By Oded Goldreich

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30) as the state m evolves. The above reasoning can be repeated for each m' E Rl,n((m, m). With m' = m+F8 we obtain for the last component of the new extended state m' that m~+l = b- AT m' = b- AT(m + F8) = (b - AT m) - ATF8. rTF] o. 31) Define the vector B+ such that B+,T := B-,T - ATF. 27) that Bt = max(-(ATF)i,O) for i = 1, ... ,m. Hence, B+ E f;F. 4. Closed-loop system of the Petri net model of the railway example together with a maximally permissive supervisor which guarantees that the number of trains within Zone 5 is not greater than one.

On the other hand, p. is the set of transitions t for which the graph contains an arc from p to t. Analogously for ~ and t· with t E T. Note that p E "t t E p. p ¢:} pEt·. 6 the enabling condition m m(p) ~ L f-(P, t)o(t) ~ F- 0 can be written as: (p E P). 11) tEp· This is because f- (p, t) = 0 for all t E P \ p •. With a similar argument the state equation m' = m + Fo can be brought into the form m'(p) = m(p) +L tEap f+(t,p)o(t) - L f-(P, t)o(t) (p E P). tEp· Both of these alternative formulations are useful.

17) are proven by induction. Suppose that (61, ... e. 18) we find that m~_i(p) - mi-i(p) = m~(p) - mo(p) So, with m'I"T' ~ and m' = m~) for p E -r'. ml"T" we have for p E rr' that (recall that m = mo L m~_i(p) ~ mi-i(P) ~ j-(p,t)6i(t). 11 that Oi E ~mi' 9. ~mi-1' Consequently, we obtain 0 Other concurrency assumptions So far we have used the transition bag assumption. The collections of events which can occur simultaneously can contain any number of events any number of times. 2). In many applications the transition bag assumption is not realistic.