3. To define the state transition induced by application of an action. The update rule for the logical state must be supplemented with an explanation of the consequences for the numeric part of the state.

4. To define when two actions can be applied concurrently and how their concurrent application affects the application of those actions individually.
The structure of the definitions is as follows. Definitions 1 to 15, given in Section 7, define what it means for a plan to be valid when the plan consists of only non-durative actions. Definitions 1 to 6 set up the basic terminology, the foundational structures and the framework for handling conditional effects and primitive numeric expressions. Definition 2 meets the first requirement identified above, defining states. Definition 9 meets the second requirement, defining when a goal description is satisfied in a state. Definition 11 defines a simple plan, extending the classical notion of a sequence of actions by adding time. Definitions 12 meets the fourth requirement, by defining when two actions cannot be executed concurrently. Definition 13 meets the third requirement, defining what we mean by execution of actions, including concurrent execution of actions. Definitions 14 and 15 define the execution of a plan and what it means for a plan to be valid, given the basis laid in the previous definitions.

In Section 8 the semantics is extended to give meaning to durative actions. We begin with Definition 16, which defines ground durative actions analogously to Definition 6 for simple (that is, non-durative) actions. Similarly, Definition 17 parallels the definition of a simple plan (Definition 11) and Definitions 19 and 20 parallel those for the execution and validity of simple plans (Definitions 14 and 15). Definition 18 is the critical definition for the semantics of plans with durative actions, supplying a transformation of temporal plans into simple plans, whose validity according to the semantics of purely **simple plans, can be** used to determine the validity of the original temporally structured plans.

## 7. The Semantics Of Simple Plans

The semantics we define in this section extends the essential core of Lifschitz' strips **semantics (1986) to handle temporally situated actions, possibly occurring simultaneously,**
with numeric and conditional effects.

Definition 1 Simple Planning Instance A simple planning instance **is defined to be a**

## Pair I = (**Dom, P Rob**)

where Dom = (F s, Rs, As, arity) is a 4-tuple consisting of (finite sets of ) **function symbols,** relation symbols, actions (non-durative), and a function arity **mapping all of these symbols** to their respective arities. P rob = (Os, Init, G) **is a triple consisting of the objects in the** domain, the initial state specification and the goal state specification.

The primitive numeric expressions of a planning instance, P NEs**, are the terms constructed from the function symbols of the domain applied to (an appropriate number of )**
objects drawn from Os. The dimension of the planning instance, dim**, is the number of** distinct primitive numeric expressions that can be constructed in the instance.