- Text
- Reachability,
- Nets,
- Transitions,
- Markings,
- Petri,
- Marking,
- Hierarchical,
- Analysis,
- Regions,
- Transition,
- Graph

Hierarchical Reachability Graph Generation for Petri Nets

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 abbreviation**for** 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 nettrans**for**mation 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 trans**for**mation follows the computation ofP-invariants, the ef**for**t is limited to the ef**for**t **for** computing P-invariants. Fortunately, theef**for**t is often much smaller, since only a subset of transitions is used in s.Example 2.The running example contains P-invariants (described as **for**mal 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 net**for** 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 per**for**med,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 in**for**mationthat retains a sufficient set of interesting properties. Be**for**e we describe a way to split an

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