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Hierarchical Reachability Graph Generation for Petri Nets

Hierarchical Reachability Graph Generation for Petri Nets


304 BUCHHOLZ AND KEMPERMixed radix numbering schemes can also be applied to number markings in RS H (PN s ).We use a two-level scheme in which the first number describes the HN marking and the secondnumber is computed from the numbers of LN markings. Thus, marking (x H , x 1 ,...,x J )receives number (x H , x L ), whereJ∑ J∏ ( )x L =n i xiH .x jj=1 i= j+1Using this numbering scheme, RG H (PN s ) can be represented using Kronecker productsof Boolean matrices. We define QtH as the adjacency matrix of the reachability graphconsidering only transition t. Using the two-level marking number, QtH has a block structurewith n 2 Hblock matrices.⎛⎞Qt H [0, 0] ··· Qt H [0, n H − 1]Qt H = ⎜⎝.. ... .. ⎟⎠Qt H [n H − 1, 0] ··· Qt H [n H − 1, n H − 1]Submatrix Q H t [x, y] includes all transitions between markings that belong to HN markingx and markings that belong to HN marking y due to transition t in the net. Each submatrixcan be represented as a Kronecker product of LN matrices.Q H t [x, y] =J⊗j=1Q jt [x j , y j ] (9)This form describes a very compact representation of a huge matrix. Since there are moretransitions than just t, the adjacency matrix of RG H (PN) results from a summation over all t:Q H = ∑ t∈TQ H t .This equation is useful as it is but we can identify special cases that can be exploitedby algorithms. Internal transitions cause a specific matrix pattern of nonzero elements thatallows to build a simpler structure by a Kronecker sum. Since Q i t [x, x] equals an identitymatrix for t ∈ LT j , j ≠ i and Q jt [x, y] = 0 for x ≠ y,Q H t [x, y] ={Il j (x) ⊗ Q jt [x j , x j ] ⊗ I u j (x) if x = y0 otherwisewhere l j (x) = ∏ j−1i=1 n i(x i ) and u j (x) = ∏ Ji= j+1 n i(x i ). By collecting internal transitionsin one matrixQ jl [x j, y j ] = ∑Q jt [x j , y j ],t∈LT jwe obtain the following representation for a submatrix of Q H , where internal transitionsare kept separately in a Kronecker sum and other transitions are represented by Kronecker

HIERARCHICAL REACHABILITY GRAPH GENERATION FOR PETRI NETS 305products as before.⎧⎪⎨Q H [x, y] =⎪⎩J⊕j=1Q jl [x j , y j ] +∑t∈T \∪ J k=1 LTkJ⊗j=1∑t∈T \∪ J k=1 LTkQ jt [x j , y j ]J⊗j=1Q jt [x j , y j ]if x = yotherwise(10)Equation (10) does not distinguish between different internal transitions of the same LN.If such a distinction is necessary, transitions that have to be visible can be excluded from thesets LT j . In this way, it is possible to keep all relevant information in the representation ofRG H (PN s ). The Kronecker representation involves only matrices that describe transitionsfor LNs. As long as the reachability sets of LNs and HN are significantly smaller thanthe complete reachability set, as it is usually the case in a non-trivial decomposition, therepresentation remains compact.However, the possibility of unreachable markings in Eq. (10) remains. To understandthis, assume that Q is the adjacency matrix of RG(PN); Q H then equals( ) Q 0ABafter reordering of markings according to reachability, i.e., markings from RS(PN) ∩RS H (PN) are followed by markings from RS H (PN)\RS(PN). If RS(PN) = RS H (PN s ),then matrices A and B disappear. Since the initial marking is part of RS(PN) ∩ RS H (PN),the above representation implies that successors of reachable markings can be computed usingmatrices Q H and, consequently, that reachability analysis can be performed using thesematrices. The following section describes an RS exploration algorithm based in Eq. (10).Kronecker-based algorithms require less memory than conventional analysis algorithmsand are often faster if applied to large examples.Example 5. The running example is rather small, so we cannot expect practical gain fromrepresenting RS or RG in a compositional way as proposed in this section. However, evenfor this simple example, the representation becomes more compact, and the example allowsus to clarify the general concepts.The following table summarizes the number of markings in RS and the number of transitionsin the RG for the various nets considered here. Obviously, the HN has an RG that issignificantly smaller than that of PN.RSRG transitionsPN 254 622HN 27 51LN 1 18 30LN 2 45 114

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