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Timeline-Based Planning: Expressiveness and Complexity

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Timeline-Based Planning: Expressiveness and Complexity University of Udine Department of Mathematics,Computer Science and Physics Ph.D. Course in Computer Science,Mathematics and Physics Dissertation TIMELINE-BASED PLANNING: EXPRESSIVENESS AND COMPLEXITY Candidate Supervisor Nicola Gigante Prof. Angelo Montanari, Ph.D. Cosupervisors Andrea Orlandini, Ph.D. Prof. Mark Reynolds, Ph.D. Cycle XXXI ǚ 2018–2019 Nicola Gigante This work is licensed under the Creative Commons 4.0 License Attribution-NonCommercial-ShareAlike R Department of Mathematics, Computer Science, and Physics University of Udine Via delle Scienze 208 Udine 33100 (UD), Italy Ǜ [email protected] Submitted to referees on 31 October 2018 Ë Approved on 16 January 2019 ß Final revision submitted on 30 January 2019 An up-to-date version of this manuscript can be found at: https://arxiv.org/abs/1902.06123 Università degli Studi di Udine Dipartimento di Scienze Matematiche,Informatiche e Fisiche Corso di Dottorato in Informatica e Scienze Matematiche e Fisiche Dissertazione TIMELINE-BASED PLANNING: EXPRESSIVENESS AND COMPLEXITY Candidato Supervisore Nicola Gigante Prof. Angelo Montanari Cosupervisori Dott. Andrea Orlandini Prof. Mark Reynolds Ciclo XXXI ǚ 2018–2019 Nicola Gigante Questo lavoro è rilasciato sotto i termini della Creative Commons 4.0 License Attribution-NonCommercial-ShareAlike R Dipartimento di Scienze Matematiche, Informatiche e Fisiche Università degli Studi di Udine Via delle Scienze 208 Udine 33100 (UD), Italia Ǜ [email protected] Sottoposta a valutazione il 31 ottobre 2018 Ë Approvata il 16 gennaio 2019 ß Revisione finale consegnata il 30 gennaio 2019 Per una versione aggiornata di questo manoscritto, visitare: https://arxiv.org/abs/1902.06123 ABSTRACT Automated planning is an area of artificial intelligence which aims at developing systems capable of autonomously reason about how to obtain a given goal, suitably interacting with their environment. Timeline-based planning is an approach originally developed in the context of space mission planning and scheduling, where problem domains are modelled as systems made of a number of independent but interacting components, whose behaviour over time, the timelines, is governed by a set of temporal constraints. This approach is different from the perspective adopted by more common planning languages, which reason in terms of which actions one or more agents can execute in order to obtain their goals. Timeline-based systems have been successfully deployed in a number of complex real-world tasks, from mission planning and control to on-board autonomy for space exploration, over the past twenty years. However, despite this practical success, a thorough theoretical understanding of the paradigm was missing. This thesis addresses this issue, by providing the first detailed account of formal and computational properties of the timeline-based approach to planning. In particular, it focuses on expressiveness and computational complexity issues. At first, we compare the expressiveness of timeline-based and action-based planning languages, showing that a particularly restricted variant of the formalism is already expressive enough to compactly capture action-based temporal planning problems. Then, we move to the characterisation of the problem in terms of computational complexity, showing that finding a solution plan for a timeline-based planning problem is EXPSPACE-complete. We also show that finding infinite solution plans is EXPSPACE-complete as well, and that finding solution plans of bounded length is NEXPTIME-complete. Then, we approach the problem of timeline-based planning with uncertainty, that is, problems that include external components whose behaviour is not under the control of the planned system. We analyse the state-of-the-art approach to these problems, based on the concept of flexible plans, identifying some key issues, and then we propose an original solution based on timeline-based games, a game-theoretic interpretation of the problem as a two-players game where the controller, in order to win, has to execute a plan guaranteeing to satisfy the problem constraints independently from the behaviour of the environment. We show that this approach is strictly more general than the current one based on flexible plans, and we show that the problem of deciding whether a winning strategy for such games exists belongs to 2EXPTIME. In the last part of the thesis, we provide a characterisation of the expressiveness of timeline-based languages in logical terms. We show that timeline-based planning problems can be expressed by Bounded TPTL with Past (TPTLb+P), a variant of the classic Timed Propositional Temporal Logic (TPTL). We introduce TPTLb+P, showing that, while TPTL with Past (TPTL+P) is known to be non-elementary, the satisfiability problem for TPTLb+P is EXPSPACE-complete. Then, we describe a tableau method for TPTL and TPTLb+P, extending a one-pass tree-shaped tableau recently introduced for Linear Temporal Logic (LTL), which is presented with a conceptually easier proof of its completeness based on a novel model-theoretic argument, and extended in order to support past modalities. SOMMARIO La pianificazione automatica (automated planning) è un’area dell’intelligenza artificiale che mira allo sviluppo di sistemi in grado di ragionare autonomamente per perseguire determinati obbiettivi, in- teragendo di conseguenza con l’ambiente circostante. Il planning basato su timeline (timeline-based planning) è un approccio originariamente sviluppato per la gestione di missioni spaziali, secondo cui i problemi vengono modellati come sistemi composti da una moltitudine di componenti indi- pendenti, ma interagenti tra loro. Il comportamento di tali componenti nel tempo, descritto dalle timeline, è governato da un insieme di vincoli temporali. Questo approccio adotta una prospettiva differente da quella dei linguaggi di pianificazione più comunemente usati, che lavorano in termini di azioni (action-based) che uno o più agenti possono eseguire per ottenere i propri obbiettivi. Negli ultimi vent’anni, i sistemi timeline-based sono stati impiegati con successo in molti scenari concreti, dal controllo di satelliti fino a missioni di esplorazione spaziale. Nonostante questo successo applicativo, manca una completa comprensione del paradigma dal punto di vista formale. Questa tesi intende riempire tale vuoto, fornendo la prima dettagliata analisi delle proprietà formali e computazionali del timeline-based planning. In particolare, ci concentriamo su questioni di espressività e complessità computazionale. Inizialmente, compariamo l’espressività dei linguaggi di modellazione timeline-based e action-based, mostrando che una particolare restrizione del formalismo è già sufficientemente espressiva da catturare compattamente il planning temporale action-based. Dopodiché, caratterizziamo il problema in termini di complessità computazionale, mostrando che trovare una soluzione per un problema di timeline-based planning è EXPSPACE- completo. Mostriamo, inoltre, che la ricerca di soluzioni infinite è anch’essa EXPSPACE-completa, e che, dato un limite alla lunghezza delle soluzioni, il problema diventa invece NEXPTIME-completo. Successivamente, approcciamo il problema del timeline-based planning con incertezza, in cui sono presenti delle componenti esterne il cui comportamento è fuori dal controllo del sistema. Analizziamo l’attuale approccio, basato sul concetto di piano flessibile, identificando alcune criticità, che approcciamo proponendo il concetto di timeline-base game. In questa interpretazione del problema basata sulla teoria dei giochi, il problema è visto come un gioco a due giocatori, in cui il controllore del sistema vince se riesce ad eseguire un piano garantendo il soddisfacimento dei vincoli, a prescindere dal comportamento dell’ambiente. Mostriamo la maggiore generalità di questo approccio rispetto all’attuale basato su piani flessibili, e dimostriamo che decidere se esiste una strategia vincente per tali giochi appartiene alla classe 2EXPTIME. Nell’ultima parte della tesi, caratterizziamo l’espressività dei linguaggi di modellazione timeline- based in termini logici. Mostriamo che i problemi di timeline-based planning possono essere espressi dalla Bounded TPTL with Past (TPTLb+P), una variante della classica logica Timed Propo- sitional Temporal Logic (TPTL). Introduciamo TPTLb+P, mostrando che, mentre la soddisfacibilità di TPTL with Past (TPTL+P) è nota essere non elementare, il problema per TPTLb+P è EXPSPACE- completo. Dopodiché, descriviamo un metodo a tableau per TPTL e TPTLb+P, estendendo un tableau ad albero recentemente introdotto per la Linear Temporal Logic (LTL), che viene presentato con una diversa e concettualmente più semplice dimostrazione di completezza, ed esteso per supportare operatori temporali al passato. ACKNOWLEDGEMENTS This thesis is the result of three years of hard work and it would not have been possible at all without the precious help of many people. My gratitude goes first of all to my parents, and my mother in particular, for their endless understanding and support, and of course to my beloved Fita: her unconditional love and esteem are my most precious resources and motivation. As everything written here is the result of team-working, I want to thank all those who made it possible. First of all, I am greatly thankful to my supervisor Angelo Montanari, who has been a wise and understanding guide during all my journey. I have learnt from him quite everything I know about how to do this job, and to him also goes the merit of identifying the satisfactory
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