Proceedings of the Sixteenth International Conference on Principles of Knowledge Representation and Reasoning (KR 2018) A Novel Automata-Theoretic Approach to Timeline-Based Planning Dario Della Monica,*†‡ Nicola Gigante,§ Angelo Montanari,§ Pietro Sala¶ * Istituto Nazionale di Alta Matematica “F. Severi” (INdAM), Italy † Università di Napoli, Italy ‡ Universidad Complutense de Madrid, Spain § Università di Udine, Italy ¶ Università di Verona, Italy Abstract possible to cope with recurring goals when the problem is interpreted over infinite timelines. Timeline-based planning is a well-established approach suc- In the timeline-based framework, the planning domain is cessfully employed in a number of application domains. A very restricted fragment, featuring only bounded temporal modelled as a set of independent, but interacting, compo- relations and token durations, is expressive enough to cap- nents. Each component is represented by a finite number of ture action-based temporal planning. As for computational state variables, whose behaviour over time (the timelines) is complexity, it has been shown to be EXPSPACE-complete governed by a set of temporal constraints (transition func- when unbounded temporal relations, but only bounded to- tions and synchronisation rules). ken durations, are allowed. In this paper, we present a novel The timeline-based approach to planning was initially in- automata-theoretic characterisation of timeline-based plan- troduced in the context of planning and scheduling of space ning where the existence of a plan is shown to be equiva- operations (Muscettola 1994). Since then, it has been suc- lent to the nonemptiness of the language recognised by a non- cessfully employed in a number of complex real-world sce- deterministic finite-state automaton that suitably encodes all the problem constraints (timelines and synchronisation rules). narios, including both long-term mission planning and on- Besides allowing us to restate known complexity results in a board spacecraft autonomy (Jónsson et al. 2000; Frank and fairly natural and compact way, such an alternative characteri- Jónsson 2003; Cesta et al. 2007; Chien et al. 2010; Barreiro sation makes it possible to finally establish the exact complex- et al. 2012). Recently, timeline-based systems have also been ity of the full version of the problem with unbounded temporal exploited in cooperative robotics applications (Umbrico et al. relations and token durations, which was still open and turns 2017). out to be EXPSPACE-complete. Moreover, the proposed tech- Compared to the well-known action-based planning nique is general enough to cope with (infinite) recurrent goals, paradigm, the timeline-based approach allows for a more which received little attention so far, despite being quite com- declarative description of the problem domain, which does mon in real-word application scenarios. not focus on what an agent has to do, but rather on what has to happen. In particular, there is no explicit separation be- Introduction tween states and actions, but, rather, the timelines represent both the current state of the system and the ongoing tasks be- In this paper, we illustrate a novel automata-theoretic ap- ing performed. This point of view makes the approach par- proach to timeline-based planning, which turns out to be ben- ticularly well suited to model the interaction of many com- eficial in a twofold perspective. On the one hand, known ponents rather than the behaviour of a single agent. complexity results can be restated in a clean and compact Despite the successful application of timeline-based plan- way by making use of nondeterministic, finite state automata ning systems, an in-depth theoretical understanding of the (NFAs); on the other hand, it allows us to fill the last com- paradigm was missing until very recently (Gigante et al. plexity gap. Moreover, an easy generalisation of the proposed 2016; 2017). solution, that replaces NFAs by Büchi automata, makes it As a first step, Gigante et al. (2016) showed that timeline- Copyright © 2018, Association for the Advancement of Artificial based planning with bounded temporal relations and to- Intelligence (www.aaai.org). All rights reserved. ken durations, and no temporal horizon, is EXPSPACE- ɫ N. Gigante and A. Montanari worked on these results mainly complete and expressive enough to capture action-based tem- while on leave at The University of Western Australia. Nicola Gi- poral planning, that is, PDDL 2 with durative actions (Fox gante was supported by the AIxIA Outgoing mobility grantΣ 2017. ¡N and Long 2003). To show this result, they first provide a This work was also partially supported by the Italian A GNCS polynomial-time reduction of action-based temporal plan- project Formal methods for verification and synthesis of discrete ning, which is known to be EXPSPACE-complete (Rintanen and hybrid systemsΣ (N. Gigante, A. Montanari, and P. Sala), a Marie Curie ¡NA -COFUND-2012 Outgoing Fellowship (D. 2007), to timeline-based planning, and then develop an EX- Della Monica), and the PRID project ENCASE - Efforts in the PSPACE procedure for the latter (a nondeterministic Turing uNderstanding of Complex interActing SystEms (N. Gigante and machine using only an exponential amount of space). A. Montanari). Later, in (Gigante et al. 2017), it has been proved 541 that EXPSPACE-completeness still holds for timeline-based Timeline-based planning planning with unbounded interval relations. The core of the This section introduces basic concepts and terminology of proof of EXPSPACE-completeness is a small model theorem timeline-based planning. The notation is mostly borrowed showing that any satisfiable timeline-based planning prob- from (Gigante et al. 2016; Della Monica et al. 2017), and was lem has a solution at most doubly exponentially long. Such originally introduced and extensively discussed in (Cialdea a result is then exploited to build a nondeterministic guess- Mayer, Orlandini, and Umbrico 2016). and-check procedure, that runs in nondeterministic exponen- tial space. The very same procedure gives a NEXPTIME Definition 1 (State variable) A state variable x is a triple bound when we restrict ourselves to exponentially sized so- .Vx;Tx;Dx/, where: lutions, that is, the problem is proved to be NEXPTIME- • Vx is the finite domain of the variable x; complete when a bound on the solution horizon is required V as part of the input. As a matter of fact, there seems to be no • Tx : Vx → 2 x is the value transition function, which easy way to accommodate such a small model argument to maps each value v Ë Vx to the set of values that x can unbounded token durations. take immediately after v; • Dx : Vx → N × .N ∪ {+∞}) is a function that maps each Here, we prove that timeline-based planning with both x=v x=v unbounded temporal relations and unbounded token dura- v Ë Vx to a pair of values 0 < dmin f dmax, which repre- tions is EXPSPACE-complete by using a completely dif- sent respectively the minimum and maximum duration of ferent automata-theoretic construction. More precisely, we an interval over which x takes value v. show that any instance of the timeline-based planning prob- Which value is taken by a state variable over a specified lem can be encoded into a suitable nondeterministic finite- time interval is stated by means of tokens. state automaton (NFA) A such that the language recognised by A, say, L.A/, is nonempty if and only if a solution plan Definition 2 (Token) A token for x is a tuple = .x; v; d/, exists. where x is a state variable, v Ë Vx, and d Ë N is the duration dx=v d dx=v The proof consists of two main steps: we first build the of the token, with min f f max. NFA A and show that it recognises exactly those finite words It is worth pointing out that, according to Definition 1, the that represent solution plans for the given problem; then, we . / duration of a token can be arbitrarily long (this is the case check the nonemptiness of L A by solving a suitable reach- when +∞ is taken as the maximum duration of the token). ability problem over A, as usual. The proof of EXPSPACE-completeness given in (Gigante et In order to respect the EXPSPACE complexity bound, we al. 2017) excludes such a possibility: it assumes the duration must control both the structural complexity of A and the of tokens to be bounded, and there is no way to generalise complexity of the reachability algorithm. To this end, we it in order to admit arbitrarily long tokens. The proof given merge the construction of the automaton and the reachability in this paper naturally handles this case, without any special check into a unique on-the-fly procedure. This technique al- treatment. lows us to finally characterise the computational complexity The time-varying behaviour of a state variable is modelled of the full problem without introducing any artificial restric- by means of a finite sequence of tokens, called a timeline. tion on the syntax of transition functions and synchronisation A for a state variable x = rules. Definition 3 (Timeline) timeline .Vx;Tx;Dx/ is a finite sequence T = ê1; § ; kë of tokens Compared to the combinatorial argument used in (Gigante for x, where vi+1 Ë Tx.vi/ for i Ë ^1; § ; k * 1`. et al. 2017), the automata-theoretic proof given here is def- initely simpler and cleaner. Moreover, it can be easily ex- Notice that the values of x in two consecutive tokens do tended to the case of infinite timelines, where synchronisa- not need to be different. tion rules can be exploited to express recurrent goals. A time interval can be associated with any token i = .x; vi; di/ in a timeline T = ê1; § ; kë by means of the Despite the existence of a number of natural application ³i*1 scenarios, to the best of our knowledge, the case of infinite functions start_time.T; i/ = j=1 dj and end_time.T; i/ = plans has not been investigated in the context of timeline- start_time.T; i/+di.