Skip to main content
SpringerLink
Log in

Language Theory of Petri Nets

  • Conference paper
Petri Nets: Central Models and Their Properties (ACPN 1986)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 254))

Included in the following conference series:

Abstract

Petri nets where multiple arcs are allows and the capacity of the places need not be bounded are here called Place/Transition systems. The restrictions of the possible finite or infinite occurence sequences of a P/T-system to the transitions are called transition sequences and give the basis to define families of formal languages related to classes of P/T-systems.

We introduce the notation and give a survey on methods and results about sets of finite transition sequences. We will compare the classes of Petri net languages we obtain with other families of languages known from automata and formal language theory. We hope to convince that these techniques and results are useful for the formulation and solution of certain questions about P/T-systems, as well as for comparing the underlying systems.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Araki, T. / Kasami, T., Decidable properties on the strong connectivity of petri net reachability sets, Theoret. Comput. Sci., 4 (1977) 99–119.

    Article  MathSciNet  MATH  Google Scholar 

  2. Baker, H., Petri nets and languages, C. S. G. Memo 68, Project MAC, M. I. T., (1972).

    Google Scholar 

  3. Baker, H. G., Equivalence problems in Petri nets, S. M. Thesis, Dept. Electr. Engin., MIT (1973).

    Google Scholar 

  4. Baker, H., Rabin’s proof of the undecidability of the reachability set inclusion problem of vector addition systems, Comput. Struct. Group Memo 79, Proj. MAC, MIT (1973).

    Google Scholar 

  5. Baker, B. S. / Book, R. V., Reversal-bounded multipushdown machines, J. Comput. Syst. Sci., 8 (1974) 315–332.

    Article  MathSciNet  MATH  Google Scholar 

  6. Berstel, J., Transductions and Context-Free Language, Teubner (1979).

    Google Scholar 

  7. Best, E. /Fernandez, C., Notations and terminology on Petri net theory, Arbeitspapiere der GMD 195, Gesellschaft für Mathematik u. Datenverarbeitung (1986), also this volume.

    Google Scholar 

  8. Borosh, I. / Flahive, M. / Treybig, B., Small solutions of linear diophantine equations, Discrete Math., 58 (1986), 215–220.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bramhoff, H. /Jantzen, M., Durch Petrinetze definierte Klassen zahlentheoretischer Funktionen, Techn. Rept. IFI-HH-B 98/83, FB-Informatik, Univ. Hamburg (1983), preliminary version: Jantzen, M., Notions of computability by Petri nets, Informatik Fachberichte 66, Springer (1983), 149-165.

    Google Scholar 

  10. Brams, G. W., Réseaux de Petri: Théorie et Pratique, Tome 1: Théorie et Analyse, Masson (1983).

    Google Scholar 

  11. Burkhard, H.-D., Ordered firing in Petri nets, EIK, 17 (1981) 71–86.

    MathSciNet  MATH  Google Scholar 

  12. Burkhard, H.-D., Two pumping lemmata for Petri nets, EIK, 17 (1981) 349–362.

    MathSciNet  MATH  Google Scholar 

  13. Cardoza, E. /Lipton, R. /Meyer, A., Exponential space complete problems for Petri nets and commutative semigroups, Proc. 8th ACM Sympos. Theory of Comput. Conf., (1976), 50-54.

    Google Scholar 

  14. Chou, T.-W. J. / Collins, G. E., Algorithms for the solution of systems of linear diophantine equations, SIAM J. Comput., 11 (1982), 687–708.

    Article  MathSciNet  MATH  Google Scholar 

  15. Clote, P., On the finite containment problem for Petri nets, Theoret. Comput. Sci., 43 (1986) 99–105.

    Article  MathSciNet  MATH  Google Scholar 

  16. Crespi-Reghizzi, S. / Mandrioli, D., Petri nets and Szilard languages, Inform, and Control, 33 (1977) 177–192.

    Article  MathSciNet  MATH  Google Scholar 

  17. Etzion, T. /Yoeli, M., The hiérarchie of labelled super-nets, Techn. Rept. # 226, Dept. of Comput. Sci., Technion, Haifa, (1981).

    Google Scholar 

  18. Etzion, T. / Yoeli, M., Super nets and their hiérarchie, Theoret. Comput. Sci., 23 (1983) 243–272.

    Article  MathSciNet  MATH  Google Scholar 

  19. Garey, M. R. / Johnson, P. S., Computers and Intractability — A Guide to the Theory of NP-Completeness, Freeman, San Francisco (1979).

    MATH  Google Scholar 

  20. Ginsburg, S., Algebraic and Automata-Theoretic Properties of Formal Languages, North-Holland (1975).

    Google Scholar 

  21. Ginzburg, A. / Yoeli, M., Vector addition systems and regular languages, J. Comput. Syst. Sci., 20 (1980), 277–284.

    Article  MathSciNet  MATH  Google Scholar 

  22. Grabowski, J., On Hack’s conjecture concerning reachability in Petri nets, EIK, 15 (1979) 339–354.

    MathSciNet  MATH  Google Scholar 

  23. Grabowski, J., On partial languages, Preprint No 40/79, Sekt. Mathematik, Humboldt-Univ. Berlin, (1979), and Fundamenta Informatikae 4, (1981) 427–498.

    MathSciNet  MATH  Google Scholar 

  24. Grabowski, I., The unsolvability of some Petri net language problems, Info. Processing Lett., 9 (1979), 60–63.

    Article  MathSciNet  MATH  Google Scholar 

  25. Grabowski, J., Lineare Methoden in der Theorie der Vektoradditionssysteme, I, II, III, Seminarberichte 24, 26, 28, Sekt. Math., Humbold-Univ., Berlin (1980)

    Google Scholar 

  26. Greibach, S. A., An infinite hiérarchie of context-free languages, J. Assoc. Comput. Mach., 16 (1969) 91–106.

    Article  MathSciNet  MATH  Google Scholar 

  27. Greibach, S. A., Remarks on blind and partially blind one-way multicounter machines, Theoret. Comput. Sci., 7 (1978) 311–324.

    Article  MathSciNet  MATH  Google Scholar 

  28. Hack, M., Decision problems for Petri nets and vector addition systems, C. S. G. Neno 95-1, Proj. MAC, MIT (1974).

    Google Scholar 

  29. Hack, M., Petri net languages, C. S. G. Memo 124, Project MAC, MIT (1975).

    Google Scholar 

  30. Hack, M., The equality problem for vector addition systems is undecidable, Theoret. Comput. Sci., 2 (1976), 77–95.

    Article  MathSciNet  MATH  Google Scholar 

  31. Harrison, M. A., Introduction to Formal Language Theory, Addison-Wesley (1978).

    Google Scholar 

  32. Hopcroft, J. E. /Ullman, J. D., Introduction to Automata Theory, Languages, and Computation, Addison-Wesley (1979).

    Google Scholar 

  33. Eopcroft, J.-E. /Pansiot, J. J., On the reachability problem for 5-dimensional vector addition systems, Tech. Rept. 76-280, Dept. of Comput. Sci., Cornell Univ. Ithaka, New York (1976), and Theoret. Comput. Sci., 8 (1979) 135-159.

    Google Scholar 

  34. Howell, R. R. /Rosier, L. E. /Yen, H.-C., An 0(n1,5) Algorithm to decide boundedness for conflict-free vector replacement systems, Internal Rept., Dept. of Comput. Sci., Univ. Texas at Austin, Texas, (1986).

    Google Scholar 

  35. Howell, R. R. /Huynh, D. T. /Rosier, L. E. /Yen, H.-C., Some complexity bounds for problems concerning finite and 2-dimensional vector addition systems with states, Techn. Rept. TR-86-08, Univ. Texas at Austin, Texas (1986).

    Google Scholar 

  36. Huynh, D. T., Commutative grammars: the complexity of the uniform word problems, Information and Control, 57 (1983), 21–39.

    Article  MathSciNet  MATH  Google Scholar 

  37. Huynh, D. T., Complexity of the word problem for commutative semigroups of fixed dimension, Acta Inforinatica, 22 (1985), 421–432.

    Article  MathSciNet  MATH  Google Scholar 

  38. Huynh, D. T., The complexity of the equivalence problem for commutative semigroups and symmetric vector addition systems, Proc. 17th ACM Symp. on Theory of Comput., (1985), 405-412.

    Google Scholar 

  39. Jantzen, M., On the hierarchy of Petri net languages, R. A. I. R. O., Informatique théorique, 13 (1979) 19–30.

    MathSciNet  MATH  Google Scholar 

  40. Jantzen, M., Eigenschaften von Petrinetzsprachen, Ph. D. Dissertation and Techn. Rept. IFI-HH-B-64/79, FB-Informatik, Iniv. Hamburg (1979).

    Google Scholar 

  41. Jantzen, M., On zerotesting-bounded multicounter machines. Lecture Notes in Comput. Sci., 67 (1979) 158–169.

    Article  MathSciNet  Google Scholar 

  42. Jantzen, M. /Valk, R., Formal properties of place transition nets, Proc. Advanced Course on GNT of Processes and Systems, Hamburg (1979), also: Lecture Notes in Comput. Sci. 84, Springer (1980) 165-212.

    Google Scholar 

  43. Jantzen, M., The power of synchronizing operations on strings, Theoret. Comput. Sci., 14 (1981) 127–154.

    Article  MathSciNet  MATH  Google Scholar 

  44. Jantzen, M., Complexity of Place/Transition nets, this volume.

    Google Scholar 

  45. Jones, N. D. / Landweber, L. H. / Lien, Y. E., Complexity of some problems in Petri nets, Theoret. Comput. Sci., 4 (1977), 277–299.

    Article  MathSciNet  MATH  Google Scholar 

  46. Kannan, R. / Bachern, A., Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix, SIAM J. Comput., 8 (1979), 499–507.

    Article  MathSciNet  MATH  Google Scholar 

  47. Kannan, R., Solving systems of linear equations over polynomials, Theoret. Comput. Sci., 39 (1985), 69–88.

    Article  MathSciNet  MATH  Google Scholar 

  48. Keller, R. M., Vector replacement systems: a formalism for modelling asynchronous systems, Techn. Rept. 117, Comput. Sci. Lab., Princeton Univ. (1972), revised (1974).

    Google Scholar 

  49. Kosaraju, S. R., Decidability of reachability in vector addition systems, Proc. 14th ACM Symp. on Theory of Computing, (1982), 267-281.

    Google Scholar 

  50. Lambert, J. L., On finding a partial solution to a linear system of equation in positive integers, Techn. Rept. no. 263, L. R. I. Univ. Paris-Sud, Orsay ( 1986).

    Google Scholar 

  51. Lambert, J. L., Consequences of the decidability of the reachability Problem for Petri nets, Techn. Rept. no 313, L. R. I. Univ. Paris-Sud, Orsay (1986).

    Google Scholar 

  52. Landweber, L. H. / Robertson, E. L., Properties of conflict free and Persistent Petri nets, J. Assoc. Comput. Mach., 25 (1978) 352–364.

    Article  MathSciNet  MATH  Google Scholar 

  53. Lipton, R. J., The reachability problem requires exponential space, Research Rept. No. 62, Dept. Comput. Sci., Yale Univ. (1976)

    Google Scholar 

  54. Mayr, E. W., Ein Algorithmus für das allgemeine Entscheidbarkeitsproblem bei Petrinetzen und damit zusammenhängende Probleme, Dissertation, Tech. Rept. TUM-I 8010, TU München (1980).

    Google Scholar 

  55. Mayr, E., Persistence of vector replacement systems is decidable, Acta Informatica, 15 (1981) 309–318.

    Article  MathSciNet  MATH  Google Scholar 

  56. Mayr, E. W., An algorithm for the general Petri net reachability problem, Siam J. Comput., 13 (1984) 441–460

    Article  MathSciNet  MATH  Google Scholar 

  57. Mayr, E. W. / Meyer, A. R., The complexity of the finite containment problem for Petri nets, J. Assoc. Comput. Mach., 28 (1981), 561–576.

    Article  MathSciNet  MATH  Google Scholar 

  58. Mayr, E. / Meyer, A., The complexity of the word problem for commutative semigroups and polynomial ideals, Adv. in Math., 46 (1982), 305–329.

    Article  MathSciNet  MATH  Google Scholar 

  59. Mazurkiewicz, A., Concurrent program schemes and their interpretation Techn. Rept., DAIMI PB 78, Univ. Aarhus (1977).

    Google Scholar 

  60. McAloon, K., Petri nets and large finite sets, Theoret. Comput. Sci., 32 (1984) 173–183.

    Article  MathSciNet  MATH  Google Scholar 

  61. Müller, H., The reachability problem for VAS, in: Advances in Petri Nets 1984, Lecture Notes in Comput. Sci. 188, Springer (1985).

    Google Scholar 

  62. Müller, H., Weak Petri net computers for Ackerman-functions, EIK, 21 (1985) 236–245.

    Google Scholar 

  63. Parigot, P. / Pelz E., A logical approach of Petri net languages, Theort.Comput. Sci., 39 (1985) 155–169.

    Article  MathSciNet  MATH  Google Scholar 

  64. Pelz, E., Closure properties of deterministic Petri nets, Techn. Rept. 297, L. R. I. Univ. Paris-Sud, Orsay (1986) and Proc. 4th Symp. on Theoretical Aspects of Comput. Sci., Passau (1987), to appear.

    Google Scholar 

  65. Peterson, J., Modeling of parallel systems, Tech. Rept. STAN-CS-74-410, Comput. Sci. Dept., Stanford Univ. (1974).

    Google Scholar 

  66. Peterson, J., Computation sequence sets, J. Comput. Syst. Sci., 13 (1976) 1–24.

    Article  MATH  Google Scholar 

  67. Peterson, J. L., Petri Net Theory and the Modeling of Systems, prentice-Hall, (1981).

    Google Scholar 

  68. Reisig, W., Petri Nets, An Introduction, EATCS Monograph on Teoret. Comput. Sci., Brauer, Rozenberg, Salomaa (Eds), Springer-Verlag (1985).

    Google Scholar 

  69. Reisig, W., Place transition systems, this volume

    Google Scholar 

  70. Rackoff, C., The covering and boundedness problems for vector addition systems, Theoret. Comput. Sci., 6 (1978), 223–231.

    Article  MathSciNet  MATH  Google Scholar 

  71. Rosier, L. E. / Yen, H.-C., A multiparameter analysis of the boundedness problem for vector addition systems, Proc. F. C. T.’ 85, Lecture Notes in Comput. Sci. 199, Springer (1985) 361–370 and J. Comput. Syst. Sci., 32 (1986) 105-135.

    Article  MathSciNet  Google Scholar 

  72. Rozenberg, G. / Verraedt, R., Subset languages of Petri nets Part I: The relationship to string languages and normal forms, Theoret. Comput. Sci., 26 (1983) 301–326; Part II: Closure properties, Theoret. Comput. Sci., 27 (1983) 85-108.

    Article  MathSciNet  MATH  Google Scholar 

  73. Schwer, S. R., Décidabilité de 1’algébricité de langages associes aux reseaux de Petri, These 3ème Cycle, Univ. Paris VII (1985).

    Google Scholar 

  74. Schwer, S. R., On the rationality of Petri net languages. Info. Processing Letters, 22 (1986) 145–146.

    Article  MathSciNet  MATH  Google Scholar 

  75. Starke, P. H., Free Petri net languages, Proc. 7th Symp. MFCS 78, Winkowski (Ed.), Lecture Notes in Comput. Sci., 64 Springer (1978) 506–515.

    Google Scholar 

  76. Starke, P. H., On the languages of bounded Petri nets, Proc. 8th Symp. MFCS 79, Becvar (Ed.), Lecture Notes in Comput. Sci. 74, Springer (1979), 425–442.

    Google Scholar 

  77. Starke, P. H., Petri-Netze, VEB Deutscher Verlag der Wissenschaften, Berlin (1980).

    Google Scholar 

  78. Starke, P. H., Processes in Petri nets, EIK 17 (1981) 389–416.

    MathSciNet  MATH  Google Scholar 

  79. Valk, R., Self-Modifying nets, a natural extension of Petri nets, Lecture Notes in Comput. Sci. 62, Springer (1978).

    Google Scholar 

  80. Valk, R. /Vidal-Naquet, G., On the rationality of Petri net languages, in Proc. Theoret. Comput. Sci., Lecture Notes in Comput. Sci. 48, Springer (1977), 319-328.

    Google Scholar 

  81. Valk, R. / Vidal-Naquet, G., Petri nets and regular languages, J. Comput. Syst. Sci., 23 (1981), 199–325.

    Article  MathSciNet  Google Scholar 

  82. Valk, R., Infinite behaviour and fairness, this volume

    Google Scholar 

  83. Vidal-Naquet, G., Rationalité et déterminisme dans les réseaux de Petri, Thèse d’Etat, Univ. Pierre et Marie Cuvie, Paris (1981)

    Google Scholar 

  84. Yoeli, M. /Ginzburg, A., Petri net languages and their applications, Research. Rept. CS-78-45, Dept. of Computer Science, University of Waterloo, ( 1978).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Informatik, Universitat Hamburg, Pothenbaumchaussee 67, D-2000, Hamburg 13, Deutschland

    Matthias Jantzen

Authors
  1. Matthias Jantzen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Technische Universität München, Postfach 202420, 8000, München 2, Federal Republic of Germany

    W. Brauer

  2. Institut für Methodische Grundlagen, Gesellschaft für Mathematik und Datenverarbeitung mbH, Postfach 1240, 5205, St. Augustin 1, Federal Republic of Germany

    W. Reisig

  3. Subfaculteit Wiskunde en Informatica, Rijksuniversiteit te Leiden, Postbus 9512, 2300 RA, Leiden, The Netherlands

    G. Rozenberg

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jantzen, M. (1987). Language Theory of Petri Nets. In: Brauer, W., Reisig, W., Rozenberg, G. (eds) Petri Nets: Central Models and Their Properties. ACPN 1986. Lecture Notes in Computer Science, vol 254. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-47919-2_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-47919-2_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17905-4

  • Online ISBN: 978-3-540-47919-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation