meaningless. This is because they reverse the expected behaviour of causality, where cause precedes effect. In any attempt to validate a plan by constructing a trace such reversed causality would be a huge problem, since we could not determine the initial effects of applying a durative action until we had seen what conditions held over the subsequent interval and conclusion of its activity, but, equally, we could not see what the effects of activity during the interval would be without seeing the initial effects of applying the durative action.

This paradox is created by the opportunity for an action to change the past.

To handle this second case we need to modify the state after the start of the durative action to "remember" whether the start conditions were satisfied and communicate this to the end of the durative action where it can then be simply looked up in the (then) **current state** to determine whether the conditional effect should be applied. We apply a transformation to conditional effects of the form (when (and (at start ps) (at end pe)) (at end q))
into a conditional effect added to the start simple action, (when ps (Mps))**, and a conditional effect added to the end simple action,** (when (and pe (Mps)) q), where Mps **is a**
special new proposition, unique to the particular conditional effect of the particular application of the durative action being transformed. By ensuring that this proposition **is unique**
in this way, there is no possibility of any other action in the plan interfering with it, so it represents an isolated memory of the fact that ps **held in the state at which the durative**
action was started. If a conditional effect does not have at end conditions, the same transformation can be applied, simply ignoring pe **in the previous discussion. Figure 15 depicts**
the transformation of a single durative action, A**, with a conditional effect, into a collection** of level 2 actions, complete with the appropriate "memory" proposition (in this case called P
∗).

The importance of the memory introduced in this transformation is explained in Figures 16 and 17. Figure 16 shows the ambiguity that results from not remembering how a state, on the trajectory of a plan, was reached. The figure illustrates that if one is in a state (P, Q,¬R) at the point when durative action A **(as described in Figure 15) ends, it is**
impossible to determine from the state alone whether R **should be added or not. This is**
because it is possible to have reached the state (P, Q,¬R**) by at least two different paths,**
with at least one path having seen A started in a state in which P **held and at least one path**
having seen A started in a state in which P did not hold (using an action, achieve-P**, with** P
as its only effect). The state (P, Q,¬R**) does not contain any information to disambiguate**
which path was used to reach it, and hence cannot determine the correct value of R **after**
A **ends.**
The third, and final, case is where the durative action has conditional effects of **the form:**
(when (and (at start ps) (over all pi) (at end pe)) (at end q)).

Again, if the effect has no at start or at end **conditions the following transformation can be**
applied simply ignoring ps or pe as appropriate. In this case we need to construct a transformation that "remembers" not only whether ps **held in the state at which the durative**
action is first applied, but also whether pi **holds throughout the interval from the start to**
the end of the durative action. Unlike the invariants of durative actions, these conditions are not required **to hold for the plan to be valid, but only determine what effects will occur** at the end of the durative action. The idea is to use intervening monitoring actions, rather as we did for invariants in definition 18. This is achieved by adding a further **effect to the start**