Studies in Action Planning Algorithms and Complexity

The action planning problem is known to be computationally hard in the general case. Propositional planning is Pspace-complete and rst-order planning is undecidable. Consequently, several methods to reduce the computational complexity of planning have been suggested in the literature. This thesis contributes to the advance and understanding of some of these methods. One proposed method is to identify restrictions on the planning problem which ensure tractability. We propose a method using a state-variable model for planning and de ne structural restrictions on the state-transition graph. We also present a planning algorithm that is correct and tractable under these restrictions as well as a map of the complexity results for planning under our new restrictions and certain previously studied restrictions. The algorithm is further extended to apply to a miniature assembly line. Another method that has been studied is state abstraction. The idea is to rst plan for the most important goals and then successively re ne the plan to also achieve the less important goals. It is known that this method can speed up planning exponentially under ideal conditions. We show that state abstraction may likewise slow down planning exponentially and even result in generating an exponentially longer solution than necessary. Reactive planning has been proposed as an alternative to classical planning. While a classical planner rst generates the whole plan and then executes it, a reactive planner generates and executes one action at a time, based on the current state. One of the approaches to reactive planning is to use universal plans. We show that polynomial-time universal plans satisfying even a very weak notion of completeness must be of exponential size in the worst case. A trade-o between classical and reactive planning is incremental planning, i.e. a planner that can output valid pre xes of the nal plan before it has nished planning. We present a correct incremental planner for a restricted class of planning problems. The plan existence problem is tractable for this class despite the fact that the plan generation problem is provably exponential. Hence, by rst testing whether an instance is solvable or not, we can avoid starting to generate pre xes of invalid plans. Acknowledgements This thesis is based on work carried out at the Department of Computer and Information Science (IDA) in Link oping since the autumn of 1993. Many people, both inside and outside IDA, have enabled and supported this work and I am very grateful to them all. However, some of them deserve being mentioned explicitly. My advisor Christer B ackstr om for three years of excellent guidance. It has been a very interesting and productive collaboration which, hopefully, will continue in the future. Erik Sandewall and Jan Ma luszy nski, my two co-advisors, for help and encouragement. My colleagues at IDA for providing a stimulating research environment; in particular, the members of Rkllab and its three daughter labs Tosca, Kplab and Taslab. Special thanks go to Magnus Andersson, Marcus Bj areland, Thomas Drakengren, Fredrik Eklund, Lars Karlsson and Simin Nadjm-Tehrani. Inger Klein in the Automatic Control Group for collaboration and interesting discussions. Lise-Lott Svensson and Lillemor Wallgren for administrative help. Leif Finmo for technical assistance. Ivan Rankin for improving my English. Naturally, remaining faults are entirely my own. Jonas Kvarnstr om who implemented some of the algorithms presented in this thesis. Karl-Johan B ackstr om for guidance and inspiration while I was writing my master's thesis. Family and friends for constant support and encouragement. The research of the author was sponsored by the Swedish Research Council for the Engineering Sciences (TFR) under grant Dnr. 93-00291. List of Papers This thesis includes the following ve papers. I. State-Variable Planning under Structural Restrictions: Algorithms and Complexity. Peter Jonsson and Christer B ackstr om. Submitted to Arti cial Intelligence. II. Tractable Planning for an Assembly Line. Inger Klein, Peter Jonsson and Christer B ackstr om. In Malik Ghallab and Alfredo Milani, editors, New Directions in AI Planning: EWSP'95|3rd European Workshop on Planning, Frontiers in AI and Applications, 1995, IOS Press. III. Planning with Abstraction Hierarchies Can Be Exponetially Less Efcient. Christer B ackstr om and Peter Jonsson. In Chris S. Mellish, editor, Proceedings of the 14th International Joint Conference on Arti cial Intelligence (IJCAI-95), 1995, Morgan Kaufmann. IV. On the Size of Reactive Plans. Peter Jonsson and Christer B ackstr om. Accepted at the 13th (US) National Conference on Arti cial Intelligence (AAAI-96), 1996. V. Incremental Planning. Peter Jonsson and Christer B ackstr om. In Malik Ghallab and Alfredo Milani, editors, New Directions in AI Planning: EWSP'95|3rd European Workshop on Planning, Frontiers in AI and Applications, 1995, IOS Press. A modi ed version of this paper has been submitted to Annals of Mathematics and Arti cial Intelligence. Paper I is a compilation of the following published papers and research reports. 1. Tractable Planning with State Variables by Exploiting Structural Restrictions. Peter Jonsson and Christer B ackstr om. In Proceedings of the 12th (US) National Conference on Arti cial Intelligence (AAAI94), 1994, AAAI Press/The MIT Press. 2. Complexity Results for State-Variable Planning under Mixed Syntactical and Structural Restrictions. Peter Jonsson and Christer B ackstr om. In Phillipe Jorrand, editor, Proceedings of the 6th International Conference on Arti cial Intelligence: Methodology, Systems, Applications (AIMSA-94), 1994, World Scienti c Publishing. 3. Tractable Planning with State Variables by Exploiting Structural Restrictions. Peter Jonsson and Christer B ackstr om. Research report LiTH-IDA-R-95-16, Department of Computer and Information Science, Link oping University, Sweden, 1995. 4. Complexity Results for State-Variable Planning under Mixed Syntactical and Structural Restrictions. Peter Jonsson and Christer B ackstr om. Research report LiTH-IDA-R-95-17, Department of Computer and Information Science, Link oping University, Sweden, 1995. 5. Complexity of State-Variable Planning under Structural Restrictions. Peter Jonsson. Licentiate thesis no 478, Department of Computer and Information Science, Link oping University, Sweden, 1995. In order to improve readability, some of the proofs in Paper I are only sketched or|in some cases|entirely omitted. The complete proofs can be found in the research reports III-V listed above. Papers II-V are also available as research reports: II. Research report. LiTH-ISY-R-1717. Department of Electrical Engineering. Link oping University, Sweden, 1995. III. Research report. LiTH-IDA-R-95-12. Department of Computer and Information Science, Link oping University, Sweden, 1995. IV. Research report. LiTH-IDA-R-96-10. Department of Computer and Information Science, Link oping University, Sweden, 1996. V. Research report. LiTH-IDA-R-95-31. Department of Computer and Information Science, Link oping University, Sweden, 1995. 1 1 Thesis Overview 1.1 About this Thesis The main topic of this thesis is the study of computational complexity in action planning1. The action planning problem has been studied by the articial intelligence community over the past thirty years. However, the intersection of complexity theory and planning has only recently become a topic of study. To our knowledge, the computational complexity of planning was rst addressed by David Chapman in the paper \Planning for Conjunctive Goals" [1987] where it was shown that rst-order planning is undecidable. Ever since, the complexity of planning has received a constantly increasing amount of interest. This thesis consists of a collection of papers which fall into two categories. Papers I, II and V are attempts to extend the class of planning problems solvable in polynomial time. Papers III and IV are critical analyses of approaches to reduce the complexity of planning proposed in the literature. All of the papers were originally written to be self-contained. Consequently some material is repeated in the various papers, such as basic de nitions. Hopefully, this will not disturb the reader too much. The papers can be read in arbitrary order but it is recommended that paper I is read before paper II and paper IV before paper V. The reader should note that the papers presented in this thesis are somewhat di erent from the original papers. Minor changes such as typographical adjustments and corrections of misspellings are left without notice whereas substantial changes are pointed out explicitly in the papers. Some familiarity with complexity theory and action planning is required to read this thesis. It does not require special mathematical knowledge; however, it presupposes an acquaintance with mathematical and logical reasoning. The complexity theory needed can be found in any standard textbook on the subject such as Garey and Johnson [1979] or Papadimitriou [1994]. Introductions to planning can be found in, for example, Tate et al. [1990], Allen [1990] and Charniak and McDermott [1985]. 1This name has been chosen in order to distinguish it from path planning [Schwartz and Sharir, 1990]. It should be noted that action planning is sometimes referred to as AI planning in the literature. Unfortunately, the term planning has been given many di erent meanings, which is why we specialize it to action planning. 21.2 Brief Summary of Papers Below, we give very brief summaries of the ve papers in this thesis. For more extensive summaries, we refer the reader to the abstracts of the papers and Section 2.4-2.7 in this introduction. Paper I Classes of tractable planning problems previously reported in the literature have almost exclusively been de ned by simple syntactical restrictions on the set of operators [Bylander, 1994;