ABSTRACTION AND SILENT STEP

As we argued in the previous chapter, we want to be able to abstract from certain actions, to hide them. It is important to note that it does not always work out well to simply remove steps (as what happens when we apply the operator θI; see for example exercise 4.6.7.2). This is because we want a process to behave precisely as it did before abstraction, apart from the actions abstracted from. For instance, a process having deadlock before abstraction, must still do so afterwards. As an example, consider the process a+bδ which has deadlock (see exercise 2.2.12.9), but after leaving out b by renaming it into θ, we obtain a+δ = a, which is without deadlock because of axiom A6 from BPAδ.

This is why we introduce the silent step τ, which can be removed in some cases, but in other cases it cannot. Abstraction now means renaming into τ. SO, abstracting from b in the process a+bδ, we obtain a+τδ, and in this case we cannot remove τ. We conclude that τa ≠ a (since τa+b ≠ a+b). However, abstracting from b in the process abc, we obtain aτc, and since this τ does not occur in a choice context, it can be removed: ac = ac.

COLOURED TRACES

Let us allow a new symbol τ to appear as a label. As mentioned earlier, this label τ refers to a silent step: by executing τ-steps the process proceeds but observers cannot record any visible actions.