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Hierarchical Reachability Graph Generation for Petri Nets

Hierarchical Reachability Graph Generation for Petri Nets


302 BUCHHOLZ AND KEMPERThe effect of transitions at the marking level is defined using Boolean matrices. As usual,we assume that multiplication of Boolean values is defined as Boolean and and summationas Boolean or. Let Q jt [˜x, ỹ] for ˜x, ỹ ∈ ˜RS( j) beann j (˜x) × n j (ỹ) matrix describingtransitions in the reachability graph of LN j due to the firing of transition t. Q jt [˜x, ỹ](x, y) =1, iff transition t is enabled in marking x ∈ RS j [˜x] and firing of t yields successor markingy ∈ RS j [ỹ]. All remaining elements in the matrices are 0. Since transitions t ∈ LT j donot modify the marking of the HN, Q jt [˜x, ỹ] = 0 for ˜x ≠ ỹ and t ∈ LT j . Furthermore,we define for t /∈ LT j ∪ ST j : Q jt [˜x, ỹ] = I n j (˜x) if ˜x = ỹ, and 0 n j (˜x),n j (ỹ) otherwise. I n isan n × n matrix with 1 at the diagonal and 0 elsewhere. 0 n,m is an n × m matrix with allelements equal to 0. The reason for this definition is that a transition t /∈ LT j ∪ ST j doesnot modify the marking of LN j and cannot be disabled by LN j. This is exactly describedby matrices I and 0.Wedefine for t ∈ T, ˜x ∈ ˜RS( j) and x ∈ RS( j, ˜x):q jt (˜x, x) =∑ ∑Q i t [˜x, ỹ](x, y).ỹ∈˜RS( j) y∈RS( j,ỹ)If q jt (˜x, x) = 1, then t is enabled by LN j in marking x. For the HN we define qtH (x) = 1ift /∈ T H or t ∈ Q H (x, y) for some y ∈ RS(HN). In all other cases qt H (x) = 0.The matrices describe the effect of transitions with respect to the HN or a single LN. Thenext step is to consider the effect of a transition with respect to the global net. Transition tis enabled in marking x ∈ RS H (PN s )iffq t (x H , x 1 ,...,x J ) = q Ht (x H )J∏j=1q j (jt xH , x j)= 1.It is straightforward to prove this enabling condition. Since q jt (·) ≡ 1 for t /∈ LT j ∪ ST jand qtH (·) ≡ 1 for t /∈ T H , enabling depends only on the marking of parts to which thetransition belongs. A transition is enabled if it is enabled in all parts simultaneously. Ina similar way we can characterize transitions between markings. Transition t is enabledin marking (x H , x 1 ,...,x J ) and its firing yields successor marking (y H , y 1 ,...,y J )ifft ∈ Q H (x H , y H ) andJ∏j=1Q jt (x j , y j ) = 1.This relation allows us to characterize the reachability graph completely. To do this in amore elegant way, we define Kronecker operations for matrices.Definition 8. The Kronecker product A ⊗ B of an n A × m A matrix A and an n B × m Bmatrix B is defined as a n A n B × m A m B matrix⎛⎞A(0, 0)B ... A(0, m A − 1)BC = ⎜. ⎝ . .. ⎟. ⎠A(n A − 1, 0)B ... A(n A − 1, m A − 1)B

HIERARCHICAL REACHABILITY GRAPH GENERATION FOR PETRI NETS 303The Kronecker sum A ⊕ B is defined for square matrices only asD = A ⊗ I n B ×n B+ I n A ×n A⊗ B.The definition of Kronecker sums/products does not include the data type of the matrixelements. Indeed, all kinds of algebraic semirings can be used. In particular, we considerhere Boolean values as needed for adjacency matrices. Since the Kronecker product isassociative, we can define a generalization for J matrices A j of dimension n j × m j .C =J⊗ J∏A j = I l j⊗ A j ⊗ I u j,j=1 j=1where l j = ∏ j−1i=1 m i and u j = ∏ Ji= j+1 n i. In the same way the Kronecker sum can bedefined for n j = m j asD =J⊗A j =j=1J∑I l j⊗ A j ⊗ I u j.j=1Observe that C is a matrix with ∏ Jj=1 n j rows and ∏ Jj=1 m j columns that is representedby n j × m j matrices. If we consider the number of non-zero elements in C in terms of thenumber of non-zero elements in A j and denote the number of non-zero elements in a matrixA as nz(A), then we obtainnz(C) =J∏nz(A j ).j=1Kronecker sums and product are a very compact way to represent huge matrices. Implicitly,Kronecker operations realize a linearization of a J-dimensional number. Row indices ofmatrix C or D are computed from the row indices of the matrices A j using the relationx =J∑J∏x jj=1 i= j+1n i ,where x is the row index in C or D, x j is the row index in A j , and n j is the number of rowsin A j . In the same way, column indices are computed from the relationy =J∑J∏y jj=1 i= j+1m i ,where y is the column index in C or D, y j is the column index in A j , and m j is the numberof columns in A j . These representations are called mixed radix number representations.Obviously, x(y) determines all x j (y j ) and vice versa. For complementary information aboutKronecker operations and mixed radix number schemes, we refer to [22] and recommendto consider an example in which with J = 3 and n j = m j = 10 for all j ∈{1, 2, 3}.

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