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 **for**mer 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 **for**mally whya hierarchical net indeed includes the reachability set or language of its N ɛ . The key issue**for** 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 **for**mally, 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.