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## 2.3 Executions

Informally, an execution of a CHiP is recursively defined as an instance of a decomposition and an ordering of its subplans' executions. Intuitively, when executing a plan, an agent chooses the plan's start time and how it is refined, determining at what points in time its conditions must hold, and then witnesses a finish time. The formalism helps us reason about the outcomes of different ways to execute a group of plans, describe state transitions, and define summary information.

An execution of CHiP is a tuple . and are positive, non-zero real numbers representing the start and finish times of execution , and . Thus, instantaneous actions are not explicitly represented. is a set of subplan executions representing the decomposition of plan under this execution . Specifically, if is an plan, then it contains exactly one execution from each of the subplans; if it is an plan, then it contains only one execution of one of the subplans; and it is empty if it is . In addition, for all subplan executions, , and must be consistent with the relations specified in . Also, the first subplan(s) to start must start at the same time as , , and the last subplan(s) to finish must finish at the same time as , . The possible executions of a plan is the set that includes all possible instantiations of an execution of , meaning all possible values of the tuple , obeying the rules just stated.

For the example in Section 1.1, an execution for the production manager's top-level plan would be some . might be , , 2.0, 9.0 where , and . This means that the execution of begins at time 2.0 and ends at time 9.0.

For convenience, the subexecutions of an execution , or , is defined recursively as the set of subplan executions in 's decomposition unioned with their subexecutions.

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