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and x 2 by subtracting 3 units from t i , respectively 1 unit from n i . That is arcswith weights (shift operators) (m, d) operate back with (−m, −d). Notice thatjoins in the direct graph are forks in the reverse graph. In the same way, forksbecome joins as for example at transition x 2 . Targets carried back from thedownstream transitions y, x 3 ,x 1 must be “added” using rules which are completelydual of those for adding informations as already discussed. Finally, wesee that we must operate with a dual algebra and with shift operators whichhave negative signs. Instead of that, we may decide that we are going to changethe signs of all equations so that we come back to the previous algebra, shift operatorsget back their positive signs, but a collection of targets is now recordedas {(−n i , −t i )}. This corresponds exactly to the coding (67) and to equations(69), where the original algebra M and the original operators A, B, C (not their“opposite”) are used, but where vectors are row vectors operating on the leftof operators (because we must anyway scan the graph from output to inputtransitions).11 Rationality, realizability and periodicity11.1 A fundamental theoremA timed event graph as that of Figure 1 can be represented as a finitedimensionallinear invariant system (63) which in turn can be represented by atransfer matrix (64).Remark 10 If there are direct arcs from input transitions to output transitions,we get an output equation of the form y = Cx ⊕ Du and a transfer matrixCA ∗ B ⊕ D. However, by redefining the “state” vector as ¯x =(x T ,u T ) T ,itis possible to come back to the form ¯CĀ∗ ¯B but at the price of increasing the“state” dimensionality. This has no importance in what follows since we are notgoing to deal with the problem of “minimal” realizations.□Observe that for a realistic (“causal”) event graph, entries of A, B, C are polynomialsof M involving only nonnegative exponents. These polynomials belongto the “dioid closure” of the “generating subset” E = {ε, e, γ, δ}. This motivatesthe following definition of rationality.Definition 11 (Rationality) A matrix or a vector with entries in M is saidrational iff its entries lie in the subdioid E ∗ ⊂ M (rational closure of E :={ε, e, γ, δ} — see Definition 8).Definition 12 (Realizability) A p × m-dimensional matrix H in M is saidrealizable iff it can be writtenH = C(γA 1 ⊕ δA 2 ) ∗ B (70)40