Infinitary languages: Basic theory and applications to concurrent systems

  • H. J. Hoogeboom
  • G. Rozenberg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 224)


The aim of this paper is to provide an outlook at this part of the theory of infinitary languages that seems to be essential for understanding the modern theory of concurrent systems. In the first part of this paper we discuss an automata-based approach to infinitary languages. In the second part we turn into applying this theory to concurrent systems as considered within the framework of Petri nets.


ω-words infinitary languages ω-regular languages limit adherence finite state automata i-acceptance determinism fair shuffle topology Petri nets firing sequences bounded control bounded i-acceptance transitional i-acceptance justice fairness 


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  1. [Ar]
    Arnold, A.: Topological characterizations of infinite behaviours of transition systems, LNCS Lecture Notes in Computer Science by Springer Verlag 154, 1983.Google Scholar
  2. [AN]
    Arnold, A. and Nivat, M.: Comportement de processus, Report LITP No. 82–12, Univ. Paris 7, 1982.Google Scholar
  3. [BN]
    Boasson, L. and Nivat, N.: Adherences of languages, JCSS 20 (1980) 285–309.Google Scholar
  4. [Bü]
    Büchi, J.R.: On a decision method in restricted second-order arithmetic, Proc. Int. Congr. Logic, Math. and Phil.Sci. 1960, Stanford University Press, Stanford, Calif., 1962.Google Scholar
  5. [CA]
    Carstensen, H.: Fairness bei Petrinetzen mit unendlichem Verhalten, Report No. 93, Fachbereich Informatik, Univ. Hamburg, 1982.Google Scholar
  6. [CV]
    Carstensen, H. and Valk, R.: Infinite behaviour and fairness in Petri nets, LNCS Lecture Notes in Computer Science by Springer Verlag 188, 1985.Google Scholar
  7. [Ch]
    Choueka, Y.: Theories of ω-automata on ω-tapes: A simplified approach, JCSS 8 (1974) 117–141.Google Scholar
  8. [CG]
    Cohen, R.S. and Gold, A.Y.: Theory of ω-languages, JCSS 15 (1977) 169–208.Google Scholar
  9. [CS]
    Culik II, K. and Salomaa, A.: On infinite words obtained by iterating morphisms, TCS 19 (1982) 29–38.Google Scholar
  10. [Ei]
    Eilenberg, S.: Automata, languages and machines, Vol. A, Academic Press, New York, 1974.Google Scholar
  11. [Gi]
    Girault-Beauquier, D.: Bilimites de langages reconnaissables, TCS, 33 (1984) 335–342.Google Scholar
  12. [GN]
    Gire, F. and Nivat, N.: Relations rationelles infinitaires, Calcolo 21 (1984) 91–125.Google Scholar
  13. [Ha]
    Hack, M.: Petri net languages, Lab.Comp.Sci., MIT, Techn.Rep. 159, Cambridge, Mass., 1972.Google Scholar
  14. [HS]
    Hartmanis, J. and Stearns, R.E.: Sets of numbers defined by finite automata, American Mathematical Monthly 74 (1967) 539–542.Google Scholar
  15. [Hd]
    Head, T.: Adherences of DOL languages, TCS 31 (1984) 139–150.Google Scholar
  16. [Hd1]
    Hedlund, G.A.: Remarks on the work of Axel Thue on sequences, Nordisk Matematisk Tidskrift 16 (1967) 148–150.Google Scholar
  17. [Ho]
    Hossley, R.: Finite tree automata and ω-automata, MAC Technical report No. 102, MIT, Cambridge, Mass., 1972.Google Scholar
  18. [Kö]
    König, D.: Theorie der endlichen und unendlichen Graphen, Chelsea, New York, 1935.Google Scholar
  19. [La]
    Landweber, L.H.: Decision problems for ω-automata, Mathematical Systems Theory 3 (1969) 376–384.Google Scholar
  20. [LPS]
    Lehman, D., Pnueli, A. and Stavi, J.: Impariality, justice and fairness: the ethics of concurrent termination, LNCS Lecture Notes in Computer Science by Springer Verlag 115, 1981.Google Scholar
  21. [LS]
    Lindner, R. and Staiger, L.: Algebraische Codierungstheorie — Theorie der sequentiellen Codierungen, Akademie, Berlin, 1971.Google Scholar
  22. [Li]
    Linna, M.; On ω-words and ω-computations, Annales Univ. Turkuensis, ser. AI 168, Turku, 1975.Google Scholar
  23. [McN]
    McNaughton, R.: Testing and generating infinite sequences by a finite automaton, Information and Control (1966) 521–530.Google Scholar
  24. [Mo]
    Morse, H.M.: Recurrent geodesics on a surface of negative curvature, Transactions of the American Mathematical Society 22 (1921) 84–100.Google Scholar
  25. [Mu]
    Muller, D.E.: Infinite sequences and finite machines, AIEE Proc 4th Ann. symp. switch. circ. th. log. design, 1963.Google Scholar
  26. [N1]
    Nivat, M.: Sur les ensembles de mots infinis engendres' par une grammaire algebrique, RAIRO Informatique Theorique 12 (1978) 259–278.Google Scholar
  27. [N2]
    Nivat, M.: Infinitary relations, LNCS Lecture Notes in Computer Science by Springer Verlag 112, 1981.Google Scholar
  28. [N3]
    Nivat, M.: Behaviours of synchronized system of processes, Report LITP No. 81–64, Univ. Paris 7, 1981.Google Scholar
  29. [NP1]
    Nivat, M. and Perrin, D.: Ensembles reconnaissables de mots biinfinis, Proc. 14th ACM symp. Th. Computing, 1982.Google Scholar
  30. [NP2]
    Automata on infinite words, Edited by M. Nivat and D. Perrin, LNCS Lecture Notes in Computer Science by Springer Verlag 192, 1985.Google Scholar
  31. [Pa]
    Park, D.: Concurrency and automata on infinite sequences, LNCS Lecture Notes in Computer Science by Springer Verlag 104, 1981.Google Scholar
  32. [Pe]
    Perrin, D.: Recent results on automata and infinite words, LNCS Lecture Notes in Computer Science by Springer Verlag 176, 1984.Google Scholar
  33. [PU]
    Prodinger, H. and Urbanek, F.J.: Language operators related to Init, TCS 8 (1979) 161–175.Google Scholar
  34. [Re]
    Reisig, W.: Petri nets, An introduction, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, Berlin, 1985.Google Scholar
  35. [RT]
    Rozenberg, G. and Thiagarajan, P.S.: Petri nets: Basic notions, structure, behaviour, this volume.Google Scholar
  36. [Sa]
    Salomaa, A.: Jewels of formal language theory, Computer Science Press, 1981.Google Scholar
  37. [SW]
    Staiger, L. and Wagner, K.: Automaten theoretische und automatenfreie Characterisierungen topologischer Klassen regulärer Folgenmengen, Elektronische Informationsverarbeitung und Kybernetik 10 (1974) 379–392.Google Scholar
  38. [TY]
    Takahashi, M. and Yamasaki, H.: A note on ω-regular languages, TCS 23 (1983) 217–225.Google Scholar
  39. [Th]
    Thue, A.: Ueber unendliche Zeichenreihen, Videnskabsselskabets Skrifter, I. Mat.-nat. Kl., Kristiania 7 (1906) 1–22.Google Scholar
  40. [Va]
    Valk, R.: Infinite behaviour of Petri nets, TCS 25 (1983) 311–341.Google Scholar
  41. [Wa]
    Wagner, K.: On ω-regular sets, Information and Control 43 (1979) 123–177.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • H. J. Hoogeboom
    • 1
  • G. Rozenberg
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of LeidenLeidenThe Netherlands

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