Complexity of Timeline-Based Planning over Dense Temporal Domains: Exploring the Middle Ground

Abstract: In this paper, we address complexity issues for timeline-based planning over dense temporal domains. The planning problem is modeled by means of 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 (synchronization rules). While the temporal domain is usually assumed to be discrete, here we consider the dense case. Dense timeline-based planning has been recently shown to be undecidable in the general case; decidability (NP-completeness) can be recovered by restricting to purely existential synchronization rules (trigger-less rules). In this paper, we investigate the unexplored area of intermediate cases in between these two extremes. We first show that decidability and non-primitive recursive-hardness can be proved by admitting synchronization rules with a trigger, but forcing them to suitably check constraints only in the future with respect to the trigger (future simple rules). More "tractable" results can be obtained by additionally constraining the form of intervals in future simple rules: EXPSPACE-completeness is guaranteed by avoiding singular intervals, PSPACE-completeness by admitting only intervals of the forms [0,a] and [b,$\infty$[.