This expression depends on the current value of energy for the rover ?x and on its static recharging rate ( recharge-rate ) specified in the initial state.  Our planner handles the dynamic duration of an action by computing and maintaining during sear an estimate of the value of the numerical quantities in the state where the action is applied. In Section 3.5 we will briefly describe how the version of LPG that took part in the 3rd IPC handles nmerical state variables, and numerical preconditions and effects involving them. However, in this paper we will not describe their treatment in detail, and in the next section we will assume that action durations are static.

Each action of a plan can be associated with a cost that may affect the plan quality.

Like action durations, in general these costs could be either static or dynamic, though the current version of LPG handles only static ones. LPG precomputes the action costs using the plan metric specified in the problem description using the PDDL2.1 field ":metric". 6  For instance, the plan metric used for a problem in the ZenoTravel-Numeric domain of the 3rd IPC is
(:metric minimize (+ (* 4 (total-time)) (* 5 (total-fuel-used))),
i.e., it is the sum of four times the plan makespan and five times the total amount of the fuel used by the actions in the plan. The cost of an action a is derived by evaluating how the value of the plan metric expression is changed by the effects of a . LPG computes an initial value mo for the expression by using the values specified in the initial state as values of the involved numerical variables. From m 0 LPG derives a new value m 1 by applying the effects of a that increase/decrease the value of one or more variables in the expression. The cost of a is defined as m 1 - mo. If this difference is zero, in order to prefer plans containing a lower nmber of actions, the cost of a is set to a small positive quantity. Notice also that in these evaluations of the metric expression the temporal value total-time is not considered (it is set to zero, if present), because the temporal aspect of the plan quality is already taken into account by the durations of the actions. In the previous example, the metric subexpression used to derive the action costs is (* 5 (total-fuel-used))). Thus, for instance, the cost of the ZenoTravel action (fly plane1 city0 city1) in the problem pfile1 of the 3rd IPC is 13560 because the effects of this action increase total-fuel-used by the following quantity
( * (distance city0 city1) (slow-burn plane1)) = 678 * 4 = 2712, which increases the metric value of the plan by 5 * 2712 = 13560.

## 3. Local Search In The Space Of Temporal Action Graphs

In this section we present some search techniques used in LPG . We start with a description of the general local search scheme in the space of action graphs.  Then we concentrate on temporal action graphs giving a detailed description of LPG 's heuristics and of its methods for computing and using reachability information, for maintaining the TA-graph representation during search, and for deriving good quality plans incrementally. In order to simplify the notation, instead of using a and [a] to indicate an action node and the action repre¬