Desel, J., Esparza, J.: Free Choice Petri Nets, vol. 40 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the corresponding logic. In defining the case for an initial marking we require that any marking, must satisfy the condition: oe enabled at such that and = In other words the Parik vector, oe , enabled at reproduces . This type of Parik vector is referred to as a T invariant [9] and a simple test to ensure that a vector is a valid T invariant is: oe such that ( 0 In order to simulate the behaviour of a net we need to apply a sequence of input vectors. These pattern independent vectors are supplied as probabilities whose task is to exercise the various paths ....

J. Desel and J. Esparza. Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science 40, Cambridge University Press, 1995.

....conditions: jp ffl j 1 implies that for all t 2 p ffl , j ffl tj = 1, or equivalently ffl t = fpg. In words, whenever two transitions share an input place, they have no other input place. Free choice nets have been extensively studied in the 70 s [9] and have regained interest recently [11, 17, 10] because they constitute a nice compromise between power of description and tractability of problems. Several semantics can be used for the resolution of conAEicts. The most common is called the race policy. Another one, called the routing policy, was introduced in [1] Let p be a place with ....

....0,3,4,5,6,7 are live Transitions 1,2,8,9 are not live. Appendix 3: The Commoner Condition In this appendix, we show that the characterization of liveness for FCNet given in the section 4.2. 3 which only uses the total ring equation also implicitly contains a proof of Commoner s theorem [9] [10]. Siphons and Traps of a FCNet The following denition is a general Petri net denition: Denition 1 A siphon is a set of places S verifying ffl S ae S ffl . A trap is a set of places T verifying T ffl ae ffl T . In [9] the following necessary and suOEcient condition of liveness for ....

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Desel J. and Esparza L. Free Choice Petri Nets, Cambridge Tracts in Theoretical Computer Science, Number 40, 1995.

....are given. Moreover, more advanced concepts such as branching bisimilarity and OLCs are presented. 2.1 Labeled Place Transition nets We de ne a variant of the classic Petri net model, namely labeled Place Transition nets. For an elaborate introduction to Petri nets, the reader is referred to [12, 18, 19]. Let L be some set of action labels. These labels correspond to methods when modeling OLCs. De nition 2.1. Labeled P T nets) 1 An L labeled Place Transition net, or simply labeled P T net, is a tuple (P; T ; F; where: 1. P is a nite set of places, 2. T is a nite set of transitions such ....

J. Desel and J. Esparza. Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science 40. Cambridge University Press, Cambridge, UK, 1995.

....marking based supervisors to monitors. Finally, section 5 concludes with directions for future research. p. 1 2 Supervisory Control of Petri Nets This section reviews the definition of ordinary Petri nets and states the supervisory control problem. For more details on Petri nets, refer to [16, 15, 1]. For more details on supervisory control refer to [4, 10, 11] 2.1 Petri nets An ordinary Petri net N is represented by the 3 tuple, S, T , F)whereS is the set of places, T is the set of transitions, F # (S T ) # (T S) is a set of input arcs (from places to transitions) and output ....

J. Desel and J. Esparza, Free Choice Petri Nets, Cambridge Tracts in Theoretical Computer Science 40, Cambridge University Press 1995.

....some techniques for avoiding this combinatorial explosion by using the concurrent nature of Petri nets. 3. 2 Analysis and control techniques for timed Petri nets For untimed Petri nets there is a vast literature with tools for verifying whether the Petri net model satis es certain properties [22, 40, 41]. Many of these techniques can be extended to timed Petri nets. Often an ecient analysis method separately analyses the properties of one component at a time. It is e.g. obvious that the number of tokens in a cyclic component like Figure 4(b) 10 is constant at all times, independently of what ....

J. Desel and J. Esparza, Free Choice Petri Nets, vol. 40 of Cambridge Tracts in Theoretical Computer Science. 1995.

....conditions: jp ffl j 1 implies that for all t 2 p ffl , j ffl tj = 1, or equivalently ffl t = fpg. In words, whenever two transitions share an input place, they have no other input place. Free choice nets have been extensively studied in the 70 s [9] and have regained interest recently [11, 17, 10] because they constitute a nice compromise between power of description and tractability of problems. Several semantics can be used for the resolution of conAEicts. The most common is called the race policy. Another one, called the routing policy, was introduced in [1] Let p be a place with ....

....liveness is given in Appendix 2 (# 6) on the FCNet given in Figure 1. 5 The Commoner Condition In this section, we show that the characterization of liveness for FCNet given in the section 4.2. 3 which only uses the total ring equation also implicitly contains a proof of Commoner s theorem [9] [10]. 5.1 Siphons and Traps of a FCNet The following denition is a general Petri net denition: RR n Sigma2839 30 F. Baccelli B. Gaujal Denition 1 A siphon is a set of places S verifying ffl S ae S ffl . A trap is a set of places T verifying T ffl ae ffl T . In [9] the following ....

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Desel J. and Esparza L. Free Choice Petri Nets, Cambridge Tracts in Theoretical Com puter Science, Number 40, 1995.

....by function tree(G) is only a necessary condition for the existence of a cyclic schedule. If we are dealing with the more general class of RSMs as defined in Def. 2.3, the predicates may restrict the set of reachable states such that no cycle exists. If the given RSM is persistent, see, e.g. [14], then the existence of a cyclic schedule is guaranteed. The vector x in (2) determines the number of transition firings but not their ordering. The large degree of freedom can be used in order to optimize further properties of the schedule such as the necessary data memory, see also [40] For ....

J. Desel and J. Esparza. Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science 40, Cambridge, 1995. 36

....the two firing vector solutions for the example PN actually provides the complete set of firing traces for the reachability marking p 1 ; p 6 oe Gamma p 1 ; p 5 . It is common to refer to the Parikh vector solutions of the difference equation with the zero right hand side as T invariants [6]. A T invariant thus corresponds to a firing sequence enabled at that reproduces . Therefore T invariants do not depend on the actual initial marking and only characterise the PN model only from the structural point of view. It should be noted that the existence of positive T invariants as ....

J. Desel and J. Esparza. Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science 40, Cambridge University Press, 1995.

....that the two firing vector solutions for the example actually provides the complete set of firing traces for the reachability marking p 1 ; p 6 oe Gamma p 1 ; p 5 . It is common to refer to the Parikh vector solutions of the difference equation with the zero right hand side as invariants [6]. A invariant thus corresponds to a firing sequence enabled at that reproduces . Therefore invariants do not depend on the actual initial marking and only characterise the model only from the structural point of view. It should be noted that the existence of positive invariants as solutions of ....

J. Desel and J. Esparza. Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science 40, Cambridge University Press, 1995.

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Desel, J., Esparza, J.: Free Choice Petri Nets, vol. 40 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1995.

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J. Esparza. Free choice Petri Nets. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1995.

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J. Desel and J. Esparza. Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science 40, Cambridge University Press, 1995.

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J. Desel and J. Esparza. Free Choice Petri Nets. Cambridge Tracts in Theoretical Computer Science 40, Cambridge University Press, 1995.

.... of global optimization is complementary to the local optimization and reduction technique that is based on applying a set of local rules for replacing a more complex fragment of a PN with a simpler one, while preserving certain semantic, behavioral, or structural properties [7] 54] 11] 41] [20]. In fact, as discussed in Section 4.5, we apply those structural reduction techniques in order to reduce the complexity of the TS representation when we derive the TS from a PN with the objective of optimizing it. It is known [3] that, for ETS, the complexity of synthesis of PNs is polynomial in ....

....be performed. 4.5.1 Encoding The dense encoding used for the markings of the PN of Fig. 13 is based on the observation that the sets of places P 1 = p 0 , p 1 , p 2 , p 3 , P 2 = p 4 , p 5 , p 6 , and P 3 = p 6 , p 7 , p 8 define three state machines, SM1, SM2, and SM3, of the PN [27] [20] with the following sets of transitions T 1 = t 1 , t 2 , t 3 , t 4 , T 2 = t 1 , t 5 , t 6 , and T 3 = t 5 , t 6 , t 7 , respectively. This information can be structurally obtained by using algebraic methods [20] State machines correspond to place invariants of the PN and preserve their ....

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# J. Desel and J. Esparza, Free-Choice Petri Nets, Cambridge Tracts in Theoretical Computer Science, vol. 40. Cambridge Univ. Press, 1995.

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