The Mathematics of Dispatchability Revisited

Abstract

Dispatchability is an important property for the efficient execution of temporal plans where the temporal constraints are represented as a Simple Temporal Network (STN). It has been shown that every STN may be reformulated as a dispatchable STN, and dispatchability ensures that the temporal constraints need only be satisfied locally during execution. Recently it has also been shown that Simple Temporal Networks with Uncertainty, augmented with wait edges, are Dynamically Controllable provided every projection is dispatchable. Thus, the dispatchability property has both theoretical and practical interest. One thing that hampers further work in this area is the under-developed theory. The existing definitions are expressed in terms of algorithms, and are less suitable for mathematical proofs. In this paper, we develop a new formal theory of dispatchability in terms of execution sequences. We exploit this to prove a characterization of dispatchability involving the structural properties of the STN graph. This facilitates the potential application of the theory to uncertainty reasoning.

Converting Simple Temporal Networks with Uncertainty into Minimal Equivalent Dispatchable Form

“…However, such a strategy requires exponential space and, therefore, is impractical. So, for practical applications, Morris (2014Morris ( , 2016 proposed converting DC STNUs into an equivalent dispatchable form. Hunsberger and Posenato (2024) showed that for a dispatchable STNU, a dynamic execution strategy can be generated incrementally during execution, using only O(n 2 ) space, where n is the number of timepoints.…”

“…It requires minimal computation during execution by propagating the effects of each real-time execution, X = v, only locally (i.e., to X's neighbors; that is, timepoints connected to X by a single edge). Morris (2016) subsequently provided a graphical characterization of STN dispatchability in terms of vee-paths.…”

“…Theorem 1. An STN is dispatchable if and only if whenever there is a path from any X to any Y , there is an SVP from X to Y (Morris 2016).…”

“…However, in positive instances, such algorithms only confirm the existence of a dynamic execution strategy; they do not output one. Since such strategies typically require exponential space, Morris (2016) extended the concept of dispatchability from STNs to extended STNUs (ESTNUs), as follows.…”

“…Generating wait edges is not required to determine the DC property (e.g., as seen in the algorithms of Morris (2014) and Cairo, Hunsberger, and Rizzi (2018)), but waits turn out to be necessary for enforcing the dispatchability of STNUs. Anticipating this, Morris (2016) defined extended STNUs and the dispatchability of ESTNUs, as follows. Definition 5.…”