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2 a) If a transition 7) is quasi-live for an initial marking m 0 , it is quasi-live for m'o > m 0 . b) If Tj is live for m 0 , it is not necessarily live for m^ & m 0 . c) If a PN is deadlock free for ir^, it is not necessarily so for m'o £ m 0 . 2a is quite clear since all firing sequences from m 0 are also firing sequences from m'o. 8 illustrates the fact that the properties of liveness and deadlock free are not maintained by increasing the marking. For marking Properties ofPetri Nets 29 m 0 = (1, 0, 0), the reachable markings are (1, 0, 0) and (0, 1, 0); thus transitions Tl and T2 are live and T3 is never enabled.

There are two repetitive components which are {Tu T2} and {T3, T4}. None of these components contains all the transitions of the PN. However, other repetitive components can be constructed using these components as a base. , all the sequences in (TXT2 + T3Tj" except the sequence E). The sequence S3 = T{T1T{TA is an element of L3. This is a complete repetitive sequence, hence the PN is consistent. 14b shows that a repetitive sequence can contain several firings of the same transition (while still being minimal).

1 Notations and Definitions Let us first introduce a few useful notations and concepts. 1 represents a PN with its initial marking m0. 1). For simplicity's sake, we shall write the markings in the transposed form in the text. We shall use square brackets to represent a matrix and round brackets to represent the transposed form. For example: 22 Chapter 2 m 0 -(1,0,0,0,0)-[1 0 0 0 Of For marking m0, there is an enabled transition which is Ty The firing of Tl from m 0 results in the marking mx = (0, 1,0, 1, 0).

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