the action. Since these cases complicate the semantics we will postpone treatment of them until the next section and begin with durative actions without conditional effects.

The mapping from durative actions to non-durative actions has the important consequence that the mutex relation implied between non-durative actions is (advantageously)
weaker than the strong mutex relation used in, for example, TGP (Smith & Weld, **1999).** Two durative actions can be applied concurrently provided that the end-points of one action do not interact either with the end-points (if simultaneous) or the invariants of **the other** action.

Definition 16 Grounded Durative Actions **Durative actions are grounded in the same**
way as simple actions (see Definition 6), by replacing their formal parameters with constants from the planning instance and expanding quantified propositions. The definition of durative actions requires that the condition be a conjunction of temporally annotated propositions. Each temporally annotated proposition is of the form (at start p), (at end p)
or (over all p), where p **is an unannotated proposition. Similarly, the effects of a durative action (without continuous or conditional effects) are a conjunction of temporally**
annotated simple effects.

The duration field of DA **defines a conjunction of propositions that can be separated into**
DCDA
start and DCDA
end , the duration conditions for the start and end of DA**, with terms being**
arithmetic expressions and ?duration**. The separation is conducted in the obvious way,**
placing at start **conditions into** DCDA
start and at end **conditions into** DCDA
end .

Each grounded durative action, DA**, with no continuous effects and no conditional effects**
defines two parameterised simple actions DAstart and DAend**, where the parameter is the**
?duration **value, and a single additional simple action** DAinv**, as follows.**
DAstart (DAend) has precondition equal to the conjunction of the set of all propositions, p**, such that** (at start p) ((at end p)) is a condition of DA**, together with**
DCDA
start (DCDA
end ), and effect equal to the conjunction of all the simple effects, e**, such that**
(at start e) ((at end e)) is an effect of DA **(respectively).**
DAinv**, is defined to be the simple action with precondition equal to the conjunction of**
all propositions, p**, such that** (over all p) is a condition of DA**. It has an empty effect.**
Every conjunct in the condition of DA **contributes to the precondition of precisely one**
of DAstart, DAend or DAinv. Every conjunct in the effect of DA **contributes to the effect of** precisely one of DAstart or DAend**. For convenience,** DAstart (DAend, DAinv**) will be used**
to refer to both the entire (respective) simple action and also to just its name.

The actions DAstart and DAend **are parameterised by** ?duration **and this parameter**
must be substituted with the correct duration value in order to arrive at the two simple actions corresponding to the start and end of a durative action.

Definition 17 Plans A plan, P, with durative actions, for a planning instance, I**, consists**
of a finite collection of timed actions which are pairs, each either of the form (t, a)**, where** t is a rational-valued time and a **is a simple action name - an action schema name together with** the constants instantiating the arguments of the schema, or of the form (**t, a**[t
′])**, where** t is a rational-valued time, a **is a durative action name and** t
′**is a non-negative rational-valued**
duration.