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Theorem 56 is valid for time Petri nets used with the extended firing rule and in which any transition has at least one input place: so, the sufficient condition for the boundedness property expressed by this theorem is also a sufficient condition for the T-bounded property (remember that theorem 56 does not involve firing domains) On the other hand, theorems 57 and 58 do not express sufficient conditions for the T-bounded property With the extended firing rule, infinite class chains in which all firing domains are distinct (because their sets of variables are distinct) may generally exist When the boundedness property cannot be demonstrated using theorem 56, it is still possible to have a bound k for the number of simultaneous enabledness states of any transition, and to construct the class graph using theorem 57 or 5.

8, while verifying straight away that any transition is at most enabled k times simultaneously Termination is thereby guaranteed, and, if the enumeration does not fail, we have demonstrated the T-bounded and bounded properties It is worth noting a final specificity about the boundedness property: a time Petri net may be unbounded with the standard firing rule but bounded with the extended firing rule! This would be the case, for example, of the net shown in Figure 510 if transitions t2 and t3 were provided with static interval [2,3] rather than [0,2]: the modified net would assume 7 state classes by using the FEFF extended firing rule, but it would assume an infinity of state classes by using the standard firing rule 5616 Alternatives An alternative for multi-enabledness processing contemplated by certain authors is to consider that tokens have an age.

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principal components analysis Table 8.3.3 S matrix 7.731562 0 0 0 0 1.070404 0 0 0 0 0.716638 0 Diagonal Matrix of Singular Values

So, transition enabling instances are ordered depending on the age of the tokens they mobilize, while our interpretation associates an age directly with enabledness Naturally, such interpretations require a different representation of time Petri net states, by considering the age of each token Such an approach was considered in [CER 99], by being limited to arcs with weight 1, in [BOY 08] which defines threshold semantics , and in [KHA 96, KHA 97], by associating intervals with places rather than with transitions..

Nelder-mead downhill simplex algorithm A nonderivative, robust local optimization method developed in 1965 Neural network An algorithm modeling biological nervous systems consisting of a large number of highly interconnected processing elements called neurons that have weighted connections called synapses Niche The survival strategy of an organism (grazing, hunting, on the ground, in trees, etc) Species in different niches (eg, one eating plants, the other eating insects) may coexist side by side without competition If two species occupy the same niche, the weaker species will be made extinct Diversity of species depends different niches or geographical separation Each peak in the cost function is analogous to a niche Objective function The function to be optimized Object variables Parameters that are directly involved in assessing the relative worth of an individual Off-line performance An average of all costs up to the present generation.

(3.9)

m and b have no common factors other than 1. a 1 is a multiple of every prime that factors m. If 4 is a factor of m, then 4 is a factor of a 1.

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