Commitment Least you haven't decided where to go shopping. Or...suppose You can get milk at the convenience store, at the dairy, or at the supermarket

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1 Planning as Search-based Problem Solving? Imagine a supermarket shopping scenario using search-based problem solving: Goal: buy milk and bananas Operator: buy <obj> Heuristic function: does <obj> = milk or bananas? The operator would be instantiated with all possible objects that can be bought! Then the heuristic function would evaluate each instantation. This is essentially a guessing game!

2 Commitment Least you haven't decided where to go shopping. Or...suppose You can get milk at the convenience store, at the dairy, or at the supermarket. Goal: buy milk and bananas Operators: go to<store>, buy <obj, store> You can only get bananas at the supermarket. If you decide where to buy milk rst (say, at the convenience store), then you will either: (a) have to backtrack, or (b) have to go to more than one store! Planners have to be exible (add actions or instantiate variables in any order) and usually follow the principle of least commitment.

3 Planning vs. Search-based Problem Solving Problem solving: actions generate successor states. Planning: actions are represented as preconditions and eects. Problem solving: state representations are complete. Planning: complete state descriptions would be enormous. Problem solving: goal test and heuristic function used to evaluate states. Planning: goals are usually represented as a conjunct of state variables. Problem solving: incrementally generates solution as a sequence of actions. Planning: can add actions to any part of the plan at any time.

4 STRIPS Language The (Stanford Research Institute Problem Solver) was a pioneering STRIPS program developed in The STRIPS \language" is still widely planning today to represent states and operators. used conjunction of atoms that must be true before the precondtions: can be applied. operator States and Goals: Conjunctions of function-free literals. Operators: action description: name for action command to environment. eects: add list and delete list

5 Operators STRIPS 1: MOVE BLOCK TO BLOCK Operator 2: MOVE BLOCK TO TABLE Operator preconds: 3: MOVE BLOCK FROM TABLE Operator preconds: preconds: block(x) ^ block(y) ^ block(z) ^ on(x y) ^ clear(x) ^ clear(z) eects: Add: on(x,z), clear(y) Delete: on(x,y), clear(z) block(x) ^ block(y) ^ on(x y) ^ clear(x) eects: Add: on(x,table), clear(y) Delete: on(x,y) block(x) ^ block(y) ^ on(x T able) ^ clear(x) ^ clear(z) eects: Add: on(x,z) Delete: on(x,table), clear(z)

6 Space Planner Searches through space of possible situations Situation like search-based problem solving. much Planner Forward search from the initial situation to the goal Progression situation. doesn't work well branching factor too high yields Usually growth of tree. exponential Planner Backward search from the goal situation to the initial Regression situation. better because goal state typically has only a few Works so only a few operators apply. But complicated conjuncts by Searching for a Satisfactory Sequence of Plan Operators because the conjunction of goals needs to be satised.

7 A Planning Example A D B C B C D Initial situation Goal situation Initial Situation: on(a,c), on(c,table), on(d,b), on(b,table), clear(a), clear(d) Goal Situation: on(a,b), on(b,c)

8 is to search through the space of plans rather than the space of Alternative situations. add a step, impose an ordering on existing steps, instantiate a Operators: unbound variable. previously Searching Plan Space Start with simple, incomplete partial plan expand until complete. Solution: nal plan.

9 Representation Plan planners use the principle of least commitment. Most order planner: represents plans in which some steps are ordered Partial other steps are unordered. and totally ordered plan that is derived from a plan P by adding ordering A is called a linearization of P. constraints Total order planner: simple list of steps.

10 Start Start Start Total Order Plans: Partial Order Plan: Start Left Sock Left Shoe Sock Right Shoe Right Finish Start Finish Sock Right Shoe Right Sock Left Shoe Left Start Left Sock Shoe Right Finish Right Sock Left Shoe Finish Sock Left Right Sock Shoe Left Right Shoe Shoe Right Finish Sock Right Left Sock Left Shoe Finish Sock Right Shoe Left Left Sock Right Shoe LeftSockOn RightSockOn LeftShoeOn, RightShoeOn Start Sock Right Shoe Right Sock Left Shoe Left Finish

11 Components of a Plan 1. A set of plan steps. Each step is one of the operators for the problem. 2. A set of step ordering constraints, S i S j. A set of variable binding constraints, v = x where v is a variable and x 3. a constant or another variable. is c set of causal links, S i! S j (S i achieves precondition c for S j.) A 4.

12 might expect that only fully instantiated, totally ordered plans would be You solutions. But this is not a good idea for several reasons: considered It makes more sense to have a planner return a partial order plan than arbitrarily choose one linearization of it. to Sometimes actions can be performed in parallel, so it is best to generate that allow for actions to happen in parallel. solutions A plan may be integrated with another plan later. Keeping the plan can help with plan integration. exible What is a solution?

13 complete plan is one in which every precondition is achieved by another A A precondition is achieved if it is an eect of a step and no other step. there is no step S k such that (:c) EFFECTS(S k ) where (2) i S k S j in some linearization of the plan. S consistent plan is a plan with no contradictions in the ordering or A of constraints. binding We consider a plan to be a solution if it is complete and consistent. step can cancel it out. Formally: i achieves precondition c of S j if S S i S j and c EFFECTS(S i ), and (1) contradiction occurs when S i S j and S j S i or if v = A A v = B for dierent constants A and B. and Under this denition, the partial plan for putting on socks is a solution!

14 Regression Planner Example Partial-Order you want a plan to buy milk, bananas, and a drill. Suppose The initial plan might be: Start At(Home) Sells(SM,Banana) Sells(SM,Milk) Sells(HWS,Drill) Have(Drill) Have(Milk) Have(Banana) At(Home) Finish

15 Operator: = Go(there) ACTION = At(here) PRECONDITIONS = At(there) ^ : At(here) EFFECTS Operator: = Buy(obj) ACTION = At(store) ^ Sells(store,obj) PRECONDITIONS = have(obj) EFFECT Operators To keep the search focused, the planner only considers adding steps that achieve a precondition that has not yet been achieved.

16 Start At(s), Sells(s,Drill) At(s), Sells(s,Milk) At(s), Sells(s,Bananas) Buy(Drill) Buy(Milk) Buy(Bananas) Have(Drill), Have(Milk), Have(Bananas), At(Home) Finish Start At(HWS), Sells(HWS,Drill) At(SM), Sells(SM,Milk) At(SM), Sells(SM,Bananas) Buy(Drill) Buy(Milk) Buy(Bananas) Have(Drill), Have(Milk), Have(Bananas), At(Home) Finish

17 The plan is extended by choosing Go actions to achieve the At preconditions. Start At(x) Go(HWS) At(x) Go(SM) At(HWS), Sells(HWS,Drill) At(SM), Sells(SM,Milk) At(SM), Sells(SM,Bananas) Buy(Drill) Buy(Milk) Buy(Bananas) Have(Drill), Have(Milk), Have(Bananas), At(Home) Finish Note that the Go actions have unachieved preconditions that interact!

18 what happens if the planner tries to achieve the preconditions of Go Look the At(home) condition in the initial state! with Start At(Home) Go(HWS) At(Home) Go(SM) At(HWS), Sells(HWS,Drill) At(SM), Sells(SM,Milk) At(SM), Sells(SM,Bananas) Buy(Drill) Buy(Milk) Buy(Bananas) Have(Drill), Have(Milk), Have(Bananas), At(Home) Finish

19 This partial plan is a dead end because one step clobbers a protected for another step. condition The planner must pay attention to the clausal links which are protected. planner must ensure that threats (steps which can clobber protected The Threats can be added by adding ordering constraints to put the threat the protected link (demotion) or after the protected link before A planner can notice dead ends preconditions) are order to come before or after the protected link. (promotion).

20 Sussman Anomaly The on one conjunct at a time can make clobbering unavoidable. Focusing A C A B B C 1998 Morgan Kaufman Publishers

21 S 3 Lc S 1 S 1 S 1 S 3 c Lc c c S 2 S 2 S 2 S 3 Lc (a) (b) (c)

22 Unfortunately, there is no way to reorder the Go threat because any will delete the At(home) condition of the other step. order Suppose we trying adding a causal link from Go(HWS) to Go(SM). Now threatens the At(HWS) precondition of Buy(drill). Go(SM) Ordering constraints aren't always enough... When a planner can't resolve a threat, it has no choice but to backtrack. We can resolve this threat by ordering Go(SM) to come after Buy(drill).

23 Start At(Home) At(HWS) Go(HWS) Go(SM) At(HWS), Sells(HWS,Drill) At(SM), Sells(SM,Milk) At(SM), Sells(SM,Bananas) At(SM) Buy(Drill) Buy(Milk) Buy(Bananas) Go(Home) Have(Drill), Have(Milk), Have(Bananas), At(Home) Finish

24 Start At(Home) Go(HWS) At(HWS) Sells(HWS,Drill) Buy(Drill) At(HWS) Go(SM) At(SM) Sells(SM,Milk) Buy(Milk) At(SM) Sells(SM,Ban.) Buy(Ban.) At(SM) Go(Home) Have(Milk) At(Home) Have(Ban.) Have(Drill) Finish

25 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-SUBGOAL( 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 return S need, c 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 c add the causal link S add ;! S need to LINKS( plan) add the 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) c for each S threat that threatens a link S i ;! S j in LINKS( plan) do choose either Promotion: Add S threat S i to ORDERINGS( plan) Demotion: Add S j S threat to ORDERINGS( plan) if not CONSISTENT( plan) then fail end

26 Threats Possible can be left unbound in plans. If an operator produces an eect that Variables cause a threat if the variable takes a certain binding, then it is a could threat. possible Resolve now by forcing a variable binding. The commitment may cause 1. later though. trouble Resolve now with an inequality constraint (x 6= home). Less 2. but more complicated for unication algorithms. commitment, Ignore possible threats and only deal with them when they become 3. threats. For example, if x = home is added then resolve the known Low commitment, but can't say for sure that the plan is a threat. solution. There are three general apporaches to dealing with possible threats:

27 missions: orchestrating observational equipment to maximize data Space while minimizing time and energy consumption. acquisition Practical Applications for Planners Job shop scheduling: assembling, manufacturing Construction: Building facilities, airplanes, spacecraft, etc. Event scheduling: Scheduling meetings, classes, and other events.

28 planning: Generating complex plans often requires abstract Hierarchical over increasingly detailed search spaces. planning state conditions: STRIPS variables have limited in their Complex For example, there is no quantication and no conditional complexity. time: The STRIPS framework assumes that everything Representing instantly. There is no way to represent duration, deadlines, happens limitations: In the real world, resources are limited. You need Resource represent the fact that the number of available workers, equipment, to Limitations of the STRIPS language statements. time windows, etc. money, etc. is constrained.