# Plan-Space Search. Searching for a Solution Plan in a Graph of Partial Plans

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1 Plan-Space Search Searching for a Solution Plan in a Graph of Partial Plans Literature Malik Ghallab, Dana Nau, and Paolo Traverso. Automated Planning Theory and Practice, chapter 2 and 5. Elsevier/Morgan Kaufmann, J. Penberthy and D. S. Weld. UCPOP: A sound, complete, partial-order for ADL. In Proceeding s of the International Conference on Knowledge Representation and Reasoning, pages , Plan-Space Search 2 1

2 State-Space vs. Plan-Space Search state-space search: search through graph of nodes representing world states plan-space search: search through graph of partial plans nodes: partially specified plans arcs: plan refinement operations solutions: partial-order plans Plan-Space Search 3 Overview The Search Space of Partial Plans Plan-Space Search Algorithms Extensions of the STRIPS Representation Plan-Space Search 4 2

3 Partial Plans plan: set of actions organized into some structure partial plan: subset of the actions subset of the organizational structure temporal ordering of actions rationale: what the action achieves in the plan subset of variable bindings Plan-Space Search 5 Adding Actions partial plan contains actions initial state goal conditions set of operators with different variables reason for adding new actions to achieve unsatisfied preconditions to achieve unsatisfied goal conditions Plan-Space Search 6 3

6 Adding Variable Bindings: Example initial state attached(pile,loc in(cont,pile top(cont,pile on(cont,pallet belong(crane,loc1 empty(crane adjacent(loc1,loc2 adjacent(loc2,loc1 at(robot,loc2 occupied(loc2 unloaded(robot 1:move(r 1,l 1,m 1 preconditions effects at(r 1,l 1 at(r 1,m 1 goal occupied(m 1 adjacent(l 1,m 1 occupied(m 1 occupied(l 1 at(robot,loc2 unloaded(robot at(r 1,l 1 variable bindings: variable = r 1 m 1 robot l 1 loc1 loc2 loc2 Plan-Space Search 11 Adding Ordering Constraints partial plan contains ordering constraints binary relation specifying the temporal order between actions in the plan reasons for adding ordering constraints all actions after initial state all actions before goal causal link implies ordering constraint to avoid possible interference Plan-Space Search 12 6

8 Plan-Space Search: Initial Search State represent initial state and goal as dummy actions init: no preconditions, initial state as effects goal: goal conditions as preconditions, no effects empty plan π 0 = ({init, goal},{(init goal},{},{}: two dummy actions init and goal; one ordering constraint: init before goal; no variable bindings; and no causal links. Plan-Space Search 15 Plan-Space Search: Initial Search State Example init attached(pile,loc in(cont,pile top(cont,pile on(cont,pallet belong(crane,loc1 empty(crane adjacent(loc1,loc2 adjacent(loc2,loc1 at(robot,loc2 occupied(loc2 unloaded(robot goal at(robot,loc2 unloaded(robot Plan-Space Search 16 8

9 Plan-Space Search: Successor Function states are partial plans generate successor through plan refinement operators (one or more: adding an action to A adding an ordering constraint to adding a binding constraint to B adding a causal link to L Plan-Space Search 17 Total vs. Partial Order Let P=(Σ,s i,g be a planning problem. A plan π is a solution for P if γ(s i,π satisfies g. problem: γ(s i,π only defined for sequence of ground actions partial order corresponds to total order in which all partial order constraints are respected partial instantiation corresponds to grounding in which variables are assigned values consistent with binding constraints Plan-Space Search 18 9

10 Partial Order Solutions Let P=(Σ,s i,g be a planning problem. A plan π = (A,,B,L is a (partial order solution for P if: its ordering constraints and binding constraints B are consistent; and for every sequence a 1,,a k of all the actions in A-{init, goal} that is totally ordered and grounded and respects and B γ(s i, a 1,,a k must satisfy g. Plan-Space Search 19 Threat: Example 1:move(robot,loc1,loc2 preconditions effects at(robot,loc1 at(robot,loc2 occupied(loc2 occupied(loc2 adjacent(loc1,loc2 occupied(loc1 at(robot,loc1 at(robot,loc2 0:goal at(robot,loc2 unloaded(robot unloaded(robot 3:move(robot,loc2,loc1 preconditions effects at(robot,loc2 at(robot,loc1 occupied(loc1 occupied(loc1 adjacent(loc2,loc1 occupied(loc2 at(robot,loc2 at(robot,loc1 2:load(crane,loc1,cont,robot preconditions belong(crane,loc1 holding(crane,cont at(robot,loc1 unloaded(robot effects empty(crane loaded(robot,cont holding(crane,cont unloaded(robot Plan-Space Search 20 10

11 Threats An action a k in a partial plan π = (A,,B,L is a threat to a causal link a i [p] a j iff: a k has an effect q that is possibly inconsistent with p, i.e. q and p are unifiable; the ordering constraints (a i a k and (a k a j are consistent with ; and the binding constraints for the unification of q and p are consistent with B. Plan-Space Search 21 Flaws A flaw in a plan π = (A,,B,L is either: an unsatisfied sub-goal, i.e. a precondition of an action in A without a causal link that supports it; or a threat, i.e. an action that may interfere with a causal link. Plan-Space Search 22 11

12 Flawless Plans and Solutions Proposition: A partial plan π = (A,,B,L is a solution to the planning problem P=(Σ,s i,g if: π has no flaw; the ordering constraints are not circular; and the variable bindings B are consistent. Proof: by induction on number of actions in A base case: empty plan induction step: totally ordered plan minus first step is solution implies plan including first step is a solution: γ(s i, a 1,,a k = γ(γ(s i, a 1, a 2,,a k Plan-Space Search 23 Overview The Search Space of Partial Plans Plan-Space Search Algorithms Extensions of the STRIPS Representation Plan-Space Search 24 12

13 Plan-Space Planning as a Search Problem given: statement of a planning problem P=(O,s i,g define the search problem as follows: initial state: π 0 = ({init, goal},{(init goal},{},{} goal test for plan state p: p has no flaws path cost function for plan π: π successor function for plan state p: refinements of p that maintain and B Plan-Space Search 25 PSP Procedure: Basic Operations PSP: Plan-Space Planner main principle: refine partial π plan while maintaining and B consistent until π has no more flaws basic operations: find the flaws of π, i.e. its sub-goals and its threats select one of the flaws find ways to resolve the chosen flaw choose one of the resolvers for the flaw refine π according to the chosen resolver Plan-Space Search 26 13

14 PSP: Pseudo Code function PSP(plan allflaws plan.opengoals( + plan.threats( if allflaws.empty( then return plan flaw allflaws.selectone( allresolvers flaw.getresolvers(plan if allresolvers.empty( then return failure resolver allresolvers.chooseone( newplan plan.refine(resolver return PSP(newPlan Plan-Space Search 27 PSP: Choice Points resolver allresolvers.chooseone( non-deterministic choice flaw allflaws.selectone( deterministic selection all flaws need to be resolved before a plan becomes a solution order not important for completeness order is important for efficiency Plan-Space Search 28 14

15 Implementing plan.opengoals( finding unachieved sub-goals (incrementally: in π 0 : goal conditions when adding an action: all preconditions are unachieved sub-goals when adding a causal link: protected proposition is no longer unachieved Plan-Space Search 29 Implementing plan.threats( finding threats (incrementally: in π 0 : no threats when adding an action a new to π = (A,,B,L: for every causal link a i [p] a j L if (a new a i or (a j a new then next link else for every effect q of a new if ( σ: σ(p=σ( q then q of a new threatens a i [p] a j when adding a causal link a i [p] a j to π = (A,,B,L: for every action a old A if (a old a i or (a j =a old or (a j a old then next action else for every effect q of a old if ( σ: σ(p=σ( q then q of a old threatens a i [p] a j Plan-Space Search 30 15

17 Implementing plan.refine(resolver refines partial plan with elements in resolver by adding: an ordering constraint; one or more binding constraints; a causal link; and/or a new action. no testing required must update flaws: unachieved preconditions (see: plan.opengoals( threats (see: plan.threats( Plan-Space Search 33 Maintaining Ordering Constraints required operations: query whether (a i a j adding (a i a j possible internal representations: maintain set of predecessors/successors for each action as given maintain only direct predecessors/successors for each action maintain transitive closure of relation Plan-Space Search 34 17

18 Maintaining Variable Binding Constraints types of constraints: unary constraints: x D x equality constraints: x = y inequalities: x y note: general CSP problem is NPcomplete Plan-Space Search 35 PSP: Data Flow π 0 plan = (A,,B,L compute threats compute open goals has flaw? return plan select flaw compute resolvers has resolver? return failure choose resolver apply resolvers maintain ordering constraints maintain binding constraints Plan-Space Search 36 18

19 PSP: Sound and Complete Proposition: The PSP procedure is sound and complete: whenever π 0 can be refined into a solution plan, PSP(π 0 returns such a plan. Proof: soundness: and B are consistent at every stage of the refinement completeness: induction on the number of actions in the solution plan Plan-Space Search 37 PSP Implementation: PoP extended input: partial plan (as before agenda: set of pairs (a,p where a is an action an p is one of its preconditions search control by flaw type unachieved sub-goal (on agenda: as before threats: resolved as part of the successor generation process Plan-Space Search 38 19

20 PoP: Pseudo Code (1 function PoP(plan, agenda if agenda.empty( then return plan (a g,p g agenda.selectone( agenda agenda - (a g,p g relevant plan.getproviders(p g if relevant.empty( then return failure (a p,p p,σ relevant.chooseone( plan.l plan.l a p [p] a g plan.b plan.b σ Plan-Space Search 39 PoP: Pseudo Code (2 if a p plan.a then plan.add(a p agenda agenda + a p.preconditions newplan plan for each threat on a p [p] a g or due to a p do allresolvers threat.getresolvers(newplan if allresolvers.empty( then return failure resolver allresolvers.chooseone( newplan newplan.refine(resolver return PSP(newPlan,agenda Plan-Space Search 40 20

21 State-Space vs. Plan-Space Planning state-space planning finite search space explicit representation of intermediate states action ordering reflects control strategy causal structure only implicit search nodes relatively simple and successors easy to compute plan-space planning finite search space no intermediate states choice of actions and organization independent explicit representation of rationale search nodes are complex and successors expensive to compute Plan-Space Search 41 Using Partial-Order Plans: Main Advantages more flexible during execution using constraint managers facilitates extensions such as: temporal constraints resource constraints distributed and multi-agent planning fit naturally into the framework Plan-Space Search 42 21

22 Overview The Search Space of Partial Plans Plan-Space Search Algorithms Extensions of the STRIPS Representation Plan-Space Search 43 Existential Quantification in Goals allow existentially quantified conjunction of literals as goal: g = x 1,,x n : l 1 l m rewrite into equivalent planning problem: new goal g = {p} where p is an unused proposition symbol introduce additional operator o = (op-g(x 1,,x n,{l 1,,l m },{p} in plan-space search: no change needed Plan-Space Search 44 22

23 DWR Example: Existential Quantification in Goals goal: x,y: on(x,c1 on(y,c2 rewritten goal: p new operator: o = (op-g(x,y,{on(x,c1,on(y,c2},{p} plan-space search goal: on(x,c1 on(y,c2 Plan-Space Search 45 Typed Variables allow typed variables in operators: name(o = n(x 1 :t 1,,x k :t k where t i is the type of variable x i rewrite into equivalent planning problem: add preconditions {t 1 (x 1,,t k (x k } to o if constant c i is of type t j, add rigid relation t j (c i to the initial state remove types from operator names Plan-Space Search 46 23

24 DWR Example: Typed Variables operator: move(r:robot,l:location,m:location precond: adjacent(l,m, at(r,l, occupied(m effects: at(r,m, occupied(m, occupied(l, at(r,l rewritten operator:move(r,l,m precond: adjacent(l,m, at(r,l, occupied(m, robot(r, loaction(l, location(m effects: at(r,m, occupied(m, occupied(l, at(r,l rewritten initial state: s i {robot(r1,container(c1,container(c2, } Plan-Space Search 47 Conditional Operators conditional planning operators: o = (n,(precond 0,effects 0,,(precond n,effects n where: n = o(x 1,,x n as before, (precond 0,effects 0 are the unconditional preconditions and effects of the operator, and (precond i,effects i for i 1 are the conditional preconditions and effects of the operator. a ground instance a of o is applicable in state s if s satisfies precond 0 let I={i [0,n] s satisfies precond i (a}; then: γ(s,a=(s - (i I effects - (a ( (i I effects + (a Plan-Space Search 48 24

25 DWR Example: Conditional Operators relation at(o,l: object o is at location l conditional move operator: move(r,l,m,c precond 0 : adjacent(l,m, at(r,l, occupied(m effects 0 : at(r,m, occupied(m, occupied(l, at(r,l precond 1 : loaded(r,c effects 1 : at(c,m, at(c,l Plan-Space Search 49 Extending PoP to handle Conditional Operators modifying plan.getproviders(p g : new action with matching conditional effect add precondition of conditional effect to agenda managing conditional threats: new alternative resolver: add negated precondition of threatening conditional effect to agenda Plan-Space Search 50 25

26 Quantified Expressions allow universally quantified variables in conditional preconditions and effects: for-all x 1,,x n : (precond i,effects i a is applicable in state s if s satisfies precond 0 Let ϭ be a substitution for x 1,,x n such that ϭ(precond i (a and ϭ(effects i (a are ground. If s satisfies ϭ(precond i (a then ϭ(effects i (a are effects of the action. Plan-Space Search 51 DWR Example: Quantified Expressions extension: robots can carry multiple containers extended move operator: move(r,l,m precond 0 : adjacent(l,m, at(r,l, occupied(m effects 0 : at(r,m, occupied(m, occupied(l, at(r,l for-all x: precond 1 : loaded(r,x effects 1 : at(x,m, at(x,l Plan-Space Search 52 26

27 Disjunctive Preconditions allow alternatives (disjunctions in preconditions: precond = precond 1 precond n a is applicable in state s if s satisfies at least one of precond 1 precond n effects remain unchanged rewrite: replace operator with n disjunctive preconditions by n operators with precond i as precondition Plan-Space Search 53 DWR Example: Disjunctive Preconditions robot can move between locations if there is a road between them or the robot has all-wheel drive extended move operator: move(r,l,m precond: (road(l,m, at(r,l, occupied(m (all-wheel-drive(r, at(r,l, occupied(m effects: at(r,m, occupied(m, occupied(l, at(r,l Plan-Space Search 54 27

28 Axiomatic Inference: Static Case axioms over rigid relations: example: l 1,l 2 : adjacent(l 1,l 2 adjacent(l 2,l 1 state-specific axioms: example: c: container(c at(c,loc1 holds in s i approach: pre-compute Plan-Space Search 55 Axiomatic Inference: Dynamic Case axioms over flexible relations: example: k,x: holding(k,x empty(k approach: divide relations into primary and secondary where secondary relations do not appear in effects transform axioms into implications where primary relations must not appear in right-hand side example: primary: holding / secondary: empty k x: holding(k,x empty(k k x: holding(k,x empty(k Plan-Space Search 56 28

29 Extended Goals not part of classical planning formalisms some problems can be translated into equivalent classical problems, e.g. states to be avoided: add corresponding preconditions to operators states to be visited twice: introduce visited relation and maintain in operators constraints on solution length: introduce count relation that is increased with each step Plan-Space Search 57 Other Extensions Function Symbols infinite domains, undecidable in general Attached Procedures evaluate relations using special code rather than general inference efficiency may be necessary in real-world domains variables must usually be bound to evaluate relations semantics of such relations Plan-Space Search 58 29

30 Overview The Search Space of Partial Plans Plan-Space Search Algorithms Extensions of the STRIPS Representation Plan-Space Search 59 30

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