Kleene Theorems for Product Systems

  • Kamal Lodaya
  • Madhavan Mukund
  • Ramchandra Phawade
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6808)


We prove Kleene theorems for two subclasses of labelled product systems which are inspired from well-studied subclasses of 1-bounded Petri nets. For product T-systems we define a corresponding class of expressions. The algorithms from systems to expressions and in the reverse direction are both polynomial time. For product free choice systems with a restriction of structural cyclicity, that is, the initial global state is a feedback vertex set, going from systems to expressions is still polynomial time; in the reverse direction it is polynomial time with access to an NP oracle for finding deadlocks.


Product System Polynomial Time Regular Expression Polynomial Time Algorithm Free Choice 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ant96]
    Antimirov, V.: Partial derivatives of regular expressions and finite automaton constructions. Theoret. Comput. Sci. 155, 291–319 (1996)CrossRefMATHMathSciNetGoogle Scholar
  2. [Arn94]
    Arnold, A.: Finite transition systems. Prentice-Hall, Englewood Cliffs (1994)Google Scholar
  3. [BeSe86]
    Berry, G., Sethi, R.: From regular expressions to deterministic automata. Theoret. Comp. Sci. 48(3), 117–126 (1986)CrossRefMATHMathSciNetGoogle Scholar
  4. [BeSh83]
    Best, E., Shields, M.: Some equivalence results for free choice and simple nets and on the periodicity of live free choice nets. In: Ausiello, G., Protasi, M. (eds.) CAAP 1983. LNCS, vol. 159, pp. 141–154. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  5. [BV84]
    Best, E., Voss, K.: Free choice systems have home states. Acta Inform. 21, 89–100 (1984)CrossRefMathSciNetGoogle Scholar
  6. [BMc63]
    Brzozowski, J.A., McCluskey, E.J.: Signal flow graph techniques for sequential circuit state diagrams. IEEE Trans. Electr. Comput. EC-12, 67–76 (1963)CrossRefMATHGoogle Scholar
  7. [CH74]
    Campbell, R.H., Habermann, A.N.: The specification of process synchronization by path expressions. In: Gelenbe, E., Kaiser, C. (eds.) Proc. Operating Systems Conference. LNCS, vol. 16, pp. 89–102 (1974)Google Scholar
  8. [CHEP71]
    Commoner, F., Holt, A.W., Even, S., Pnueli, A.: Marked directed graphs. J. Comp. Syst. Sci. 5(5), 511–523 (1971)CrossRefMATHMathSciNetGoogle Scholar
  9. [DE95]
    Desel, J., Esparza, J.: Free choice Petri nets, Cambridge (1995)Google Scholar
  10. [DR95]
    Diekert, V., Rozenberg, G. (eds.): The book of traces. World Scientific, Singapore (1995)Google Scholar
  11. [EZ76]
    Ehrenfeucht, A., Zeiger, P.: Complexity measures for regular expressions. J. Comp. Syst. Sci. 12, 134–146 (1976)CrossRefMATHMathSciNetGoogle Scholar
  12. [GR92]
    Garg, V.K., Ragunath, M.T.: Concurrent regular expressions and their relationship to Petri nets. Theoret. Comp. Sci. 96(2), 285–304 (1992)CrossRefMATHMathSciNetGoogle Scholar
  13. [GL73]
    Genrich, H.J., Lautenbach, K.: Synchronisationsgraphen. Acta Inform. 2, 143–161 (1973)CrossRefMATHMathSciNetGoogle Scholar
  14. [Gra81]
    Grabowski, J.: On partial languages. Fund. Inform. IV(2), 427–498 (1981)MATHMathSciNetGoogle Scholar
  15. [GH08]
    Gruber, H., Holzer, M.: Finite automata, digraph connectivity, and regular expression size. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 39–50. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. [Hack72]
    Hack, M.H.T.: Analysis of production schemata by Petri nets. Project MAC Report TR-94. MIT (1972)Google Scholar
  17. [Hoa85]
    Hoare, C.A.R.: Communicating sequential processes. Prentice-Hall, Englewood Cliffs (1985)MATHGoogle Scholar
  18. [JR91]
    Jiang, T., Ravikumar, B.: A note on the space complexity of some decision problems for finite automata. Inf. Proc. Lett. 40(1), 25–31 (1991)CrossRefMATHMathSciNetGoogle Scholar
  19. [Kle56]
    Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, Princeton, pp. 3–41 (1956)Google Scholar
  20. [Lod06a]
    Lodaya, K.: Product automata and process algebra. In: Pandya, P.K., Hung, D.v. (eds.) Proc. 4th SEFM, Pune, pp. 128–136. IEEE, Los Alamitos (2006)Google Scholar
  21. [Lod06b]
    Lodaya, K.: A regular viewpoint on processes and algebra. Acta Cybernetica 17(4), 751–763 (2006)MATHMathSciNetGoogle Scholar
  22. [LRR03]
    Lodaya, K., Ranganayakulu, D., Rangarajan, K.: Hierarchical structure of 1-safe petri nets. In: Saraswat, V.A. (ed.) ASIAN 2003. LNCS, vol. 2896, pp. 173–187. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. [LW00]
    Lodaya, K., Weil, P.: Series-parallel languages and the bounded-width property. Theoret. Comp. Sci. 237(1-2), 347–380 (2000)CrossRefMATHMathSciNetGoogle Scholar
  24. [Maz77]
    Mazurkiewicz, A.: Concurrent program schemes and their interpretations. DAIMI Report PB-78 (1977)Google Scholar
  25. [Mil89]
    Milner, R.: Communication and concurrency. Prentice-Hall, Englewood Cliffs (1989)MATHGoogle Scholar
  26. [Moh99]
    Mohalik, S.K.: Local presentations for finite state distributed systems, PhD thesis, University of Madras (1999)Google Scholar
  27. [MR02]
    Mohalik, S., Ramanujam, R.: Distributed automata in an assumption-commitment framework. Sādhanā 27, part 2, 209–250 (2002)MATHMathSciNetGoogle Scholar
  28. [MS97]
    Mukund, M., Sohoni, M.A.: Keeping track of the latest gossip in a distributed system. Distr. Comp. 10(3), 117–127 (1997)CrossRefGoogle Scholar
  29. [Och85]
    Ochmański, E.: Regular behaviour of concurrent systems. Bull. EATCS 27, 56–67 (1985)Google Scholar
  30. [Pet62]
    Petri, C.-A.: Fundamentals of a theory of asynchronous information flow. In: Popplewell, C.M. (ed.) Proc. IFIP, Munich, pp. 386–390. North-Holland, Amsterdam (1962)Google Scholar
  31. [RL03]
    Ranganayukulu, D., Lodaya, K.: Infinite series-parallel posets of 1-safe nets. In: Thangavel, P. (ed.) Proc. Algorithms and Artificial Systems, Chennai, pp. 107–124. Allied (2003)Google Scholar
  32. [TT07]
    Tesson, P., Thérien, D.: Logic meets algebra: the case of regular languages. Log. Meth. Comput. Sci. 3(1) (2007)Google Scholar
  33. [TV84]
    Thiagarajan, P.S., Voss, K.: A fresh look at free choice nets. Inf. Contr. 61(2), 85–113 (1984)CrossRefMATHMathSciNetGoogle Scholar
  34. [Thi96]
    Thiagarajan, P.S.: Regular trace event structures. BRICS Report RS-96-32. Dept. of Computer Science, Aarhus University (1996)Google Scholar
  35. [Weil04]
    Weil, P.: Algebraic recognizability of languages. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 149–175. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  36. [Zie87]
    Zielonka, W.: Notes on finite asynchronous automata. RAIRO Inform. Th. Appl. 21(2), 99–135 (1987)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kamal Lodaya
    • 1
  • Madhavan Mukund
    • 2
  • Ramchandra Phawade
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.Chennai Mathematical InstituteSiruseriIndia

Personalised recommendations