Planning is one of the most studied problems in computer science. In this paper, we focus on the timeline-based approach, where the domain is modeled by a set of independent, but interacting, components, each one represented by a number of state variables, whose behavior over time (timelines) is governed by a set of temporal constraints (transition functions and synchronization rules). Whereas the time domain is usually assumed to be discrete, here we address decidability and complexity issues for timeline-based planning (TP) over dense time. We first prove that dense TP is undecidable in the general case; then, we show that decidability can be recovered by restricting to synchronization rules with a suitable future semantics. More “tractable” settings can be obtained by additionally constraining the form of intervals used in rules: EXPSPACE-completeness is obtained by avoiding singular intervals, and PSPACE-completeness by admitting only intervals of the forms [0,a] and [b,+∞[. Finally, NP-completeness can be proved for dense TP with purely existential rules only.
