Liveness of weighted circuits and the diophantine problem of Frobenius
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.
KeywordsStructure theory Weighted circuits Diophantine problem of Frobenius Liveness Petri nets
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