Analysis of relative control

In this section, we will prove that the relative control implementation is precise with respect to a given relative schedule. Consider a constraint graph G that is derived from a sequencing graph G, under timing constraints, and a relative schedule 12(G), we will prove that the difference between T(w ) and T(vo) is equal to the control delay, thereby proving the relative control implementation to be precise. 

Proof The proof is analogous to the proof for Theorem 8.1.1. Specifically, we need to show that for a given input sequence, the precise transition and precise restarting criteria are satisfied. If this is true, thoi since the properties of stateless and direct-sink are dynamically determined, the control implementation is precise. 

We now show that the precise restarting criterion is satisfied. The completea signal for any anchor a E A is asserted during the final cycle of its execution. Furthermore, dsinka is asserted if its completion may result in the completion of the entire graph. With these two results in mind, let us consider the expression 

Since the precise transition and precise restarting criteria are both satisfied for a particular input sequence, and since statelessa and dsinka for all anchors are dynamically evaluated, the relative control implementation for G is precise with respect to the schedule 12(G).  

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