By Geert Stremersch (auth.)

*Supervision of Petri Nets* offers supervisory keep an eye on thought for Petri nets with a felony set because the keep watch over target. Petri nets version discrete occasion structures - dynamic platforms whose evolution is totally decided through the prevalence of discrete occasions. keep watch over legislation, which be sure that the method meets a collection of requirements within the presence of uncontrollable and unobservable occasions, are studied and built, utilizing software components comparable to computerized production and transportation structures. *Supervision of Petri Nets* introduces a brand new and mathematically sound method of the topic. latest effects are unified by way of presenting a normal mathematical language that makes broad use of order theoretical principles, and diverse new effects are defined, together with ready-to-use algorithms that build supervisory keep watch over legislation for Petri nets. *Supervision of Petri Nets* is a wonderful reference for researchers, and will even be used as a supplementary textual content for complex classes on keep watch over theory.

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E. 6. = f\f \ {o}. (i! The legal set is A = {m E N3 I m(pl) ::; 2}. In the definition of we then have that m + F6' E A if and only if A(m + F6') ::; b, where A = (1,0,0), b = 1 and F = ° -1 -1]° . ° ° 1 [ -1 1 1 -1 1 This condition is further equivalent to 6'(td + 6'(t2) - 6' (t3) - 6'(t4) ::; 1. 15), 6 E if and only if (i! 6'(td + 6'(t2) - 6'(t3) - 6'(t4) ::; 1 for all 6' E n with 6' ::; O. This condition can be simplified to 6'(td or to 6(tl) + 6(t2) + 6'(t2) ::; 1 for all 6' E n with 6' ::; 6, ::; 1.

2. 4. and the control set lJ = {{(1,0)},{(1,0),(2,0)},{(0,1)},{(0,1),(0,2)}}. In this case is lJ~l = {{(1,0)},{(0, I)}, {(O, 1), (0,2)}}, [{(I,O)}]m = {{(1,0)}} and [{(O,I)}]m = {{{1,0)},{{0,1),{0,2)}}. ::::. permissive control values are {(I, O)}, {(O, I)} and {(O, 1), (0, 2)}. Note that £~ can be an infinite set for which the property does not hold for all A and mEA. 1O) holds for all nonempty, proper subsets A of l'fl' and for all mEA. 10) is that lJ is a finite set. In that case £~ is a finite set for all A ~ l'fl' and all mEA.

To express this formally, we introduce a number of definitions. 7 introduces an equivalence relation in U. 7. Let m E ~ and (1, (2 E U. Then (1 Om n (1 "'m (2 if = Om n (2. Define [elm := {(' E U I( "'m (} as the set of control values which are "'m-equivalent to ( E U. 5 it follows that 'R 1((1,m) = 'R1((2,m) if (1 "'m (2, and thus that [elm ~ U;' for all ( E U;.. 9). The quotient set of U;' modulo "'m is denoted £;.. Next, we introduce a partial order relation in this quotient set £;.. 8. Let mEA and Then, 6 :::;m 6 if 6,6 E £;.