300 BUCHHOLZ AND KEMPER6. **Hierarchical** representations of RS and RGIn the previous section, a method has been introduced to decompose a flat net into partsand to describe the dynamic behavior of the parts by generating the reachability sets andgraphs **for** the LNs and the HN. Some results can be proved or disproved on the componentreachability sets/graphs with techniques like those described in [4]. However, to provegeneral properties, the complete reachability set/graph has to be known. In the followingtwo sections, we present a methodology to generate and represent RS(PN) and RG(PN)in space- and time-efficient way. The idea is to describe markings or states of PN by acomposition of LN markings related to an HN marking. In a similar way, the reachabilitygraph of PN is described by a set of adjacency matrices, which are defined as the Kroneckerproduct of small adjacency matrices **for** the LNs. The Kronecker representation is a veryconvenient way to represent huge graphs in a compact **for**m. Additionally, and often moreimportantly, very time- and space-efficient analysis algorithms have been developed **for**Kronecker structures, cf. Section 7.For notational convenience, we assume that PN is decomposed into one HN and J LNs thatare consecutively numbered 1 through J. Furthermore, we assume that all reachability sets**for** the HN and the extended LNs are finite. Consequently, reachability sets are isomorphic tofinite sets of consecutive integers. Let RS(HN) ={0,...,n H −1}. Marking M x correspondsto the x’th marking in RS(HN), and we use x and M x interchangeably. We can representRG(HN) by an n H ×n H matrix Q H with Q H (x, y) = t iff M x [t > M y **for** transition t ∈ T H ,provided that between any two markings in RG(HN) at most one transition exists. If morethan one transition between M x and M y exists, Q H (x, y) describes a list of transition indexes.We use **for** generality the notation t ∈ Q H (x, y) **for** all t with M x [t > M y . Q H contains thecomplete description of RS(HN).The reachability set RS( j) ofLN j depends on the environment given by HN. Hence,we consider the EN e that corresponds to j, and define RS( j) as a projection of RS(e) onthe places of j. Since any LN j and the HN share some places P j ∩ P H , we also define˜RS( j) as a projection of RS( j) onto places from P j ∩ P H . Markings from ˜RS( j) are macromarkings and allow the partition of RS( j), i.e., markings from ˜RS( j) represent disjointsubsets of RS( j). Macro markings are useful **for** the generation of RS( j). Full details of anHN are irrelevant at the LN level, and one can redefine transitions of Tjin such that their firingis marking-dependent with respect to RS(HN)| P j ∩P ⊇ ˜RS( j). There**for**e, EN is definedHto help obtain a clear notion of RS( j). In practice, however, computation of RS( j) canbe per**for**med more efficiently by using only macro-marking-dependent transitions of Tj in ,since transitions local to HN are ignored. The resulting set RS( j) might contain markingsthat are not in RS(PN), but they can be eliminated in a subsequent step, cf. Section 7.Let ˜RS( j) ={0,...,ñ j − 1} and let RS( j, ˜x) with ˜x ∈{0,...,ñ j − 1} denote the set ofmarkings from RS( j) that belong to marking ˜x in ˜RS( j). All markings from a set RS( j, ˜x)are indistinguishable in the HN, i.e., the marking of the places from P H ∩ P j is the same.Since reachability sets are assumed to be finite, each set RS( j, ˜x) can be represented bya set of integers {0,...,n j (˜x) − 1}. A marking M x ∈ RS(HN) uniquely determines themacro markings **for** all LNs. We use x j to denote the macro marking of LN j belonging tomarking x, and obtain M x j = M x | P H ∩P . j

HIERARCHICAL REACHABILITY GRAPH GENERATION FOR PETRI NETS 301Markings of PN s can be characterized using J + 1-dimensional integer vectors (x H ,x 1 ,...,x J ), where 0 ≤ x H < n H and 0 ≤ x j < n j (x j H ) **for** j ∈{1,...,J}. x H describes amarking from RS(HN) and x j a marking from RS( j, x j H ), where x j Hdescribes the macromarking of LN j when the marking of HN equals x H . This implies that M xH | P H ∩P = jM xj | P H ∩P j . Since the previous relation holds, each integer vector of the previously introduced**for**m determines a marking of the extended net. We define a hierarchically generatedreachability set:RS H (PN s ) =⋃x H × J j=1 RS( j, x j )H(8)x H ∈RS(HN)Observe that the number of markings in RS H (PN s ) equalsn∑H −1x=0J∏n j (x j ),j=1but at most ∑ n H −1x=0 (1 + ∑ Jj=1 n j(x j )) markings have to be stored to represent RS H (PN s ). Afew reachability sets with a few hundred markings are enough to describe sets with severalmillion or billion markings. To keep the representation compact, it has to be assured thatreachability sets of the LNs are roughly of the same size. Of course, this is hard to assure apriori, but it is possible to generate regions in such a way that they include a similar numberof places and transitions; that is often sufficient to yield reachability sets of a similar size**for** the different regions. The following lemma shows that RS H (PN s ) contains all markingsin RS(PN s ).Lemma 4. The hierarchically generated reachability set and the reachability set arerelated as follows.RS(PN s ) ⊆ RS H (PN s ) and RS(PN) ⊆ RS H (PN s )| PProof: The previous lemmas imply that RS(PN s )| P H ⊆ RS(HN) = RS H (PN s )| P H ,RS(PN s )| P j ⊆ RS( j) = RS H (PN s )| P j **for** all j ∈{1,...,J} and **for** each M ∈ RS(PN s ):M| P H ∈ RS(HN), M| P j ∈ RS( j). By construction of the hierarchically generated reachabilityset, M ∈ RS H (PN s ) follows. The second relation follows since RS(PN) = RS(PN s )| P .Un**for**tunately, the reachability set of the original net is not necessarily equal to RS H (PN s );it is only included in the hierarchically generated reachability set. Be**for**e we computeRS(PN s ) as part of RS H (PN s ), the reachability graph is represented in a compact **for**msimilar to the compact representation of the reachability set.First of all, we define the effect of transitions locally **for** LNs. Two different classes oftransitions have to be distinguished with respect to LN j.• LT j = T j \T H is the set of internal transitions in LN j.• ST j = Tjin ∪ Tjout is the set of transitions that describe the communication between LNj and HN.