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

## Summary

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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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