Download PDF by Takeyuki Hida; Masuyuki Hitsuda: Gaussian processes

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T3 is live since M1[t3i. t4 is live since M2[t4i. t5 is live since M2[t5i. Since every transition of (N, M0) is live, (N, M0) is live. Besides, (N, M0) is not reversible since, for a reachable marking M1, the condition M1 [N,*i M0 does not hold. Siphons and traps were first introduced by Commoner [7]. Commoner defined a special property, called siphon-trap property or Commoner’s property, for characterizing the deadlock freeness of a PT-net. For some subclasses of Petri nets, the siphon-trap property can also be used for characterizing the liveness and reversibility of a PT-net.

13 An augmented marked graph is live and reversible if it satisfies the siphon-trap property [1]. 14 An augmented marked graph (N, M0; R) is live and reversible if every R-siphon contains a trap marked by M0 [1]. 9 Consider the augmented marked graph (N, M0; R), where R ¼ { r1, r2 }, shown in Fig. 5. 5, there are 12 minimal siphons in (N, M0; R). S1 ¼ { p1, p3, p7, p9 }, S2 ¼ { p1, p3, p7, p10 }, S3 ¼ { p1, p4, p7, p9 } and S4 ¼ { p1, p4, p7, p10 } are NR-siphons. S5 ¼ { r1, p2, p3, p6, p7, p9 }, S6 ¼ { r1, p2, p3, p6, p7, p10 }, S7 ¼ { r1, p2, p4, p6, p7, p9 }, S8 ¼ { r1, p2, p4, p6, p7, p10 }, S9 ¼ { r2, p3, p5, p7, p8, p9 }, S10 ¼ { r2, p3, p5, p7, p8, p10 }, S11 ¼ { r2, p4, p5, p7, p8, p9 } and S12 ¼ { r2, p4, p5, p7, p8, p10 } are R-siphons.

We have αV ¼ 0, where V is the incidence matrix of N, as shown below. Hence, α is a place invariant in N. 1 2 1 1 0 1 -1 1 -1 0 0 0 1 -1 1 0 0 0 0 0 -1 1 1 0 0 0 0 -1 0 1 0 0 1 0 -1 -1 = 0 0 0 0 0 Let β ¼ (1, 1, 1, 1, 1) ! 0 be a transition vector. We have Vβ ¼ 0, where V is the incidence matrix of N, as shown below. Hence, β is a transition invariant of N. 2 19 1 -1 1 0 0 0 0 0 -1 1 1 0 0 0 0 -1 0 1 0 0 1 0 -1 -1 1 1 1 1 1 = 0 0 0 0 0 0 Properties of Petri Nets The properties of a Petri net can be categorized as behavioural properties and structural properties.

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