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Hierarchical Reachability Graph Generation for Petri Nets

Hierarchical Reachability Graph Generation for Petri Nets

296 BUCHHOLZ AND

296 BUCHHOLZ AND KEMPERis dead (due to M 0 (p13) < I − (p13, t)) or whether the place can safely be omitted (sinceM(p13) ≥ I − (p13, t) for all M ∈ RS). If the former is the case, it is clear that the net isnot live.So far we have described a way to add places to a net without changing its reachability setor language. The notion of an extended net is only introduced to help explain formally whya hierarchical net indeed includes the reachability set or language of its N ɛ . The key issuefor a hierarchy is abstraction. At a higher level the state of a subsystem must be representedin less detail than at a lower level. We use aggregated places to obtain an aggregated staterepresentation, and the notion of a subsystem is built on the concept of region. In thefollowing we describe how to split an extended net into a high-level net HN and a set oflow-level nets L 1 ,...,L J via the notion of regions.Let R(N ɛ ) denote the set of minimal regions w.r.t. an extended net (N ɛ , V ɛ , A ɛ ). When weextend this net for a transition t ∈ Trout of a region N r ∈ R(N ɛ ), then the new places connectN r with regions that contain t•. Consequently, we merge all these regions with N r ; this yieldsa new region N r ′ according to Proposition 1. Since we start from a partition into regions, theresulting set of regions is a partition again, but this partition is less fine. Transition t becomesinternal in N r ′ , because •t ∪ t• ⊆P′ r , and new places give an aggregated description of theinternal behavior of N r ′ w.r.t. transition t. Following this procedure over a sequence s oftransitions yields (N s , V s , A s ) and a partition into regions, where some regions have internaltransitions and aggregated places. In this situation a decomposition of an extended net intoa high-level net using the aggregated description and a set of low-level nets resulting fromregions with internal behavior gives the two-level hierarchy we aim for. In the high-levelnet, transitions that are internal to some region are no longer visible, and places that areonly connected to internal transitions are also invisible.More formally, a high-level net for a given extended net (N s , V s , A s ) results from aprojection with respect to A s .Definition 6. Let (N s , V s , A s ) be an extended net; its corresponding high-level net HN =(P H , T H , I H− , I H+ , M 0H )isdefined by:P H ={p | v p ∈ A s } (5)T H =•P H ∪ P H • (6)and I H− , I H+ , M 0H are the corresponding projections of I − , I + , and M 0 w.r.t. P H ∪ T H .Example 3.in figure 3.Figure 4 shows the high-level net for the extended net of our running exampleThe following Lemma shows that the reachability set of the high-level net contains allmarkings of places from P H in the original net.Lemma 3. Let RS ′ (N ɛ ) ={M ′ |∃M ∈ RS(N ɛ ):∀p ∈ P s : M ′ (p) = AG(v p , M)} for (N s ,V s , A s ), thenRS ′ (N ɛ ) ⊆ RS(HN) (7)

HIERARCHICAL REACHABILITY GRAPH GENERATION FOR PETRI NETS 297Figure 4.High-level net for s = t1t6t7.Proof: The proof uses the previous lemma about equality for N ɛ and N s , and the fact thatHN is deduced by omitting places (releases enabling conditions and thus increases RS) andby omitting transitions that are isolated (since all elements in the pre- and postset of omittedtransitions are used in linear combinations and thus are not contained in A s anymore).Isolated transitions have no effect on RS.P-invariants of HN are linear combinations of P-invariants of N ɛ ; hence, if N ɛ is coveredby P-invariants, so is HN. Consequently, we can guarantee finiteness of RS(HN) ifN ɛ iscovered by P-invariants.Lemma 3 states that the HN indeed considers a more abstract net such that the detailednet can only behave in a way that is consistent with this abstraction/aggregation. The nextstep is to consider single regions in detail. If a region is trivial (i.e., it contains no internaltransitions), then it is part of the HN. For non-trivial regions we define a description thattakes into consideration the detailed region plus the places from P agg that describe theaggregated state of the region. The resulting net for region r is a low-level net for region r.Definition 7. A low-level net is a net LN = (P L , T L , I L− , I L+ , M 0L ) that belongs to aregion r in an extended net (N s , V s , A s ). Let P L = (P r ∩ P H ) ∪ (P r ∪ P ɛ ), T L = P L •,(• with respect to N ɛ ).⎧⎪⎨ I H− (p)(t) if p ∈ P H \P ɛI L− (p)(t) = I − (p)(t) if p ∈ P ɛ⎪⎩0 otherwise⎧⎪⎨ I H+ (p)(t) if p ∈ P H \P ɛI L+ (p)(t) = I + (p)(t) if p ∈ P ɛ⎪⎩0 otherwiseIf LN and the corresponding region r H in HN do not differ in their transitions, LN istrivial and can be neglected. Otherwise LN is non-trivial.

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