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

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,$\infty$[.Comment: In Proceedings GandALF 2018, arXiv:1809.0241

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

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,$\infty$[.Comment: In Proceedings GandALF 2018, arXiv:1809.0241

Taming the complexity of timeline-based planning over dense temporal domains

The problem of timeline-based planning (TP) over dense temporal domains is known to be undecidable. In this paper, we introduce two semantic variants of TP, called strong minimal and weak minimal semantics, which allow to express meaningful properties. Both semantics are based on the minimality in the time distances of the existentially-quantified time events from the universally-quantified reference event, but the weak minimal variant distinguishes minimality in the past from minimality in the future. Surprisingly, we show that, despite the (apparently) small difference in the two semantics, for the strong minimal one, the TP problem is still undecidable, while for the weak minimal one, the TP problem is just PSPACE-complete. Membership in PSPACE is determined by exploiting a strictly more expressive extension (ECA+) of the well-known robust class of Event-Clock Automata (ECA) that allows to encode the weak minimal TP problem and to reduce it to non-emptiness of Timed Automata (TA). Finally, an extension of ECA+(ECA++) is considered, proving that its non-emptiness problem is undecidable. We believe that the two extensions of ECA (ECA+ and ECA++), introduced for technical reasons, are actually valuable per sé in the field of TA

Undecidability of future timeline-based planning over dense temporal domains

The present work focuses on timeline-based planning over dense temporal domains. In automated planning, the temporal domain is commonly assumed to be discrete, the dense case being dealt with by resorting to some form of discretization. In the last years, the planning problem over dense temporal domains has been finally addressed both in the timeline-based setting and, very recently, in the action-based one. Dense timeline-based planning, in its full generality, has been shown to be undecidable. Decidability has been recovered by imposing suitable syntactic and/or semantic restrictions (the complexity of decidable fragments varies a lot, spanning from non-primitive recursive hardness to NP-completeness, passing through EXPSPACE- and PSPACE-completeness). In this paper, we proved that restricting to the future fragment is not enough to get decidability

Model Checking Timeline-based Systems over Dense Temporal Domains?

In this paper, we introduce an automaton-theoretic approach to model checking linear time properties of timeline-based systems over dense temporal domains. The system under consideration is specified by means of (a decidable fragment of) timeline structures, timelines for short, which are a formal setting proposed in the literature to model planning problems in a declarative way. Timelines provide an interval-based description of the behavior of the system, instead of a more conventional point-based one. The relevant system properties are expressed by formulas of the logic MITL (a well-known timed extension of LTL) to be checked against timelines. In the paper, we prove that the model checking problem for MITL formulas (resp., its fragment MITL(0,∞)) over timelines is EXPSPACE-complete (resp., PSPACE-complete)

Controller Synthesis for Timeline-based Games

In the timeline-based approach to planning, originally born in the space sector, the evolution over time of a set of state variables (the timelines) is governed by a set of temporal constraints. Traditional timeline-based planning systems excel at the integration of planning with execution by handling temporal uncertainty. In order to handle general nondeterminism as well, the concept of timeline-based games has been recently introduced. It has been proved that finding whether a winning strategy exists for such games is 2EXPTIME-complete. However, a concrete approach to synthesize controllers implementing such strategies is missing. This paper fills this gap, outlining an approach to controller synthesis for timeline-based games

Controller Synthesis for Timeline-based Games

In the timeline-based approach to planning, the evolution over time of a set of state variables (the timelines) is governed by a set of temporal constraints. Traditional timeline-based planning systems excel at the integration of planning with execution by handling temporal uncertainty. In order to handle general nondeterminism as well, the concept of timeline-based games has been recently introduced. It has been proved that finding whether a winning strategy exists for such games is 2EXPTIME-complete. However, a concrete approach to synthesize controllers implementing such strategies is missing. This paper fills this gap, by providing an effective and computationally optimal approach to controller synthesis for timeline-based games.Comment: arXiv admin note: substantial text overlap with arXiv:2209.1031