Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

Definition 10.4: A transition to place connectivity (TPC) matrix P is a
binary matrix whose element pij =1 if pi ∈ O (trj), otherwise pi j = 0. With n
places and m transitions in the FPN, the P matrix is of (n × m) dimension.
Since the binary product (AND-OR composition) (P o Q) represents mapping
from places to their immediately reachable places, therefore, the presence of a
'one' in the matrix M1 = (P o Q) at position (j, i) represents that pj ∈ IRS(pi).
Analogously, a 'one' at position (j, i) in matrix Mr = (P o Q) r for positive
integer r represents pj ∈ IRS r (pi), i.e., pj is reachable from pi with a degree of
reachability r.
Definition 10.5: If an element mij of matrix Mk = (P o Q) k is unity for
positive integer k, then pi and pj are called associative places with respect
to mij .
Theorem 10.1: If the diagonal elements mii of the matrix Mk = (P o Q) k are
unity, then the associative places pi for all i lie on cycles through k
transitions in each cycle.
Proof: The proof is presented in Appendix C.
Corollary 1: In a purely cyclic FPN [34], where all transitions and places lie
on a cycle, Mk = (P o Q) k = I, where k is the number of transitions (places) in
the FPN.
For identifying cycles in a FPN, the matrix Mk =(P o Q ) k for k = 1 to m
is to be computed, where m is the number of transitions in the network. Then
by theorem 10.1, the associative places corresponding to the diagonal
elements of Mk will lie on a cycle with k transitions on each cycle. However,
if more than k number of diagonal elements are unity, then places lying on a
cycle are to be identified by finding immediate reachability of places on the
cycle using M1 = (P o Q) matrix.
The algorithm for cycle-detection consists of several procedures.
Procedure Find-places-on-cycle determines the set of places Sk that lie on
cycles with k transitions. Procedure Find-IRS-places saves in Lk the
connectivity between pairs of immediately reachable set of places, lying on
cycles with k transitions. Procedure Find-connected-places-on-cycles
determines the list of places ordered according to their immediate reachability
on cycles with k transitions and saves them in Newlistk.. Procedure Puttransitions
positions appropriate transitions in the list of places in Newlistk,
so that places preceding and following a transition in the modified list
Finallistk are its input and output places on a cycle with k-transitions. The
variable 'cycles' in procedure Cycle-detection denotes the list of cycles.