the proposition is true, false, undefined or inconsistently both true and false at the instant of transition. Clearly the last of these possibilities is undesirable. The solution we adopt is a combination of the pragmatic and the philosophically principled. The pragmatic **element**
is that we choose to model actions as instantaneous transitions with effects beginning at the instant of application. Thus, the actions mark the end-points of intervals of persistence of state which are closed on the left and open on the right. This ensures that the intervals nest together without inconsistency and the truth values of propositions are always defined. The same half-open-half-closed solution is adopted elsewhere. For example, Shanahan (1999) observes that a similar approach is used in the event calculus, although there the intervals are closed on the right. Although the two choices are effectively equivalent, we slightly prefer the closed-on-the-left choice since this allows the validation of a **plan to conclude** with the state at the point of execution of its final action, making the determination of the temporal span of the plan unambiguous.

From a philosophical point of view the truth value of the proposition at the instant of application of an action cannot be exploited by any other action, by virtue both of the no moving targets rule and our position, outlined above, that a valid plan cannot depend on precise synchronisation of actions. This forces actions that require a proposition as a precondition to sit at the open end of a half-open interval in which the proposition holds.

## 11.4 Planning With Time

In classical planning models, time is treated as relative. That is, the only temporal structuring in a plan, and in reasoning about a plan, is in the ordering between actions. This is most clearly emphasised by the issues that dominated planning research in the **late 1980s and**
early 1990s, when classical planning was mainly characterised by the exploration of partial plan spaces, in planners such as tweak (Chapman, 1987), snlp **(McAllester & Rosenblitt,**
1991) and ucpop **(Penberthy & Weld, 1992). Partial plans include a collection of actions**
representing the activity thus far determined to be part of a possible plan and a set of temporal constraints on those actions. The temporal constraints used in a partial plan are all of the form A < B where A and B **are time points corresponding to the application of**
actions.

Classical linear planners (Fikes & Nilsson, 1971; Russell & Norvig, 1995) rely on the simple fact that a total ordering on the points at which actions are applied can be trivially embedded into a time line. Again, the duration between actions is not considered. The role of time in planning becomes far more significant once metric time is introduced. With metric time it is possible to associate specific durations with actions, to set deadlines or windows of opportunity. The problems associated with relative time have still to be **resolved in a** metric time framework, but new problems are introduced. In particular, durations become explicit, so it is necessary to decide what the durations attach to: actions **or states. Further,** explicit temporal extents make it more important to confront the issue of concurrency in order to best exploit the measured temporal resources available to a planner.

In contrast to the simple ordering constraints required for relative time, metric time requires more powerful constraint management. Most metric time constraint **handlers are** built around the foundations laid by Dechter, Meiri and Pearl (1991). For example, IxTeT uses extensions of temporal constraint networks (Laborie & Ghallab, 1995). The language