Fast Strong Planning for FOND Problems with Multi-Root DAGs Andres Calderon Jaramillo - Dr. Jicheng Fu Department of Computer Science, University of Central Oklahoma We present a planner for addressing a difficult, yet under-investigated class of planning problems: Fully Observable Non-Deterministic (FOND) planning problems with strong solutions. Our strong planner implements two novel ideas. First, we employ a new data structure, MRDAG (multi-root directed acyclic graph), to define how the solution space should be expanded. We further equip a MRDAG with two heuristics to ensure planning towards the relevant search direction. Results show that our strong algorithm achieves impressive performance on a variety of benchmark problems: it runs more than three orders of magnitude faster than MBP and Gamer and demonstrates significantly better scalability. ABSTRACT In its broadest terms, artificial intelligence planning deals with the problem of designing algorithms to find a plan in order to achieve a goal under certain constraints. In this context, a domain is a structure that describes the possible actions that can be used in finding a plan. A planning problem for a given domain specifies the initial state of a system and a set of goals to achieve. A planner is an algorithm that solves a planning problem by finding a suitable set of actions in the domain to take the system from the initial state to at least one goal state. FOND problems assume that each state in a system can be fully observed and that the actions in the domain may have more than one possible outcome (non-determinism). Solutions can be classified in three categories [Cimatti et al., 2003]: weak plans, strong cyclic plans, and strong plans. See Figure 1 and Figure 2. BACKGROUND Our planner finds a strong plan if one exists. At each stage, states with a single applicable action are expanded until states with more than one applicable action are encountered. A set of actions is then selected to be applied to the latter. OUR PLANNER AC B pick-up(B, A) put-down(B) Figure 2. Example of a simple strong plan. The action pick-up(x, y) is non-deterministic as it can succeed or fail (block x may fall on the table). The action put- down(x) is deterministic. Initial State Goal … … … Figure 3. Expansion of the solution space. This graph illustrates how MRDAGs are structured and expanded. Dark green nodes are roots of a MRDAG. Light green nodes are states with exactly one applicable action. Initial State Goal MRDAG 2 MRDAG 1 MRDAG 3 MRDAG 4 Figure 1(a). A weak plan. There is at least one successful path to the goal. Figure 1(b). A strong cyclic plan. Plan may use actions that can cause cycles but will likely succeed eventually. Figure 1(c). A strong plan. Goal is achieved from any state without using actions that cause cycles. Initial State Goal Initial State Goal Initial State Goal The procedure continues until the only non-expanded states are goal states, in which case a strong plan is returned. If dead-ends are encountered, the algorithm backtracks to a previous stage. If the algorithm has to backtrack from the initial state, a strong plan can not exist. At each expansion, the planner checks that no cycle is produced. Each stage produces a multi-root directed acyclic graph (MRDAG), where the roots of the graph are the states with more than one applicable action. See Figure 3. We use two heuristics to inform our planner: Most-Constrained State (MCS): expands states with fewer applicable actions first. Least-Heuristics Distance (LHD): uses applicable actions with the least estimated distance to the goal first. OUR PLANNER (CONT’D) AC B ACB Among the planners that are capable of solving strong FOND problems, the two that are most well-known are arguably MBP [Cimatti et al., 2003] and Gamer [Kissmann and Edelkamp, 2009]. We used domains derived from the FOND track of the 2008 International Planning Competition [Bryce and Buffet, 2008]. Gamer outperformed MBP in all domains. Nevertheless, our planner could perform 2 to 4 orders of magnitude faster than Gamer. EVALUATION [Bryce and Buffet, 2008] Daniel Bryce and Olivier Buffet. International Planning Competition Uncertainty Part: Benchmarks and Results, In Proceedings of International Planning Competition, [Cimatti et al., 2003] Alessandro Cimatti, Marco Pistore, Marco Roveri, and Paolo Traverso. Weak, strong, and strong cyclic planning via symbolic model checking, Artificial Intelligence, 147(1-2):35–84, [Kissmann and Edelkamp, 2009] Peter Kissmann and Stefan Edelkamp. Solving Fully-Observable Non-Deterministic Planning Problems via Translation into a General Game, In Proceedings of the 32nd Annual German Conference on Advances in Artificial Intelligence (KI'09), pages 1–8, Berlin, Heidelberg: Springer-Verlag. REFERENCES