences, to appear.

  9. H. Ehrig, H.-J. Kreowski, A. Maggiolo-Schettini, B. K. Rosen, and J. Winkowski, Deriving structures from structures,Lecture Notes in Computer Science, 64 (1978), pp. 177–190.

    Google Scholar 

  10. H. Ehrig, M. Pfender, and H. J. Schneider, Graph grammars: an algebraic approach,Proc. 14th Ann. IEEE Symp. on Switching and Automata Theory, Iowa City, October 1973, pp. 167–180.

  11. H. Ehrig and B. K. Rosen,Commutativity of independent transformations on complex objects, IBM Research Report RC 6251, Yorktown Heights NY, October 1976.

  12. H. Ehrig and B. K. Rosen, The mathematics of record handling,SIAM J. Computing, 9 (1980), 441–469.

    Google Scholar 

  13. H. Ehrig and B. K. Rosen,Commutativity, parallelism, and concurrency for transformations of structures, Bericht 79–21, Fachbereich Informatik, Tech. U. Berlin, October 1979.

  14. H. Ehrig and K. W. Tischer, Graph grammars and applications to specialization and evolution in biology,J. Computer and System Sci., 11 (1975), pp. 212–236.

    Google Scholar 

  15. S. Eilenberg and J. Wright, Automata in general algebras,Information and Control, 11 (1967), 452–470.

    Google Scholar 

  16. R. Farrow, K. Kennedy, and L. Zucconi, Graph grammars and global program data flow analysis,Proc. 17th Ann. IEEE Symp. on Foundations of Computer Sci., Houston, October 1976, pp. 42–56.

  17. H. Herrlich and G. Strecker,Category Theory, Allyn and Bacon, Rockleigh, New Jersey, 1973.

    Google Scholar 

  18. H.-J. Kreowski,Ein Pumpinglemma für Kanten-Kontextfreie Graph-Sprachen, Bericht 77–15, Fachbereich Informatik, Tech. U. Berlin, September 1977. (See also [4].)

  19. M. J. O'Donnell, Computing in systems described by equations,Lecture Notes in Computer Science, 58 (1977).

  20. G. Pacini, C. Montangero, and F. Turini, Graph representation and computation rule for typeless recursive languages,Lecture Notes in Computer Science, 14 (1974), pp. 157–169.

    Google Scholar 

  21. Padawitz, P.,Graph grammars and operational semantics, Bericht 78–33, Fachbereich Informatik, Tech. U. Berlin, October 1978. (See also [4].)

  22. V. Rajlich, Dynamics of discrete systems ⋯,J. Computer and System Sci., 11 (1975), pp. 186–202.

    Google Scholar 

  23. B. K. Rosen, Deriving graphs from graphs by applying a production,Acta Informatica, 4 (1975), pp. 337–357.

    Google Scholar 

  24. H. J. Schneider and H. Ehrig, Grammars on partial graphs,Acta Informatica, 6 (1976), pp. 297–316.

    Google Scholar 

  25. J. Staples, A class of replacement systems with simple optimality theory,Bull. Austral. Math. Soc., 17 (1977), p. 335–350.

    Google Scholar 

  26. J. Staples, Computation on graph-like expressions,Theoretical Computer Sci., 10 (1980), 171–185.

    Google Scholar 

  27. J. Staples, Optimal evaluations of graph-like expressions,Theoretical Computer Sci., 10 (1980), 297–316.

    Google Scholar 

  28. J. Staples, Speeding up subtree replacement systems,Theoretical Computer Sci., 11 (1980), 39–47.

    Google Scholar 

  29. S. A. Vere, Relational production systems,Artificial Intelligence, 8 (1977), pp. 47–68.

    Google Scholar 

  30. J. Vuillemin, Correct and optimal implementations of recursion in a simple programming language,J. Computer and System Sci., 9 (1974), pp. 332–354.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Informatik, Technische Universität Berlin, 1000, Berlin 10, Federal Republic of Germany

    Hans-Jörg Kreowski

  2. Istituto di Scienze dell'Informazione, Universita di Pisa, 56100, Pisa, Italy

    Andrea Maggiolo-Schettini

  3. Computer Sciences Department, IBM Thomas J. Watson Research Center, 10598, Yorktown Heights, NY, USA

    Barry K. Rosen

  4. Instytut Podstaw Informatyki PAN, Box 22, 00-901, Warszawa, PKiN, Poland

    Jozef Winkowski

Authors
  1. Hartmut Ehrig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans-Jörg Kreowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrea Maggiolo-Schettini
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Barry K. Rosen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jozef Winkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

A condensation of an earlier version of this paper was presented as [9] at the 7th Symposium on Mathematical Foundations of Computer Science, September 1978, Zakopane, Poland. Work by Ehrig was partially supported by IBM Germany and IBM World Trade Corp. Work by Rosen and Maggiolo-Schettini was partially supported by the Laboratorio di Cibernetica del C.N.R.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ehrig, H., Kreowski, HJ., Maggiolo-Schettini, A. et al. Transformations of structures: An algebraic approach. Math. Systems Theory 14, 305–334 (1981). https://doi.org/10.1007/BF01752403

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01752403

Keywords

Search

Navigation