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Hierarchical Reachability Graph Generation for Petri Nets

Hierarchical Reachability Graph Generation for Petri Nets

292 BUCHHOLZ AND KEMPERA

292 BUCHHOLZ AND KEMPERA ɛ ) is an extended net where N ɛ = PN, V ɛ =∪ p∈P v p , A ɛ = V ɛ and⎧⎫⎪⎨ 1 if x = p ⎪⎬v p (x) = −I − (p, t) + I + (p, t) if x = t⎪⎩⎪⎭0 otherwiseFor better readability, let New ={v | v = LC(v a , v b , t), v a , v b ∈ A s } be an abbreviationfor the resulting new vectors of a linear combination w.r.t. transition t, and Used ={v a , v b | v = LC(v a , v b , t), v a , v b ∈ A s } denote the vectors used to generate New for anextended net (N s , V s , A s ).(N st , V st , A st ) is an extended net if (N s , V s , A s ) is an extended net, s ∈ (T \{t}) ∗ andV st = V s ∪ NewA st = (A s \Used) ∪ NewP st = P s ∪{p |∃v p ∈ New}T st = T⎧ s⎨ IIst − s − (x, y)(x, y) = ∑if⎩ v x (z)Is −z∈P ɛ⎧⎨ II st + s + (x, y)(x, y) = ∑if⎩ v x (z)I s +z∈P ɛMst 0 (p) = AG(v p, M 0 )(z, y) if(z, y) ifx ∈ P sx ∉ P sx ∈ P sx ∉ P sNote that c a , c b ≥ 0 by construction.We additionally distinguish ordinary places Psord= P ɛ from those generated in the exten-∪ P aggsion sequence denoted by P aggs= P s \P ord , such that P s = P ordsThe definition separates available vectors A s from the total set of vectors V s in order toensure that vectors are used in at most one step of a sequence s. This restriction is madein order to focus on those linear combinations that are relevant in the following. The nettransformation basically mimics the computation of P-invariants according to [35]. Theextreme case (N s , V s , A s ), in which s ∈ T ∗ contains all transitions t ∈ T exactly once,describes an extended net, in which each P-invariant is realized by a place p ∈ P agg .Ifan aggregated place p represents a P-invariant, its marking is constant (i.e., I + (p, t) =I − (p, t) for all t ∈ T ). We can interpret the marking of an aggregated place as a macromarking that includes an abstract view of the detailed marking. Since the complete net doesnot exchange tokens with the environment, macro markings that represent P-invariants areinvariant. However, if sequence s contains only a subset of transitions, then it is possiblethat the marking of an aggregated place will represent only a macro marking for a subset ofplaces belonging to a P-invariant. In this case, the marking of the aggregated place changeswhenever tokens are added to or removed from the partial P-invariant it represents. Onlys .

HIERARCHICAL REACHABILITY GRAPH GENERATION FOR PETRI NETS 293transitions that are not hidden (i.e., that are not part of sequence s) can remove tokens fromor add tokens to places p ∈ P agg . Since the net transformation follows the computation ofP-invariants, the effort is limited to the effort for computing P-invariants. Fortunately, theeffort is often much smaller, since only a subset of transitions is used in s.Example 2.The running example contains P-invariants (described as formal sums):p1 + p2 + p3 = 2 (1)p4 + p5 = 2 (2)p6 + p7 = 2 (3)p8 + p9 + p10 = 2 (4)and a T -invariant (2 t1, 2 t2, t3, 2 t4, 2 t5, 2 t6, 2 t7). Figure 3 shows the extended netfor sequence s = t1t6t7; Psagg ={p11, p12, p13} are hatched and arcs are dotted to indicatethat they are different from N ɛ . After the operations for sequence s are performed,the transitions t1, t6, and t7 become internal; all remaining transitions describe interfacesbetween different regions. The minimal regions that are connected via t1, t6, and t7 aremerged; shaded polygons denote the new and larger regions in N s . p-vectors and correspondinglinear combinations are given in the table below.p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 t1 t2 t3 t4 t5 t6 t7v1 1 0 0 0 0 0 0 0 0 0 1 −1 0 0 0 0 0v2 0 1 0 0 0 0 0 0 0 0 0 1 −2 0 0 0 0v3 0 0 1 0 0 0 0 0 0 0 −1 0 2 0 0 0 0v4 0 0 0 1 0 0 0 0 0 0 0 −1 0 1 0 0 0v5 0 0 0 0 1 0 0 0 0 0 0 1 0 −1 0 0 0v6 0 0 0 0 0 1 0 0 0 0 0 0 −2 0 1 0 0v7 0 0 0 0 0 0 1 0 0 0 0 0 2 0 −1 0 0v8 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 −1 0v9 0 0 0 0 0 0 0 0 1 0 0 0 0 −1 −1 1 1v10 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 −1– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –v11 = v1 + v3 1 0 1 0 0 0 0 0 0 0 0 −1 2 0 0 0 0v12 = v8 + v9 0 0 0 0 0 0 0 1 1 0 0 0 0 0 −1 0 1v13 = v10 + v12 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0Note that the definition of extended nets defines arc weights for new arcs connected to newplaces as weighted arcs of the original net. This makes it possible to consider bidirectionalarcs (self-loops) appropriately, as illustrated by the arcs connected to places p12 and p13in figure 3.The problem in derivation of a hierarchy is finding an appropriate aggration of informationthat retains a sufficient set of interesting properties. Before we describe a way to split an

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