therefore the order in which the functions are applied is irrelevant. Therefore, the value of x
′**is well-defined in this last definition. The requirement that the activity for a happening**
must have the same number of elements as the happening itself is simply a constraint that ensures that each action name in the happening leads to a valid action that is applicable in the appropriate state. We have already seen that conditional effects induce the construction of families of grounded actions, but that at most one of each family can be applicable in a state. If none of them is applicable for a given name, then this must mean that the precondition is unsatisfied, regardless of the conditional effects. In this case, we are asserting that the attempt to apply the action has undefined interpretation.

Definition 14 Executability A simple plan, SP, for a planning instance, I**, is executable**
if it defines a happening sequence, {ti}i=0...k**, and there is a sequence of states,** {Si}i=0...k+1 such that S0 is the initial state for the planning instance and for each i = 0 **. . . k**, Si+1 is the result of executing the happening at time ti in SP.

The state Sk+1 is called the final state produced by SP **and the state sequence** {Si}i=0...k+1 is called the trace of SP**. Note that an executable plan produces a unique trace.**
Definition 15 Validity of a Simple Plan A simple plan (for a planning instance, I**) is**
valid if it is executable and produces a final state S**, such that the goal specification for** I is satisfied in S.

## 8. The Semantics Of Durative Actions

Plans with durative actions with discrete effects can be given a semantics in terms of the semantics of simple plans. Handling durative actions that have continuous effects is more complex - we discuss this further in Section 9.

Durative actions appearing in a plan must be given with an additional field indicating the duration. This is given with the syntax:
t : (action p1 . . . pn) [d]
where d **is a rational valued duration, written in floating point syntax.**
Durative actions are introduced into the framework we have defined so far by generalising Definition 1 to include durative action schemas. The definition of the grounded action must now be extended to define the form of grounded durative actions. However, this **definition**
can be given in such a way that we associate with each durative action two simple (nondurative) actions, corresponding to the end points of the durative action. These simple actions can, together, simulate almost all of the behaviour of the durative action. The **only** aspects that are not captured in this pair of simple actions are the duration of the durative action and the invariants that must hold over that duration. These two elements **can,** however, be simply handled in a minor extension to the semantics of simple plans, and this is the approach we adopt. By taking this route we avoid any difficulties in establishing the effects of interactions between durative actions - this is all handled by the semantics for the concurrent activity within a simple plan. As we will see, one difficulty in this account is the handling of durative actions with conditional effects that contain conditions and effects that are associated with different times or conditions that must hold over the entire duration of