By P. Berthelot

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**Extra info for Cohomologie Cristalline des Schemas de Caracteristique po**

**Example text**

Consequently, even if ∀i ∈ ((A+ 0 ⊗ A0 ) )i,i ≤ 0, the term + ∗ ((A− 0 ⊕ A0 ) )i,i can be positive. 2 [VAN 05]), the associated graph of + matrix (A− 0 ⊕ A0 ) can have circuits with positive weights. 8] is consistent. In conclusion, this part shows that the consistency depends on the circuits in an associated graph. This analysis will now be generalized to an arbitrary initial marking. 6] will first be rewritten on a short horizon in order to simplify the analysis. Then, this new form will be used to calculate extremal trajectories and to analyze the consistency in the following sections.

Moreover, the results obtained in step (a) for a given horizon can be reused in the calculations for a new horizon. The same remark can be made for step (b) if the initial starting point is identical. The determination of the maximal horizon of temporal consistency is the second objective. The technique is based on the analysis of convergence of matrices wk∗ : each entry can converge to a stable finite value or the infinite value +∞. For a given P-time event graph, the case of a convergence to a constant matrix after a transitory period hmax facilitates the Consistency 51 storage and reuse in the calculation of a new trajectory for any horizon.

Extremal acceptable trajectories by series of matrices Unlike the class of timed event graphs, which define a unique trajectory on assumption of earliest behavior, P-time 32 Discrete Event Systems event graphs define a set of trajectories that depend on matrices A= , A− and A+ . The aim of this section is the determination of the lowest (respectively, greatest) acceptable trajectories satisfying an initial condition given by X (0) ∈ [X0− , X0+ ]. In the following text, it is shown that the existence of a trajectory depends on special new matrices denoted wk .