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Kamel Barkaoui (CEDRIC-CNAM – Paris, France)

Copyright: © 2013
|Pages: 14

DOI: 10.4018/978-1-4666-3922-5.ch003

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TopThis section contains the basic definitions and notations of Petri nets’ theory (Reisig, 1985) which will be needed in the rest of this chapter.

**Definition 1:***A P/T net is a weighted bipartite digraph N*=(*P*,*T*,*F*,*V*)*where: P*≠ ∅*is a finite set of node*places*; T*≠ ∅*is a finite set of node*transitions*; F*⊆ (*P*×*T*) ∪ (*T*×*P*)*is the*flow*relation; V*:*F*→ IN^{+}*is the*weight function (*valuation*).**Definition 2:***Let N*= (*P*,*T*,*F*,*V*)*be a P/T net*.*The*preset*of a node x*∈(*P*∪*T*)*is defined as*{^{•}x =*y*∈ (*P*∪*T*)*s.t.*(*y, x*)∈*F*},*The*postset*of a node x*∈ (*P*∪*T*)*is defined as x*{^{•}=*y*∈(*P*∪*T*)*s.t.*(*x, y*)∈*F*}, the preset (resp. postset) of a set of nodes is the union of the preset (resp. postset) of its elements.*The*sub-net*induced by a sub-set of places P’*⊆*P is the net N’=*(*P’, T’, F’, V’*)*defined as follows: T =*∪^{•}P*P*((^{•}; F = F ∩*P × T*)∪(*T × P*))*; V is the restriction of V on F’. The sub-net induced by a sub-set of transitions T’*⊆*T is defined analogously.***Definition 3:***Let N =*(*P, T, F, V*)*be a P/T net. A shared place p*(*|p*)^{•}|≥2*is said to be*homogenous*iff:*∀*t, t’*∈*p*(^{•}, V*p, t*)*= V*(*p, t’*).*A place p*∈*P is said to be non-blocking iff: p*⇒^{•}≠ ∅*M in*{_{t∈•p}*V*(*t, p*)}*≥ M in*{_{t∈p•}*V*(*p, t*)}.*If all shared places of P are homogenous, then the valuation V is said to be homogenous. The valuation V of a P/T net N can be extended to the application W from*(*P × T*)∪(*T × P*)*→*IN*defined by:*∀*u*∈(*P × T*)∪ (*T × P*),*W*(*u*)*=V*(*u*)*if u*∈*F and W*(*u*)*=*0*otherwise.***Definition 4:***The matrix C indexed by P*×*T and defined by C*(*p, t*) =*W*(*t, p*)*− W*(*p, t*)*is called the incidence matrix of the net. An integer vector f ≠ 0 indexed by P*(*f*∈Z)^{P}*is a P-invariant iff f*0^{t}·C =(^{t}. An integer vector g ≠ 0 indexed by T*g*∈**Z**)^{T}*is a T-invariant iff C·g =*0. ||*f*||*=*{*p*∈*P/f (p) =*0}*(resp.*||*g*||*=*{*t*∈*t/g*(*t*)*= 0*}*) is called the support of f (resp. of g). We denote by*||*f*||{^{+}=*p*∈*P/f (p) > 0*}*and by*||*f*||{^{−}=*p*∈*P/f*(*p*)*< 0*}.*N is said to be conservative iff there exists a P-invariant f such that*||*f*||*=*||*f*||^{+}= P.

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