Intro to AI Fall 2002 © L. Joskowicz 1 Introduction to Artificial Intelligence LECTURE 13: Advanced Planning Motivation: least commitment principle Partial-order planning (POP) Planning with partially instantiated operators Hierarchical decomposition Other extensions “An Introduction to Least Commitment Planning” D. Weld, Artificial Intelligence Magazine, Winter 1994, pp

Intro to AI Fall 2002 © L. Joskowicz 2 Least commitment principle Make choices only when necessary, leaving the decision for the time it is required –variable binding: most-general unifier is a least commitment strategy Prefer buy(Store,drill) to buy(store55,drill) –partial ordering: assume operators can be performed simultaneously unless there is a requirement to do otherwise if S 1 deletes precondition c and c is needed by S 2, perform S 2 before S 1

Intro to AI Fall 2002 © L. Joskowicz 3 Example: putting on shoes Start: {} Goal: {RightShoeOn, LeftShoeOn} Operators: Op(Action: RightShoeOn, Precond: RightSockOn, Effect: RightShoeOn) Op(Action: LeftShoeOn Precond: LeftSockOn, Effect: LeftShoeOn) Op(Action: RightSockOn Effect: RightSockOn) Op(Action: LeftSockOn, Effect: LeftSockOn)

Intro to AI Fall 2002 © L. Joskowicz 4 Partial vs. total order plans

Intro to AI Fall 2002 © L. Joskowicz 5 Operator representation Operator name, precondition, effect (both add and delete lists) Op(Action: action-name, Precond: conjunction of literals (positive) Effect: conjuction of literals (positive and negative) Graphically action-name p 1 p p n e 1 e e m preconditions effects

Intro to AI Fall 2002 © L. Joskowicz 6 Plan representation (1) Plan steps: a sequence of operators Step ordering constraints: indicate step precedence relations S i < S j “S i must be executed sometime before S j ” Variable binding constraints: indicate variable assignmentsX = a, Y  b, etc Causal links: record the purpose of the step S i -- c --> S j “S i achieves precondition c for S j ”

Intro to AI Fall 2002 © L. Joskowicz 7 Plan representation (2) Initially, the plan consists of two steps, Start and Finish, with null actions associated to them, with ordering Start < Finish and with the desired goal (g 1 /\ g 2 /\ … /\ g n ) as precondition Plan(Steps:{S 1 : Op(Action: Start), S 2 : Op(Action: Finish, Precond: (g 1 /\ g 2 … /\g n ))} Orderings: {S i < S j }, Bindings: {}, Links: {})

Intro to AI Fall 2002 © L. Joskowicz 8 Example of plan representation Start Finish LeftShoeOn /\ RightShoeOn Ordering: Left Sock < Left Shoe Right Sock < Right Shoe Start “all” links

Intro to AI Fall 2002 © L. Joskowicz 9 Complete plans A plan is complete iff each precondition of each step is achieved by some other step. A step achieves a precondition if the condition is one of the effects of the step and if no other step can cancel out the condition: S i achieves precondition c of S j iff 1. ( S i < S j ) /\ (c in Effects(S i )) 2. ~  S k ( ~ c in Effects(S k )) /\ (S i < S k < S j )

Intro to AI Fall 2002 © L. Joskowicz 10 Consistent plans A plan is consistent iff there are no contradictions in the ordering or binding constraints. A contradiction occurs when: 1. ( S i < S j ) and ( S j < S i ) or 2. (X = a) /\ (X = b) or (X=a) /\ (X  a)

Intro to AI Fall 2002 © L. Joskowicz 11 Solutions as plans A solution is a complete and consistent plan that achieves the desired goal. Any linearization of a partial plan is also a solution Partially ordered plans are better solutions than totally ordered plans because: –no arbitrary choice of ordering –parallel execution of branches – easier to combine plan fragments

Intro to AI Fall 2002 © L. Joskowicz 12 Partial Order Planner: Overview Regression planning: work from goal to start Start from the initial plan, add one step (operator) in each iteration Add only steps that serve to achieve a precondition that has not been achieved yet. Keep track of interactions with causal links. When a conflict occurs, resolve it by imposing an order between steps Keep track of all choice points and backtrack as necessary

Intro to AI Fall 2002 © L. Joskowicz 13 Example: shopping for groceries SM = Supermarket HWS = Hardware Store Steps: {Start: Op(Action: Start, Effect: At(Home) /\ Sells(HWS,Drill) /\ Sells(SM,Milk) /\ Sells(SM,Banana), Finish: Op(Action: Finish, Precond: At(Home) /\ Have(Drill) /\ Have(Milk) /\ Have(Banana)}

Intro to AI Fall 2002 © L. Joskowicz 14 Actions: Go and Buy Op(Action: Go(there) Precond: At(here) Effect:At(there) /\ ~ At(here)) Op(Action: Buy(x) Precond: At(store) /\ Sells(store,x) Effect:Have(x) Go(there) At(here) At(there) ~At(here) Buy(x) At(store) Sells(store(x) Have(x)

Intro to AI Fall 2002 © L. Joskowicz 15 Plan to achieve three preconditions Have(Milk)Have(Ban.) Have(Drill) At(Home) Sells(HWS,Drill) Sells(SM,Milk) Sells(SM,Ban.) Bold links are causal links Light links are ordering links

Intro to AI Fall 2002 © L. Joskowicz 16 Instantiation and causal links {s/SM} {s/HWS} Causal links can be added because there is no conflict! No ordering is necessary At(Home) Sells(HWS,Drill) Sells(SM,Milk) Sells(SM,Ban.) Have(Milk) Have(Ban.)Have(Drill)

Intro to AI Fall 2002 © L. Joskowicz 17 Next step: get to the store At(Home) Sells(HWS,Drill) Sells(SM,Milk) Sells(SM,Ban.) At(HWS) ~At(x) At(SM) ~At(x) Have(Milk)Have(Ban.) Have(Drill)

Intro to AI Fall 2002 © L. Joskowicz 18 Instantiation and causal links {x/Home} At(Home) Sells(HWS,Drill) Sells(SM,Milk) Sells(SM,Ban.) At(HWS) ~At(Home)At(SM) ~At(Home) Have(Milk)Have(Ban.)Have(Drill) Flawed plan! Causal links conflict: cannot be in two places simultaneously ! Re-ordering is necessary

Intro to AI Fall 2002 © L. Joskowicz 19 Soving causal link conflicts ccc Promotion and demotion sequentialize actions

Intro to AI Fall 2002 © L. Joskowicz 20 After threat resolution (demotion) At(Home) Sells(HWS,Drill) Sells(SM,Milk) Sells(SM,Ban.) At(HWS)At(SM) Have(Milk)Have(Ban.)At(Home)Have(Drill)

Intro to AI Fall 2002 © L. Joskowicz 21 Final Solution Have(Milk)Have(Ban.) At(SM) At(Home) Have(Drill) At(HWS) At(Home) Sells(HWS,Drill) Sells(SM,Milk) Sells(SM,Ban.)

Intro to AI Fall 2002 © L. Joskowicz 22 POP algorithm (1) function POP(initial,goal,operators) returns plan plan := Make-Minimal-Plan(initial,goal) loop do if Solution?(plan) then return plan (S-need,c) := Select-Sub-Goal(plan) Choose-Operator(plan,operators,S-need,c) Resolve-Threats(plan) end function Select-Subgoal(plan) returns (S-need,c) pick a plan step S-need from STEPS( plan) with a precondition c that has not been achieved returns (S-need,c)

Intro to AI Fall 2002 © L. Joskowicz 23 POP algorithm (2) procedure Choose-Operator(plan,operators,S-need,c) choose (a step S-add from operators) or ( STEPS (plan) that has c as an effect) if there is no such step then fail add causal link (S-add -- c --> S-need) to LINKS (plan) add ordering constraint S-add < S-need to ORDERINGS( plan) if S-add is a newly added step from operators then add S-add to STEPS( plan) add Start < S-add < Finish to ORDERINGS( plan) procedure Resolve-Threats(plan) for each S-threat that threatens a link (Si -- c --> Sj) in LINKS (plan) do choose either Promotion: add S-threat < Si to ORDERINGS (plan) Demotion: add Sj < S-threat to ORDERINGS (plan) if not Consistent(plan) then fail

Intro to AI Fall 2002 © L. Joskowicz 24 POP is sound and complete POP constructs a proof that each precondition of the goal step is achieved: –Choose-Operator selects an action to get subgoal –Resolve-Threats sequentializes to ensure no interference between operations POP is sound and complete: every plan it returns is a solution, and if there is a solution, it will be found (assuming complete search -- BFS or iterative deepening search) It is also sound and complete with partially instantiated operators (see next slides)

Intro to AI Fall 2002 © L. Joskowicz 25 Resolving conflicts with partially instantiated operators: is At(x) a threat to ~ At(Home)? It is a possible threat, which can be dealt with by 1. resolve now with an equality constraint add binding x = HWS 2. resolve now with an inequality constraint add the clause x  Home 3. resolve later: do nothing. It is not a threat until x becomes instantiated. When it does, use promotion and demotion to resolve the conflict. Partially instantiated operators

Intro to AI Fall 2002 © L. Joskowicz 26 Extended notion of achieving A step achieves a precondition if the condition is one of the effects of the step, and if no other step can cancel out the condition for all instantiations. S i achieves precondition c of S j iff 1. ( S i < S j ) and S i has an effect c’ that necessarily unifies with c 2. ~  S k (S i < S k < S j ) in some linearization of the plan and S k has an effect c’ that possibly unifies with ~ c. SiSi SjSj c c’

Intro to AI Fall 2002 © L. Joskowicz 27 Modified Choose-Operator* procedure Choose-Operator(plan,operators,S-need,c) choose (a step S-add from operators) or ( STEPS (plan) that has c-add as an effect) such that u = Unify(c,c-add,Bindings(plan)) if there is no such step then fail add u to Bindings(plan) add causal link (S-add -- c --> S-need) to LINKS (plan) add ordering constraint S-add < S-need to ORDERINGS( plan) if S-add is a newly added step from operators then add S-add to STEPS( plan) add Start < S-add < Finish to ORDERINGS( plan) * for resolving later -- least commitment strategy

Intro to AI Fall 2002 © L. Joskowicz 28 Modified Resolve-Threats* procedure Resolve-Threats(plan) for each (Si -- c --> Sj) in LINKS (plan) do for each S-threat in STEPS (plan) do for each c’ in EFFECTS (S-threat) do if Subst(Bindings(plan),c) = Subst(Bindings(plan), ~ c’) then choose either Promotion: add S-threat < Si to ORDERINGS (plan) Demotion: add Sj < S-threat to ORDERINGS (plan) if not Consistent(plan) then fail end * for resolving later -- least commitment strategy

Intro to AI Fall 2002 © L. Joskowicz 29 Blocks world revisited b c a a c b Follow POP on blocks world examples!

Intro to AI Fall 2002 © L. Joskowicz 30 Advanced planning topics Hierarchical plans steps at different levels of resolution More complex conditions universal quantification, conditionals Dealing with time constraints incorporate time intervals an deadlines Resources and costs choose the plan that satisfies resource and cost constraints

Intro to AI Fall 2002 © L. Joskowicz 31 Hierarchical decomposition POP does not distinguish between different levels of abstraction of operators: go(home,airport) vs. go(bed,living_room) Typical plans usually have many steps! Figure out first how to get to the airport, then find out how to exit the house! Operators should describe actions at different levels of abstraction, so “big” goals get solved first

Intro to AI Fall 2002 © L. Joskowicz 32 Example

Intro to AI Fall 2002 © L. Joskowicz 33 Abstract operators Decompose operators into a group of more detailed operators that form a plan to implement it. The decomposition ends with primitive operators which are not decomposed Build(House) is decomposed into Build(Foundation), Build(Floor), Build(Walls), Build(Roof), …..

Intro to AI Fall 2002 © L. Joskowicz 34 Decomposition methods (1) Specify that a nonprimitive operator that unifies with it can be decomposed into a plan Decompose(operation,p) is a new structure akin to a subroutine or a macro for operators: Decompose(Construction, Plan(Steps:{S 1 : Build(Foundation), S 2 : Build(Frame), S 3 : Build(Roof), S 4 : Build(Walls), S 5 : Build(Interior)} Orderings:{S 1 <S 2 ; S 2 <S 3 ; S 2 <S 4 ; S 3 <S 5 ; S 4 <S 5,..... }, Bindings: {}, Foundation Frame Frame Roof Walls Links: {S 1 --->S 2, S >S 3, S 2 --->S 4, S 3 --->S 5, S 4 --->S 5 }))

Intro to AI Fall 2002 © L. Joskowicz 35 Decomposition methods (2) Plan p correctly implements an operator o if it is a complete and consistent plan for the problem of achieving the effects of o given the preconditions of o: –p must be consistent (no ordering or assignment contradiction). –every effect of o must be asserted by at least one step of p and not denied by a later step. –every precondition of steps in p must be achieved by a step in p or be one of the preconditions of o.

Intro to AI Fall 2002 © L. Joskowicz 36 Hierarchical POP algorithm function HD-POP(plan,operators,methods) returns plan loop do if Solution?(plan) then return plan (S-need,c) := Select-Sub-Goal(plan) Choose-Operator(plan,operators,S-need,c) S-nonprimitive := Select-Nonprimitive(plan) Choose-Decomposition(plan,methods,S-nonprimitive) Resolve-Threats(plan) end

Intro to AI Fall 2002 © L. Joskowicz 37 HD-POP subroutines Solution? must check that every step of the plan is primitive Select-Nonprimitive arbitrarily selects a non- primitive step of the plan (no backtracking) Choose-Decomposition: when a method is chosen: 1. Steps: add all method steps, remove S-nonprimitive 2. Bindings: add all bindings of method 3. Orderings: place new constraints latest or earliest 4. Links: explicitly add all links Fail if 1 or 2 introduce a contradiction!

Intro to AI Fall 2002 © L. Joskowicz 38 Detailed decomposition of a step

Intro to AI Fall 2002 © L. Joskowicz 39 Analysis of hierarchical decomposition HD helps prune branches in the search tree. Two useful properties of solutions are: –Downward solution property: if p is an abstract solution, then there is a primitive solution of which p is an abstraction. Once an abstract solution is found, all other branches can be pruned! –Upward solution property: if p is an inconsistent abstract plan, then there is no primitive solution of which it is an abstraction Prune away all descendants of inconsistent plans!

Intro to AI Fall 2002 © L. Joskowicz 40 Solution space properties Bold boxes are solutions Dotted boxes are inconsistent Boxes marked with “X” can be pruned abstract primitive plan

Intro to AI Fall 2002 © L. Joskowicz 41 Complexity of hierarchical decomp. For a plan with n steps and an average of b choices at each step (branching factor), the complexity of search is O(b n ) Let d be the depth of the hierarchical plan, and s average number of decomposition steps. When only searching for abstract solutions, one of every b decompositions is a solution. If each decomposition has s steps, the planner looks at bs i steps at depth d =i. The complexity is O(bs d ) << O(b n )

Intro to AI Fall 2002 © L. Joskowicz 42 Quantitative example abstract primitive

Intro to AI Fall 2002 © L. Joskowicz 43 Is completeness preserved? The upward and downward solution properties are not necessary correctness conditions for decompositions! To avoid loosing completeness, no pruning can take place -- still can be used to guide search There is an abstract solution that is inconsistent, but the decomposition solves the problem. “a couple has two possesions: he a gold watch and her beautiful hair. They each plan to buy presents to make each other happy. He wants to trade the watch for a comb, she wants to trade her hair for a watch chain. Can they execute their plans?

Intro to AI Fall 2002 © L. Joskowicz 44 Ex: no upward solution property Cannot be ordered!

Intro to AI Fall 2002 © L. Joskowicz 45 Solution: unique main subaction To guarantee the upward solution property, require that there is one step of the decomposed plan to which all preconditions and effects of the abstract operator are attached In the previous example, the unique main subaction condition does not hold!

Intro to AI Fall 2002 © L. Joskowicz 46 Approximation Another way of guiding the search is to rank goals by order of importance (criticality level). Op(Action: Buy(x) Effect: Have(x) /\ Have(MoneyAmount) Precond: 1. Sells(store,x) /\ 2. At(store) /\ 3. Have(MoneyAmount) Solve the problem by considering ONLY preconditions with criticality less or equal than 1, than 2, etc.

Intro to AI Fall 2002 © L. Joskowicz 47 Other extensions More expressive operator descriptions –conditional effects: add when conditions –universal quantification: preconditions with “forall” quantifier Resource constraints: consider costs of each action -- leads to optimization problems Time constraints: can be handled as resources

Intro to AI Fall 2002 © L. Joskowicz 48 Conditional effects Previous scheme sometimes forces premature commitment that can lead to inefficiencies Solution: extend the operator language to include conditional effects: “condition c must hold when p holds”. Such type of clauses will be added to the effects of an action. If later p appears, the condition c will be added and handled. Extend Select-SubGoal and Resolve-Threats to deal with this new type of conditionals

Intro to AI Fall 2002 © L. Joskowicz 49 Conditionals: example Two different actions for picking a block: Op(Action: move(B,X,Y), Precond: on(B,X) /\ clear(B) /\ clear(Y) Effect: on(B,Y) /\ ~ on(B,X) /\ clear(X) /\ clear(B) /\ ~ clear(Y)) Op(Action: movetotable(B,X), Precond: on(B,X) /\ clear(B) Effect: on(B,table) /\ ~ on(B,X) /\ clear(X) /\ clear(B) Start: on(a,b) Goal: clear(b) Problem : two operators for the same type of action!

Intro to AI Fall 2002 © L. Joskowicz 50 Conditionals: solution One operator with conditional effect : Op(Action: move(B,X,Y), Precond: on(B,X) /\ clear(X) /\ clear(Y) Effect: on(B,Y) /\ ~ on(B,X) /\ clear(X) /\ clear(B) /\ ( ~ clear(Y)) when Y  table) When Y gets instantiated, the condition ~clear(Y) will be added if appropriate. To resolve threats: if a step has the effect ( ~c’ when p) is a possible threat to the causal link S i -- c --> S j when c’ and c unify. Resolve threat by confrontation: ensuring that p does not hold.

Intro to AI Fall 2002 © L. Joskowicz 51 Resolve-Threats with conditionals procedure Resolve-Threats(plan) for each (Si -- c --> Sj) in LINKS (plan) do for each S-threat in STEPS (plan) do for each c’ in EFFECTS (S-threat) do if Subst(Bindings(plan),c) = Subst(Bindings(plan),~c’) then choose either Promotion: add S-threat < Si to ORDERINGS (plan) Demotion: add Sj < S-threat to ORDERINGS (plan) Confrontation: if c’ is of the form (c’ when p) then Choose-Operator(plan,operators,S-threat, ~ p) Resolve-Threats(plan) if not Consistent(plan) then fail end end~

Intro to AI Fall 2002 © L. Joskowicz 52 Negated and disjuctive preconditions Choose-Operator introduced a negated literal Can be handled by checking for effects that match the goal, and ensure that unification between p and ~~p is possible. Also, must deal with special “closed-world assumption” requirements for start, where no negative literals are present. Disjunctive preconditions p \/ q introduce nondeterministic choices. Disjunctive effects are harder to deal with... Ex: Flip(coin)

Intro to AI Fall 2002 © L. Joskowicz 53 Universal quantification (1) Extend the language and algorithms to handle general statements In preconditions: instead of clear(b), write  X block(X) => ~ on(X,B) “ no block is on top of b” In effects: Op(Action: carry(Bag,X,Y), Precond: bag(Bag) /\ at(Bag,X), Effect: at(Bag,Y) /\ ~ at(Bag,X) /\  I item(I) => (at(I,Y) /\ ~at(I,X)) when in(Y,Bag)) “all objects that are in a bag are in location Y after the bag has been carried from location X to location Y”

Intro to AI Fall 2002 © L. Joskowicz 54 Universal quantification (2) Note that adding universal quantification does NOT turn the language into FOL. The restrictions are: –worlds with finite, static, typed universe –universally quantified conditions satisfied by simple enumeration  X t(X) => c(X) is c(x 1 ) when t(x 1 ) /\ c(x 2 ) when t(x 2 ) /\..... c(x n ) when t(x n )

Intro to AI Fall 2002 © L. Joskowicz 55 Universal quantification (3) The planner must expand universally quantified preconditions to eliminate the quantifier (possibly inefficient, but no better solution…) Universally quantified effects need not be expanded, since it might be that many literals are irrelevant. Instead, leave as is but make sure that Resolve-Threats and Choose-Operator are properly modified. Modified routines for POP-DUNC ( POP with disjunction, universal quantification, negation, and conditionals). It is sound and complete.

Intro to AI Fall 2002 © L. Joskowicz 56 Extended-POP Select-Subgoal function Select-Subgoal(plan) returns (plan,precondition conjunct) pick a plan step S-need from STEPS( plan) with a precondition c that has not been achieved if c is a universally quantified expression then return (S-need, Expansion(c)) else if c is a disjunction c1 \/ c2.. \/ cn then return (S-need, choose(c1,c2,…,cn)) else returns (S-need,c)

Intro to AI Fall 2002 © L. Joskowicz 57 Extended-POP Choose-Operator procedure Choose-Operator(plan,operators,S-need,c) choose (a step S-add from operators) or ( STEPS (plan) that has c-add as an effect such that u = Unify(c,c-add,Bindings(plan)) if there is no such step then fail u’ := u without universally quantified variables of c-add add u’ to Bindings(plan) add causal link (S-add -- c --> S-need) to LINKS (plan) add ordering constraint S-add < S-need to ORDERINGS( plan) if S-add is a newly added step from operators then add S-add to STEPS( plan) add Start < S-add < Finish to ORDERINGS( plan)

Intro to AI Fall 2002 © L. Joskowicz 58 Extended-POP Resolve-Threats procedure Resolve-Threats(plan) for each (Si -- c --> Sj) in LINKS (plan) do for each S-threat in STEPS (plan) do for each c’ in EFFECTS (S-threat) do if Subst(Bindings(plan),c) = Subst(Bindings(plan), ~ c’) then choose either Promotion: add S-threat < Si to ORDERINGS (plan) Demotion: add Sj < S-threat to ORDERINGS (plan) Confrontation: if c’ is of the form (c’ when p) then Choose-Operator(plan,operators,S-threat, ~ p) Resolve-Threats(plan) if not Consistent(plan) then fail end

Intro to AI Fall 2002 © L. Joskowicz 59 Resource constraints Need to deal with quantities: cost, time, etc. Ex: buying an object decreases the amount of cash we have: New construct: measures, which are global variables that can be compared and updated Check inequality constrains each time an operator is chosen ==> a CSP problem! buy(X,Store) have(X) /\ Cash := Cash - price(X,Store) at(Store) /\ sells(Store,X) /\ Cash > price(X,Store)

Intro to AI Fall 2002 © L. Joskowicz 60 Planning and acting Up to now, we assumed that we first plan, and then execute the plan. All the necessary knowledge is available to do the plan. Sometimes, we need gather additional information: to see if the bus station is open, we need to go there first! Conditional or contingency planning: generate plan alternatives that account for each possible outcome of a contingency. Include sensing operators (see Chapter 13).

Intro to AI Fall 2002 © L. Joskowicz 61 Conditional plan for fixing flat tire Fixing a flat tire by either inflating it or replacing with the spare tire. Since we don’t know why the tire is flat, we need to generate two contingency plans

Intro to AI Fall 2002 © L. Joskowicz 62 Planning in practice Job shop scheduling: Plan production and assembly schedule Hitachi’s Tosca: 350 products, 35 assembly machines, 2,000 operations. Plan 30-day schedule for three 8-hour shifts. Follows a partial-order, least commitment approach Space mission scheduling: plans order of experiments and resource use used for Hubble space telescope, Voyager, etc. SIPE planner: planning maintenance and materials logistics military operations for US Air Force.