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Hierarchical Reachability Graph Generation for Petri Nets

Hierarchical Reachability Graph Generation for Petri Nets

294 BUCHHOLZ AND

294 BUCHHOLZ AND KEMPERFigure 3.Extended net.extended net into a high-level net and a set of low-level nets to obtain the desired hierarchy,we formalize the aggregation and then consider the reachability and language invariance ofthe net extension. These are the properties, our hierarchy construction needs to preserve.A place p ∈ P agg represents a set of places in P ord . The following Lemma shows that themarking of each aggregated place can be computed as the weighted sum of the markingsof the places from the original net. This result implies that the marking of the original netuniquely determines the marking of all places from P agg , and that each marking of theplaces P agg describes a set of markings of places P ord .Lemma 1. M(p) = AG(v p , M) = ∑ z∈P ɛv p (z)M(z) for all p ∈ P aggs and all s ∈ T ∗ .Proof: We consider an induction over transition sequences s; initially, all p ∈ Pɛord triviallyfulfill the lemma and Pɛ agg =∅. For the induction step we consider a p ∈ P st (s ∈ T ∗ , t ∈ T ,t not in s) that results from v p = c a · v a + c b · v b , where v a , v b ∈ New when extending s by t(the case p ∈ P s is trivial in N st , since for these places M 0 , I − , and I + remain unchanged).We further consider in the extended net N st an induction over firing sequences σ . InitiallyM 0 (p) = ∑ z∈P ɛv p (z)M 0 (z) holds by definition. For the induction step, we considerM 0 [σ>M[κ>M ′ where the induction assumption ensures that M(p) = ∑ z∈p ɛv p (z)M(z), and we have to show M ′ (p) = ∑ z∈P ɛv p (z)M ′ (z) after extending σ by a transition κ.M ′ (p) = M(p) − I − (p,κ) + I + (p,κ)= M(p) − ∑ v p (z)I − (z,κ) + ∑ v p (z)I + (z,κ)z∈P ɛ z∈P ɛ= M(p) + ∑ z∈P ɛ(c a · v a (z) + c b · v b (z))(−I − (z,κ) + I + (z,κ))according to the definitions of successor marking and extended net. By inductionassumptionM(p) = ∑ z∈P ɛv p (z)M(z) = ∑ z∈P ɛ(c a · v a (z) + c b · v b (z))M(z)

HIERARCHICAL REACHABILITY GRAPH GENERATION FOR PETRI NETS 295so we can replace M(p) in the equation above. Observe that I − , I + remain invariant forz ∈ P ɛ , so we obtain:M ′ (p) = ∑ z∈P ɛ(c a · v a (z) + c b · v b (z))(M(z) − I − s (z,κ) + I + s (z,κ))= ∑ z∈P ɛ(c a · v a (z) + c b · v b (z))(M(z) − I − st (z,κ) + I + st (z,κ))= ∑ z∈P ɛv p (z)M ′ (z)The following lemma shows that the extension of the net by places P agg does not modifythe behavior in terms of transition sequences (i.e., the language of the net) or markings ofplaces P ord . Thus, all places from P agg are redundant.Lemma 2.holds:For all s ∈ T ∗ for which an extended net (N s , V s , A s ) is defined, the followingRS(PN) = RS(PN s )| PL(PN) = L(PN s )Proof: The proof is done by induction over sequences s. Initially s = ɛ is triviallyfulfilled. For the induction step we start with the special case New = Used =∅, whichdirectly implies equality. This case can occur, e.g., if A s =∅or if ̸∃v a , v b ∈ A s : v a (t) > 0,v b (t) < 0.For the general case, assume that the lemma holds for a sequence s, we consider theinduction step for sequence st, and we give a proof by contradiction:case RS(PN ɛ ) = RS(PN s )| P ⊂ RS(PN st )| P∃σ ∈ T ∗ : Mst 0[σ>M st but σ not possible in PN s . Hence ∃t ′ ∈ σ , which is not enabledin PN s . I.e., there is are fewer tokens in a place p ∈ P s than in P st . This contradicts thedefinition of the extended nets, because M 0 , I − , I + are modified only with respect to newplaces.case RS(PN ɛ ) = RS(PN s )| P ⊃ RS(PN st )| P∃σ ∈ T ∗ : Ms 0[σ > M s but σ not possible in PN st . Hence ∃t ′ ∈ σ , which is not enabledin PN st . I.e., there is an insufficient number of tokens in a place p ∈ P st \P s . Accordingto Lemma 1, M(p) = ∑ z∈P ɛv p (z)M(z). Since Ist −(p, t) = ∑ z∈P ɛv p (z)Iz,t − and for eachz ∈ P ɛ : M(z) ≥ Iz,t − , we obtain a contradiction.In summary, equality holds. Equivalence of languages follows by the same line of argumentation.A direct consequence of Lemma 2 is that invariants remain valid, T -invariants because oflanguage equivalence and P-invariants because of additivity of invariants, cf. Theorem 1.Furthermore, for places like p13 in our example, where I + (p13, t) = I − (p13, t) for alltransitions t, we can decide whether a given initial marking M 0 (p13) ensures that a transition

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