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  1. On State Management in Plan-Space Planning from CP Perspective Pavel Surynek Charles University, Prague

  2. Problem we are solving • a set of actions of the form • (name; preconditions; effects) example: action that moves robot from A to B (move-robot-AB; prec={at(robot, A)}; effects={at(robot, B), at(robot,A}) • a goal is a set of literals example: {at(robot_1,B); at(robot_2,C)} • find a sequence of actions satisfying a specified goal starting from a specified state ICAPS 2006 Pavel Surynek

  3. State Variables • instead of saying that something is true or false say: • a certain property takes a certain value example: location(robot)=A • change actions and goals according to the above scheme: example of an action with state variables: (move-robot-AB; prec={location(robot)=A)}; effects={location(robot)=B}) example of a goal with state variables: {location(robot_1)=B; location(robot_2)≠C} ICAPS 2006 Pavel Surynek

  4. State-Space Planning • maintain current state of the planning world at every stage of the planning process example of a state (set of propositions): {location(pack_X)=robot; location(robot)=B; ...} • forward/backward search algorithms • evolve the initial/goal state using actions • systematically try alternative actions (backtracking) • domain specific heuristics extract information from current state ICAPS 2006 Pavel Surynek

  5. Plan-Space Planning • partial plan is a structure of the form: • (actions, precedence relations, causal links) • precedence relations are between actions example: action move-robot-AB must be performedbeforeload-pack-B • causal links are relations of the form: action1 →(e) action2 • action1 has an effect e which is required by action2 • algorithms refine partial plan until a valid plan can be extracted from it ICAPS 2006 Pavel Surynek

  6. State-Space x Plan-Space • State-Spaceapproach: • world states are explicitly present through the whole reasoning process • allows an easier incorporation of domain specific heuristics • Plan-Space approach: • lack of explicit world states • provides a framework for search space pruning through imposing of constrains ICAPS 2006 Pavel Surynek

  7. CSP and Plan-Space • partial plan has similar structure asCSP • can be expressed as CSP • partial plan is changed during search for solution → dynamic CSP • standard definition - sequence of CSPs, description of changes, all CSPs have to be solved • our augmented definition - only the final CSP in the sequence of dynamic CSP must be solved ICAPS 2006 Pavel Surynek

  8. Constraint Model • Constraint model for modeling partial plans (of length at most l) • Variables, constraints and evolution in time (dynamic problem) • Variables for actions • domains contain possible moments of execution example: move-robot-AB{1,2,3,...,l} • Constraints for expressing precedence relations • standard ordering constraints example: move-robot-AB<load-package-B ICAPS 2006 Pavel Surynek

  9. Incorporation of States • Special precondition and effect variables • associated with every action in the partial plan • a precondition/effect variable for every state variable function • domain of the variable is same as the range of corresponding state variable function example: location(robot)={A,B,C} (st.var.func.)move_robot_AB-location-precond{A,B,C}move_robot_AB-location-effect{A,B,C} ICAPS 2006 Pavel Surynek

  10. State Constraints • State transition constraints • precedence and effect variables must be consistent with corresponding action • Frame axiom constraints • precedence and effect variables not affected by an action must not change its value • Threat resolving constraints • if an action a is a threat for a pair of actionsc →(e) d then it must be before or after both example: (a<c) or (d<a) ICAPS 2006 Pavel Surynek

  11. Strong State Constraints • Unique action time constraints • every action must be performed at a unique moment • implemented using allDifferent constraint • stronger propagation than standard ordering constraints • State sequencing constraints • if it is known that two actions must be executed right after each other: • then effect variables of the first must take the same values as precondition variables of the second ICAPS 2006 Pavel Surynek

  12. The Utility of States • States managed within the model allowsto deduce more information • unique action time constraints and state sequencing constraints removes values from the domains of action variables • maximum plan length l is very tight • searching of the shortest plan • unique action time constraint propagates well • state sequencing constraints are activated frequently ICAPS 2006 Pavel Surynek

  13. Algorithm • Constraint model changed dynamically by the solving algorithm • Algorithm consists of four interleaved phases: • enforcing arc-consistency in the model • threat resolutionphase • treats are resolved by adding threat resolving constraints • open goal resolution phase • unsatisfied preconditions → add new actions (add variables and constraints) or causal link • labeling of variables of the model • standard method how CSP is solved (select a variable and a valueand assign) ICAPS 2006 Pavel Surynek

  14. Details of the algorithm • Arc consistency • maintained incrementally • special algorithm for maintaining arc-consistency of the allDifferent constraint (unique action time) • Threat and open goals • must be detected by the algorithm in the CSP model • the algorithm actively modifies the CSP model • Labeling (no threat, no open goals) • performed by adding of constraints (alternatives X=1; X≠1) ICAPS 2006 Pavel Surynek

  15. Preliminary Experiments ICAPS 2006 Pavel Surynek

  16. Conclusions • Plan-Space partial plans expressed using CP • a model for stronger constraint propagation is proposed • planning world states are maintained within the constraint model • allows additional types of constraints (state sequencing and unique time constraints)→ stronger propagation • The model was used to design an algorithm • state management reduces the total number of steps (preliminary experiments) ICAPS 2006 Pavel Surynek