Handling Soft Constraints and Goals Preferences in SGPlan * Chih-Wei Hsu and Benjamin W. Wah Department of Electrical and Computer Engineering and the Coordinated Science Laboratory University of Illinois, Urbana-Champaign Urbana, IL 61801, USA {chsu,wah}@manip.crhc.uiuc.edu Ruoyun Huang and Yixin Chen Department of Computer Science and Engineering Washington University in St Louis St Louis, MO 63130, USA rh11@cec.wustl.edu chen@cse.wustl.edu Abstract In this paper, we present the partition-and-resolve strat- egy in SGPlan (hereafter called SGPLan5) for fully sup- porting all language features in PDDL3.0. Based on the architecture of SGPlan that supported PDDL2.2 (here- after called SGPLan4), SGPLan5 partitions a large plan- ning problem into subproblems, each with its own sub- goal, and resolves those inconsistent solutions using our extended saddle-point condition. Subgoal partitioning is effective for solving large planning problems because each partitioned subproblem involves a substantially smaller search space than that of the original problem. In SGPLan5, we generalize subgoal partitioning so that the goal state of a subproblem is no longer one goal fact as in SGPLan4, but can be any fact with loosely coupled constraints with other subproblems. We have further de- veloped methods for representing a planning problem in a multi-valued form and for carrying out partitioning in the transformed space. The multi-valued representation leads to more efficient heuristics for resolving trajectory and temporal constraints and goal preferences. INTRODUCTION In this paper, we present the partition-and-resolve strategy in SGPLan 5 for fully supporting all language features in PDDL3.0 (Gerevini & Long 2005). By extending the archi- tecture of SGPLan 4 (Chen, Wah, & Hsu 2006) that supports PDDL2.2 (Edelkamp & Hoffmann 2004), SGPLan 5 parti- tions a large planning problem into subproblems, each with its own subgoal, and resolves those inconsistent solutions of subgoals using our extended saddle-point condition. Inspired by real applications, Smith recently introduced the over-subscription planning problem (Smith 2004) that has a number of soft goals with different violation costs. Un- like PDDL2.2 domains whose goal state is a conjunctive list of facts, the planning task in an over-subscription planning problem entails the selection of an appropriate subset of soft goals when it is infeasible to achieve the entire set of goals. The idea has been extended in PDDL3.0 in such a way there are soft constraints over intermediate states. * Research supported by the National Science Foundation Grant IIS 03-12084. Copyright c  2006, American Association for Artificial Intelli- gence (www.aaai.org). All rights reserved. To develop a planner that fully supports the PDDL3.0 planning models and to accommodate its new features on trajectory constraints and goal preferences, we have trans- formed each planning problem into a multi-valued domain formulation and have revised our partitioning strategy in SGPLan 4 based on the new representation. We have de- veloped new search techniques, both at the global and the subproblem levels, for optimizing goal preferences and for resolving trajectory and temporal constraints. DESIGN GOALS PDDL3.0 extends the previous PDDL2.2 specifications by introducing several new features: a) simple preferences over only action preconditions or goals, b) qualitative preferences that are logical preferences over trajectory constraints, c) complex constraints that are trajectory constraints with met- ric time and possibly numeric fluents, and d) complex pref- erences that are preferences over trajectory constraints with metric time and possibly numeric fluents. We have devel- oped new components in SGPLan 5 to support these features. Given a plan π, an initial state I , a sequence of actions and possibly their schedule, we can derive the trajectory of π under the domain definition. We can compute for this trajec- tory its violated constraints, which include mutex constraints as in PDDL2.2, inconsistent state-variable assignments, and trajectory constraints introduced in PDDL3.0. The objective is to satisfy all the hard constraints or goals as well as to optimize the soft-constraint violations and the plan quality. SGPLan 5 uses a multi-valued domain formulation (MDF) based on the SAS+ formalism. MDF has been used in several planners, including Fast Downward (Helmert 2004) and the IP planner (van den Briel, Vossen, & Kambhampati 2005). Its advantage is that it allows a more compact rep- resentation of facts and their dependencies. For example, in the traditional representation of binary facts, the location of truck1 in the TPP domain is represented in binary facts, such as at(truck1, location1), ... , at(truck1, location8). These facts can be denoted more compactly in MDF by one vari- able location(truck1) that takes multiple values: location1, ... , location8. The MDF variables further allows us to derive the possible transitions among their values. For in- stance, location1 -→ location2 represents the connection between two locations. Based on the MDF formulation, we can derive causal