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The residue of vector sets with applications to decidability problems in Petri nets

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A set K of integer vectors is called right-closed, if for any elementmεK all vectors m′≧m are also contained in K. In such a case K is a semilinear set of vectors having a minimal generating set res(K), called the residue of K. A general method is given for computing the residue set of a right-closed set, provided it satisfies a certain decidability criterion.

Various right-closed sets wich are important for analyzing, constructing, or controlling Petri nets are studied. One such set is the set CONTINUAL(T) of all such markings which have an infinite continuation using each transition infinitely many times. It is shown that the residue set of CONTINUAL(T) can be constructed effectively, solving an open problem of Schroff. The proof also solves problem 24 (iii) in the EATCS-Bulletin. The new methods developed in this paper can also be used to show that it is decidable, whether a signal net is prompt [23] and whether certain ω-languages of a Petri net are empty or not.

It is shown, how the behaviour of a given Petri net can be controlled in a simple way in order to realize its maximal central subbehaviour, thereby solving a problem of Nivat and Arnold, or its maximal live subbehaviour as well. This latter approach is used to give a new solution for the bankers problem described by Dijkstra.

Since the restriction imposed on a Petri net by a fact [11] can be formulated as a right closed set, our method also gives a new general approach for „implementations” of facts.

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7. References

  1. Brinch Hansen, P.: Operating System Principles. Englewood Cliffs: Prentice-Hall 1973

    Google Scholar 

  2. Brams, G.W.: Réseaux de Petri: Théorie et pratique. Paris: Masson 1983

    Google Scholar 

  3. Burkhard, H.D.: Two Pumping Lemmata for Petri nets. EIK 17, 349–362 (1981)

    Google Scholar 

  4. Byrn, H.W.: Sequential processes, deadlocks and semaphore primitives. Harvard Univ., Techn. Rep. 7-75, Cambridge 1975

  5. Carstensen, H.: Fairneß bei Petrinetzen mit unendlichem Verhalten. Univ. Hamburg, Fachbereich Informatik, Report B-93/82 (1982)

  6. Conway, J.H.: Regular Algebra and Finite Machines. London: Chapman and Hall 1971

    Google Scholar 

  7. Carstensen, H., Valk, R.: Infinite behaviour and fairness in Petri nets. Fourth European Workshop on Application and Theory of Petri Nets, Toulouse, France (1983)

  8. Dijkstra, E.W.: Co-operating sequential Processes. In: Programming Languages 43–112 F. Genuys (ed.). Academic Press, London: 1968

    Google Scholar 

  9. Best, E, Thiagarajan, P.S.: P24 (iii). In: EATCS Bulletin 20, 310 (1983)

    Google Scholar 

  10. Eilenberg, S., Schützenberger, M.P.: Rational sets in communicative monoids. J. Algebra 13, 173–191 (1969)

    Google Scholar 

  11. Genrich, H.J., Lautenbach, K.: Facts in place/transition-nets. Lect. Notes Comput. Sci. No 64, pp. 213–231. Berlin-Heidelberg-New York: Springer

  12. Grabowski, J.: Linear methods in the Theory of Vector addition systems I. EIK 16, 207–236 (1980)

    Google Scholar 

  13. Hack, M.: Decision Problems for Petri Nets and Vector Addition Systems. MIT, Proj. MAC, Comput. Struct. Group Memo 95-1 (1974)

  14. Hack, M.: Petri net languages. MIT, Proj. MAC, Comp. Struct. Group Memo 124 (1975)

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

    Google Scholar 

  16. Hauschildt, D., Valk, R.: Safe states in banker like resource allocation problems. Proc. 5th. European Workshop Appl. Theory of Petri Nets, Aarhus, 1984

  17. Jantzen, M., Valk, R.: Formal properties of place/transition nets, In: Net Theory and Applications. W. Brauer (ed.), pp. 165–212. Lect. Notes Comput. Sci. No 84. Berlin-Heidelberg-New York: Springer 1979

    Google Scholar 

  18. Keller, R.M.: Vector Replacement Systems: A Formalism for Modelling Asynchronous Systems. Comput. Sci. Lab., Princeton Univ., Techn. Rep. 117 (1972, revised 1974)

  19. Karp, R.M., Miller, R.E.: Parallel Program Schemata. J. Comput. Syst. Sci. 3, 147–195 (1969)

    Google Scholar 

  20. Landweber, L.H.: Decision problems for ω-automata. Math. Syst. Theory 3, 376–384 (1969)

    Google Scholar 

  21. Lipton, R.J.: The Reachability Problem Requires Exponential Space. Yale Univ., Dept. of Comp. Sci., Research Report #62 (1976)

  22. Nivat, M., Arnold, A.: Comportements de processus. Lab. Informatique Théor. et Programm., Univ. Paris 6 and 7, Paris (1982)

    Google Scholar 

  23. Patil, S.S., Thiagarajan, P.S.: unpublished manuscript

  24. Rackoff, C.: The Covering and Boundedness Problems for Vector Addition Systems. Theor. Comput. Sci. 6, 223–231 (1978)

    Google Scholar 

  25. Schroff, R.: Vermeidung von totalen Verklemmungen in bewerteten Petrinetzen. Ph.D. Thesis, Techn. Univ. München (1974)

    Google Scholar 

  26. Schroff, R.: Vermeidung von Verklemmungen in bewerteten Petrinetzen. Lect. Notes Comput. Sci. No. 26, pp. 316–325 Berlin-Heidelberg-New York: Springer 1975

    Google Scholar 

  27. Valk, R.: Prévention des bloquages aux systèmes paralleles, Lecture notes, Univ. Paris VI (1976)

  28. Valk, R.: Infinite behaviour of Petri nets. Theor. Comput. Sci. 25, 311–341 (1983)

    Google Scholar 

  29. Valk, R., Vidal-Naquet, G.: Petri Nets and Regular Languages. J. Comput. Syst. Sci. 23, 299–325 (1981)

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Informatik (TGI), Universität Hamburg, Rothenbaumchaussee 67, D-2000, Hamburg 13, Bundesrepublik Deutschland

    Rüdiger Valk & Matthias Jantzen

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  1. Rüdiger Valk
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  2. Matthias Jantzen
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Valk, R., Jantzen, M. The residue of vector sets with applications to decidability problems in Petri nets. Acta Informatica 21, 643–674 (1985). https://doi.org/10.1007/BF00289715

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