Download Dynamic Analysis of Petri Net-Based Discrete Systems by Andrei Karatkevich PDF

By Andrei Karatkevich

Design of contemporary electronic platforms and of complicated software program platforms is sort of continuously hooked up with parallelism. for instance, execution of an object-oriented p- gram might be regarded as parallel functioning of the co-operating items; all glossy working structures are multitasking, and the software program has a tendency to be multithread; many advanced calculation projects are solved in allotted manner. yet designers of the regulate structures most likely need to face parallelism in additional glaring and direct approach. Controllers infrequently take care of only one managed item. frequently a approach of numerous items is to be managed, after which the regulate set of rules clearly turns to be parallel. So, classical and intensely deeply investigated version of discrete machine, Finite country computer, isn't expressive sufficient for the layout of keep an eye on units and platforms. Theoretically in such a lot of instances habit of a controller could be defined through an FSM, yet frequently it isn't handy; such FSM description will be even more advanced, than a parallel specification (even as a community of numerous speaking FSMs).

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Extra info for Dynamic Analysis of Petri Net-Based Discrete Systems

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Reduced Reachability Graphs p2 p1 p3 t2 t1 t3 t4 p4 p5 p6 Fig. 5. 24. TP n } be a set of non-intersecting persistent sets at M0 . , tkn } such that tki ∈ TP i . Repeat the operation for every newly obtained marking. Every deadlock reachable from M0 in Σ will be reached by such search. Proof. Let Md be a reachable deadlock. Then exists sequence M0 σMd . As far as no transition is enabled in Md and no transition outside a persistent set can disable a transition in the persistent set, for every TP i ∈ U there is a transition ti ∈ σ, such that all previous transitions in σ are independent of it; all such transitions (belonging to different persistent sets) are mutually independent.

9. 1,3 1,3 t2 t2 t1,t2 2,4 1,2 1,2 t3 t1 t4 t3 2,4 6 5 5 t4 6 a) b) Fig. 7. Reachability graphs generated by OPT (a) and by PSS (b) for the net from Fig. 5 1,3,4,5 t1,t2,t5,t6 2,6,7,8 t4,t7 7,10,11 t2,t5,t6 1,2,7,8 t3,t7 7,9,11 Fig. 8. Reachability graphs generated by OPT for the net from Fig. 6 The essential differences between PSS and OPT are: 1. OPT generates shorter firing sequences, but PSS simulates in some cases less transition firings. 2. OPT is defined only for the state machine decomposable (SMD) nets, PSS works for any safe Petri net (probably for any ordinary Petri net).

For the first condition it is enough to prove, that transition tk ∈ TwS such that tk is enabled in M and has no common input place with any transition outside TwS , is independent of any transition ti in the marking Mi−1 in the sequence mentioned above. Suppose there are transitions in the sequence, dependent on tk , and let tm be the first such transition. Then tm has a common input place with tk (as far as both tm and tk are enabled in Mm−1 , / enabled(M ) (otherwise tm ∈ there is no other possibility), tm ∈ TwS , tm ∈ Te ), and for any input place p of tm empty at M there is a transition ti (1 ≤ i < m) such that p ∈ t•i .

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