Elsevier

Theoretical Computer Science

Volume 901, 12 January 2022, Pages 87-113
Theoretical Computer Science

Complexity issues for timeline-based planning over dense time under future and minimal semantics

https://doi.org/10.1016/j.tcs.2021.12.004Get rights and content

Highlights

  • We prove that the restriction to future semantics does not recover decidability of timeline based planning over dense time.

  • We introduce two semantic variants (weak and strong minimal) of timeline based planning expressing meaningful properties.

  • The problem is undecidable for the strong minimal semantics and PSPACE-complete for the weak minimal one.

  • Membership in PSPACE is determined by introducing an original strictly more expressive extension of Event-Clock Automata.

Abstract

The problem of timeline-based planning (TP) over dense temporal domains is known to be undecidable in the general case. We first prove that the restriction to the future semantics does not suffice to recover decidability. Then, we introduce two semantic variants of TP, called strong minimal and weak minimal semantics, and show that they allow one 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 one, but the weak minimal variant distinguishes minimality in the past from minimality in the future. Surprisingly, we show that, despite the (apparently) small differences between the two semantics, the TP problem is still undecidable for the strong minimal one, while it is PSPACE-complete for the weak minimal one. 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 us 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.1

Introduction

Timeline-based planning. Timelines provide an approach to planning alternative to the classic action-based one [3], [4]. In the action-based approach of classical planning, the task of the planner is to find a sequence of actions that, applied from an initial state, allow an actor to achieve a given goal. Timeline-based planning (TP), instead, originates from the integration of planning and scheduling concepts in the context of space operations. Unlike action-based planning, timeline-based one does not explicitly distinguish among states, actions, and goals. It models the domain as a set of independent, but interacting, components, whose behavior over time (the timelines) is ruled by a set of temporal constraints, called synchronization rules. In such a framework, a solution plan is a set of timelines expressing a behavior of the system components that satisfies all the rules. Compared to classical action-based temporal planning, TP adopts a more declarative paradigm which focuses on the constraints that sequences of actions have to fulfil to reach a given goal. The declarative flavor allows knowledge engineers to focus on what has or has not to happen, instead of on what the agent has to do to achieve a goal. Moreover, the modular structure makes it possible to separately model distinct system components. Over the years, TP has been successfully applied in many complex tasks, ranging from long- to short-term mission planning to on-board autonomy [5], [6], [7], [8], [9], [10].

In TP, the planning domain is modeled as a set of independent, but interacting, components, each one modeled by a state variable. The temporal behavior of a single state variable (component) is described by a sequence of tokens (timeline), where each token specifies a value of the variable (state) and the period of time during which it takes that value. The overall temporal behavior (set of timelines) is constrained by a set of synchronization rules that specify quantitative temporal requirements between the time events (start-time and end-time) of distinct tokens. Synchronization rules have a very simple format: either trigger rules, expressing invariants and response properties (for each token in a given state, called trigger, there exist some other tokens satisfying some mutual temporal relations), or trigger-less ones, expressing goals (there exist some tokens satisfying some mutual temporal relations). Notice that the way in which requirements are specified by synchronization rules corresponds to the “freeze” mechanism in the well-known timed temporal logic TPTL [11], which uses the freeze quantifier to bind a variable to a specific temporal context (a token in the TP setting).

TP has been successfully exploited in a number of application domains, including space missions, constraint solving, and activity scheduling (see, e.g., [12], [13], [14], [15], [16], [17]). A systematic study of expressiveness and complexity of TP has been undertaken only very recently in both the discrete-time and the dense-time settings [18], [19], [20], [21], [22].

In the discrete-time case, the TP problem turns out to be EXPSPACE-complete, and expressive enough to capture action-based temporal planning (see [21], [22]).

In this paper we will consider TP over a dense temporal domain, without having recourse to any form of discretization, which is quite a common trick. A reason for assuming this different version of time domain is, basically, to increase expressiveness: in this way one can abstract from unnecessary (or even “forced”) details, often artificially added due to the necessity of discretizing time, and can suitably represent actions with duration, accomplishments and temporally extended goals. However, despite the simple format of synchronization rules, the shift to a dense-time domain dramatically increases expressiveness and complexity, depicting a scenario which resembles that of the well-known timed linear temporal logics MTL and TPTL, under a point-wise semantics, which are undecidable in the general setting [11], [23]. Known results about the TP problem over dense time are reported in Table 1. The problem in its full generality is undecidable [20], undecidability being caused by the high expressiveness of trigger rules (if only trigger-less rules are used, it is just NP-complete [24]). Decidability can be recovered by imposing suitable syntactic/semantic restrictions on the trigger rules. In particular, two significant restrictions have been considered [18], [19]: (i) the first one limits the comparison to tokens whose start times follow the start time of the trigger (future semantics of trigger rules); (ii) the second one imposes that a non-trigger token can be referenced at most once in the time constraints of a trigger rule (simple trigger rules). By imposing the above two restrictions, the TP problem becomes decidable with a non-primitive recursive complexity [19] and can be solved by reducing it to model checking of Timed Automata (TA) [25] against MTL specifications over finite timed words, the latter being a known decidable problem [26]. It is worth pointing out that both restrictions effectively contribute to decidability. Indeed, the TP problem is still undecidable when restricted to simple trigger rules [20] and when the future semantics is assumed, but rules are not constrained to be simple. The latter result is illustrated in the next section; a preliminary account of it was given in [1]. As in the case of MTL [27], in the setting of simple trigger rules, better complexity results can be obtained if, in addition, we restrict the type of intervals used to compare tokens in simple trigger rules [18], [19]. In particular, the problem is EXPSPACE-complete when only intervals with a non-null duration are considered (non-singular intervals) and PSPACE-complete for intervals which start at time 0 or are unbounded (the set of these intervals is denoted by Intv(0,)).

Paper contributions. The first contribution of the paper is the already-mentioned proof of undecidability of the TP problem under the future semantics. Its most relevant contributions are the introduction and systematic investigation of alternative semantics for the trigger rules in the dense-time setting, called minimal semantics. In the standard semantics of trigger rules, if there are many occurrences of non-trigger tokens carrying the same specified value, say v, nothing forces the choice of a specific occurrence for satisfying the given constraints. As an example, suppose that the trigger token represents a prompt for which a v-valued token is required in response. If many v-valued tokens occur in the timeline, the chosen one is not guaranteed to be the first token occurring after issuing the prompt. In a reactive context, one is usually interested in relating an issued prompt to the first response to it and not to an arbitrarily delayed one. In this paper, we define and study semantics requiring that the rule constraints are satisfied by the suitably-valued tokens occurring close to the trigger one. A similar idea is exploited by Event-Clock Automata (ECA) [28], a well-known robust subclass of Timed Automata (TA) [25]. In ECA, each symbol a of the alphabet is associated with a recorder, or past clock, recording (at the current time) the time elapsed since the last occurrence of a, and a predictor, or future clock, measuring the time required for the next occurrence of a.

In more detail, the minimal semantics of trigger rules is based on the minimality in the time distances of the start times of existentially quantified tokens from the start time of the trigger token in a trigger rule. In fact, the minimality constraint can be used to express two alternative semantics: the weak minimal semantics, which distinguishes minimality in the past, with respect to the trigger token, from minimality in the future, and the strong minimal semantics, which considers minimality over all the start times (both in the past and in the future). Surprisingly, this apparently small difference in the definitions of weak and strong minimal semantics leads to a dramatic difference in the complexity-theoretic characterization of the TP problem: while the TP problem under the strong minimal semantics is still undecidable, the TP problem under the weak minimal semantics turns out to be PSPACE-complete (which is the complexity of the emptiness problem for TA and ECA [25], [28]). PSPACE membership of the weak minimal TP problem is shown by a non-trivial exponential-time reduction to non-emptiness of TA. To handle the trigger rules under the weak minimal semantics, we exploit, as an intermediate step in the reduction, a strictly more expressive extension of ECA, called ECA+. This novel extension of ECA is obtained by allowing a larger class of atomic event-clock constraints, namely, diagonal constraints between clocks of the same polarity (past or future) and sum constraints between clocks of opposite polarity. In [29], these atomic constraints are used in event-zones to obtain symbolic forward and backward analysis semi-algorithms for ECA, which are not guaranteed to terminate. We show that, in analogy to ECA, ECA+ are closed under language Boolean operations and can be translated in exponential time into equivalent TA with an exponential number of control states, but a linear number of clocks. We also investigate an extension of ECA+, called ECA++, where the polarity requirements in the diagonal and sum constraints are relaxed, and we show that the nonemptiness problem for such a class of automata is undecidable.

To summarize, the proposed weak minimal semantics allows one to solve the TP problem in the dense-time setting with a reasonable computational complexity, without imposing any syntactic restriction to the format of synchronization rules. Moreover, it turns out to be still quite expressive and relevant for practical applications. As a by-product, two original extensions of ECA (ECA+ and ECA++) have been introduced to prove the main complexity results, which are interesting per se, as they shed new light on the landscape of event-clock and timed automata.

Outline. The paper is organized as follows. In Section 2, we recall the TP framework, we proof the undecidability of the TP problem under the future semantics and, then, we introduce the strong and weak minimal semantics. In Section 3, we prove that the TP problem under the strong minimal semantics is still undecidable. Next, in Section 4, we introduce ECA+ and ECA++and study their expressiveness and complexity. Finally, in Section 5, by exploiting the results for ECA+, we prove PSPACE-completeness of the weak minimal TP problem. Conclusions provide an assessment of the work done and outline future research themes.

Section snippets

The timeline-based planning problem

In this section, we first recall the standard TP framework, as described in [9], [21], [18], and then we introduce the strong and weak minimal semantics.

Undecidability of the strong minimal TP problem

In this section, we prove the undecidability of the strong minimal TP problem by a polynomial-time reduction from the halting problem for Minsky 2-counter machines [30]. The key feature in the reduction is the ability of expressing for a given value v, a temporal equidistance requirement w.r.t. the start point of the trigger token for the start points of the last token before the trigger with value v and the first token after the trigger with value v (the same feature exemplified in Example 5).

Some novel extensions of Event-Clock Automata (ECA)

As we shall prove in Section 5, the TP problem under the weak minimal semantics is PSPACE-complete. Such a complexity result is proved by a non-trivial exponential-time reduction to the non-emptiness of TA. To handle the trigger rules under the weak minimal semantics, we shall exploit, as an intermediate step in the reduction, a strictly more expressive extension of ECA (Event Clock Automata) [28], called ECA+, which is introduced and studied in this paper. This novel extension of ECA is

Decidability of the weak minimal TP problem

In this section, by exploiting the results of Section 4, we show that the weak minimal TP problem is decidable and PSPACE-complete. The upper bound is obtained by an exponential-time reduction to nonemptiness of Timed Automata (TA) [25]. In order to handle the trigger rules under the weak minimal semantics, we exploit as an intermediate step the class of ECA+, introduced and investigated in Section 4.

In the following, we fix a TP domain D=(SV,R). We construct a TA accepting suitable encodings

Conclusions

In this paper, we addressed the TP problem in the dense-time setting. First, we negatively answered the question of decidability of the TP problem with future semantics, which was left open in [18]. Then, we introduced and investigated two novel semantics in the dense-time domain (the weak and strong minimal semantics) aimed at overcoming the structural restrictions on rule formats introduced in [18] to recover decidability. Surprisingly, we showed that, despite the apparently small difference

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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