K. Lautenbach. Linear algebraic calculation of deadlock and traps. Concurrency and Nets -- Advances in Petri Nets, pages 315--336, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a generalization of the Commoner s Theorem for asymmetric choice nets [3] Thus liveness in an asymmetric choice Petri nets can be related to siphon control. Our approach involves the computation of a special class of minimal siphons. Methods of siphon computation have been given for instance in [12, 4, 7]. 1.3 Paper Structure The document is organized as follows. Section 2 reviews basic Petri net properties and describes the notations which are used throughout the paper. Section 3 presents some deadlock and liveness properties. We emphasize the supervisory control aspect of enforcing liveness and ....

K. Lautenbach. Linear algebraic calculation of deadlock and traps. Concurrency and Nets -- Advances in Petri Nets, pages 315--336, 1987.

.... to construct another net (with only a linear size increase of the original size) for which the previously considered multisets of circuits of live and safe free choice nets can be algebraically characterized (in fact, computed as P semiflows) This can be done in a similar way to that presented in [Lau87] for the polynomial computation of the minimal traps of a net. In the next section we formalize the concept of multiset of circuits and present a net transformation for the efficient computation of mean interfiring time on multisets. 5 Polynomial computation: multisets of circuits A multiset is ....

....i in the original net) and b s(t) 0 if t 2 b T n T . For marked graphs, a graph theoretical concept (circuit) is related with another one of algebraic nature (minimal P semiflow) Now, multisets of circuits are related with (non necessarily minimal) P semiflows, for marked graphs. Lemma 5. 2 [Lau87] Let N = hP; T ; P re; P osti be a marked graph. Then Y is a P semiflow of N iff there exists a multiset M of circuits of N such that Y (p) M(p) for all p 2 P . Now, the following result can be derived from lemma 5.2. Theorem 5.1 Let M be a multiset of circuits of a free choice net N . M is a ....

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K. Lautenbach. Linear algebraic calculation of deadlocks and traps. In K. Voss, H. Genrich, and G. Rozenberg, editors, Concurrency and Nets, pages 315--336. Springer-Verlag, Berlin, 1987.

....The improvement is based on the application of a linear programming problem to a net obtained from the original one after a transformation of linear size increasing. The transformation, that is a modification of the Lautenbach transformation for the computation of minimal traps in a net [Lau87] will not be presented here (interested readers are referred to [CC91] The application of this method to the net in figure 8.a gives exactly the mean interfiring time of t 7 , given by (30) for deterministic service times of transitions. 5.3 Some derived results Linear programming problems ....

K. Lautenbach. Linear algebraic calculation of deadlocks and traps. In K. Voss, H. Genrich, and G. Rozenberg, editors, Concurrency and Nets, pages 315--336. Springer-Verlag, Berlin, 1987.

....checks whether the net can be covered by positive P or T invariant. If not, the set of uncovered places, resp. transitions is presented. deadlock trap condition For the class of simple nets it is possible to ensure liveness by checking the deadlock trap condition. The algorithm is taken from [16] and generates the set of minimal deadlocks and checks if any minimal deadlock contains a marked trap. check state machine decomposability The algorithm is taken from [13, 14] It allows to recognise live and bounded Free Choice nets by checking the net structure. This is highly efficient ....

K. Lautenbach. Linear algebraic calculation of deadlocks and traps. In K. Voss, H.J. Genrich, and G. Rozenberg, editors, Concurrency and Nets, Advances of Petri Nets, Berlin, 1987. Springer.

....ffl H holds. 3. A deadlock (trap) H is minimal iff there is no deadlock (trap) contained in H as a proper subset. Once a deadlock lost all tokens it remains unmarked. Once a trap gained at least one token it remains marked. The support kik of all p invariants i is both a deadlock and a trap. In [10][11] for strongly connected p t nets N = S; T ; F; W ) where W (f) 1 for all f 2 F , it is shown that deadlocks (traps) can be calculated as special multisets of circuits. We will slightly generalise this method (definition 7, theorem 1) For calculating deadlocks and traps in general see ....

....divisor of all entries of C(s) is 1. Theorem 1 Let N = S; T ; F; W ) be a strongly connected p t net. 1. If C is a d system (t system) then kCk is a deadlock (trap) 2. If H S is a minimal deadlock (trap) then there is a minimal d system (t system) C such that kCk = H . Proof see [10][11] The next example shows the method presented in [10] to calculate deadlocks on the base of linear algebraic techniques. a1 a2 a b c1 c2 c3 c4 c r s 11 12 13 2 31 32 33 34 35 4 2 6 Figure 1: p t system (N 0 ; M 0 ) a1 a2 a b c1 c2 c3 c4 c 11 12 13 2 31 32 33 34 35 4 r1 r2 r3 r4 r5 ....

[Article contains additional citation context not shown here]

K. Lautenbach, "Linear algebraic calculation of deadlocks and traps," in Concurrency and Nets (Voss, Genrich, and Rozenberg, eds.), pp. 315--336, Springer--Verlag, 1987.

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Lautenbach K., (1987) "Linear Algebraic Calculation of Deadlocks and Traps," in Concurrency and Nets, Springer-Verlag 1987.

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K. Lautenbach. Linear algebraic calculation of deadlocks and traps. In: K. Voss, H.J. Genrich, and G. Rozenberg (eds.). Concurrency and Nets, Springer (1987).

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