Liveness of weighted circuits and the diophantine problem of Frobenius

  • Piotr Chrzastowski-Wachtel
  • Marek Raczunas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 710)


The paper deals with the problem of liveness of weighted conservative circuits. The weight of a marking gives us an important information about its liveness. Some weights correspond only to live markings, some only to dead ones and some to both. The simultaneous presence of live and dead markings with the same weight is associated with the presence of several equivalence classes generated by the solutions of the state equation. We call such classes orbits. It is impossible to reach from a given marking a state belonging to a different orbit and this creates an opportunity of coexistence of a dead and a live marking with the same weight. Different orbits are also associated with the presence of a kind of frozen tokens. The diophantine problem of Frobenius is used to determine a formula for the least live weight.


Structure theory Weighted circuits Diophantine problem of Frobenius Liveness Petri nets 


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  1. [Brauer 42]
    A. Brauer. On a problem of partitions. Amer.J.Math., 64 (1942)Google Scholar
  2. [ChR 93]
    P. Chrzastowski-Wachtel, M. Raczunas. Liveness of Weighted Circuits and the diophantine problem of Frobenius Technical Report, Humboldt University, Berlin 1993Google Scholar
  3. [RCh 92]
    M. Raczunas, P. Chrzastowski-Wachtel. A diophantine problem of Frobenius in terms of the least common multiple submitted to Acta ArithmeticaGoogle Scholar
  4. [Selmer 77]
    E. S. Selmer. On the linear diophantine problem of Frobenius J.reine angew. Math. 293/294 (1977)Google Scholar
  5. [TCCS 92]
    E. Teruel, P. Chrzastowski-Wachtel, J. M. Colom, M. Silva. On weighted T-systems. Advances in Petri Nets, Lecture Notes in Computer Science, vol.616, Springer Verlag (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Piotr Chrzastowski-Wachtel
    • 1
  • Marek Raczunas
    • 2
  1. 1.Institute of InformaticsWarsaw UniversityWarszawaPoland
  2. 2.Institute of MathematicsUniversity of GdańskGdańskPoland

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