Planning (under complete Knowledge)

Transcription

1 CSC384: Intro to Artificial Intelligence Planning (under complete Knowledge) We cover Sections11.1, 11.2, Section 11.3 talks about partial-order planning, an interesting idea that hasn t really done well in practice talks about planning by converting planning problems in to SAT and then using backtracking search to solve the problem. This also is an interesting idea, but its practical performance has been lacking. Chapter 12 talks about some practical extensions to planning needed in real world applications. We will not cover that material. But it is interesting reading. 1

2 Why Planning Intelligent agents must operate in the world. They are not simply passive reasoners (Knowledge Representation, reasoning under uncertainty) or problem solvers (Search), they must also act on the world. We want intelligent agents to act in intelligent ways. Taking purposeful actions, predicting the expected effect of such actions, composing actions together to achieve complex goals. 2

3 Why Planning E.g. if we have a robot we want robot to decide what to do; how to act to achieve our goals 3

4 A Planning Problem How to change the world to suit our needs Critical issue: we need to reason about what the world will be like after doing a few actions, not just what it is like now GOAL: Craig has coffee CURRENTLY: robot in mailroom, has no coffee, coffee not made, Craig in office, etc. TO DO: goto lounge, make coffee, 4

5 Planning Reasoning about what the world will be like after doing a few actions is similar to what we have already examined. However, now we want to reason about dynamic environments. in(robby,room1), lighton(room1) are true: will they be true after robby performs the action turnofflights? in(robby,room1) is true: what does robby need to do to make in(robby,room3) true? Reasoning about the effects of actions, and computing what actions can achieve certain effects is at the heart of decision making. 5

6 Planning under Uncertainty One of the major complexities in planning that we will deal with later is planning under uncertainty. Our knowledge of the world probabilistic. Sensing is subject to noise (especially in robots). Actions and effectors are also subject to error (uncertainty in their effects). 6

7 Planning But for now we will confine our attention to the deterministic case. We will examine: Determining the effects of actions. finding sequences of actions that can achieve a desired set of effects. This will in some ways be a lot like search, but we will see that representation also plays an important role. 7

8 Situation Calculus First we look at how to model dynamic worlds within first-order logic. The situation calculus is an important formalism developed for this purpose. Situation Calculus is a first-order language. Include in the domain of individuals a special set of objects called situations. Of these s 0 is a special distinguished constant which denotes the initial situation. 8

9 Situation Calculus Situations are used to index states of the world. When dealing with dynamic environments, the world has different properties at different points in time. e.g., in(robby,room1, s 0 ), in(robby,room3,s 0 ) in(robby,room3,s 1 ), in(robby,room1,s 1 ). Different things are true in situation s 1 than in the initial situation s 0. Contrast this with the previous kinds of knowledge we examined. 9

10 Fluents The basic idea is that properties that change from situation to situation (called fluents) take an extra situation argument. clear(b) clear(b,s) clear(b) is no longer statically true, it is true contingent on what situation we are talking about 10

11 Blocks World Example. robot hand clear(c,s 0 ) on(c,a,s 0 ) clear(b,s 0 ) handempty(s 0 ) C A B 11

12 Actions in the Situation Calculus Actions are also part of language A set of primitive action objects in the (semantic) domain of individuals. In the syntax they are represented as functions mapping objects to primitive action objects. pickup(x) function mapping blocks to actions pickup(c) = the primitive action object corresponding to picking up block c stack(x,y) stack(a,b) = the primitive action object corresponding to stacking a on top of b 12

13 Actions modify situations. There is a generic action application function do(a,s). do maps a primitive action and a situation to a new situation. The new situation is the situation that results from applying A to S. do(pickup(c), s 0 ) = the new situation that is the result of applying action pickup(c) to the initial situation s 0. 13

14 What do Actions do? Actions affect the situation by changing what is true. on(c,a,s 0 ); clear(a,do(pickup(c),s 0 )) We want to represent the effects of actions, this is done in the situation calculus with two components. 14

15 Specifying the effects of actions Action preconditions. Certain things must hold for actions to have a predictable effect. pickup(c) this action is only applicable to situations S where clear(c,s) handempty(s) are true. Action effects. Actions make certain things true and certain things false. holding(c, do(pickup(c), S)) X. handempty(do(pickup(x),s)) 15

16 Specifying the effects of actions Action effects are conditional on their precondition being true. S,X. ontable(x,s) clear(x,s) handempty(s) holding(x, do(pickup(x),s)) handempty(do(pickup(x),s)) ontable(x,do(pickup(x,s)) clear(x,do(pickup(x,s)). 16

17 Reasoning with the Situation Calculus. 1. clear(c,s 0 ) 2. on(c,a,s 0 ) 3. clear(b,s 0 ) 4. ontable(a,s 0 ) 5. ontable(b,s 0 ) 6. handempty(s 0 ) Query: Z.holding(b,Z) 7. ( holding(b,z), ans(z)) does there exists a situation in which we are holding b? And if so what is the name of that situation. C A B 17

18 Resolution Convert pickup action axiom into clause form: S,Y. ontable(y,s) clear(y,s) handempty(s) holding(y, do(pickup(y),s)) handempty(do(pickup(y),s)) ontable(y,do(pickup(y,s)) clear(y,do(pickup(y,s)). 8. ( ontable(y,s), clear(y,s), handempty(s), holding(y,do(pickup(y),s)) 9. ( ontable(y,s), clear(y,s), handempty(s), handempty(do(pickup(x),s))) 10. ( ontable(y,s), clear(y,s), handempty(s), ontable(y,do(pickup(y,s))) 11. ( ontable(y,s), clear(y,s), handempty(s), clear(y,do(pickup(y,s))) 18

19 Resolution 12. R[8d, 7]{Y=b,Z=do(pickup(b),S)} ( ontable(b,s), clear(b,s), handempty(s), ans(do(pickup(b),s))) 13. R[12a,5] {S=s 0 } ( clear(b,s 0 ), handempty(s 0 ), ans(do(pickup(b),s 0 ))) 14. R[13a,3] {} ( handempty(s 0 ), ans(do(pickup(b),s 0 ))) 15. R[14a,6] {} ans(do(pickup(b),s 0 )) 19

20 The answer? ans(do(pickup(b),s 0 )) This says that a situation in which you are holding b is called do(pickup(b),s 0 ) This name is informative: it tells you what actions to execute to achieve holding(b). 20

21 Two types of reasoning. In general we can answer questions of the form: on(b,c,do(stack(b,c), do(pickup(b), s 0 ))) S. on(b,c,s) on(c,a,s) The first involves predicting the effects of a sequence of actions, the second involves computing a sequence of actions that can achieve a goal condition. 21

22 The Frame Problem Unfortunately, logical reasoning won t immediately yield the answer to these kinds of questions. e.g., query: on(c,a,do(pickup(b),s 0 ))? is c still on a after we pickup b? Intuitively it should be Can logical reasoning reach this conclusion? 22

23 The Frame Problem 1. clear(c,s 0 ) 2. on(c,a,s 0 ) 3. clear(b,s 0 ) 4. ontable(a,s 0 ) 5. ontable(b,s 0 ) 6. handempty(s 0 ) 8. ( ontable(y,s), clear(y,s), handempty(s), holding(y,do(pickup(y),s)) 9. ( ontable(y,s), clear(y,s), handempty(s), handempty(do(pickup(x),s))) 10. ( ontable(y,s), clear(y,s), handempty(s), ontable(y,do(pickup(y,s))) 11. ( ontable(y,s), clear(y,s), handempty(s), clear(y,do(pickup(y,s))) 12. on(c,a,do(pickup(b),s 0 )) {QUERY) Nothing can resolve with 12! 23

24 Logical Consequence Remember that resolution only computes logical consequences. We stated the effects of pickup(b), but did not state that it doesn t affect on(c,a). Hence there are models in which on(c,a) no longer holds after pickup(b) (as well as models where it does hold). The problem is that representing the non-effects of actions very tedious and in general is not possible. Think of all of the things that pickup(b) does not affect! 24

25 The Frame Problem Finding an effective way of specifying the noneffects of actions, without having to explicitly write them all down is the frame problem. Very good solutions have been proposed, and the situation calculus has been a very powerful way of dealing with dynamic worlds: logic based high level robotic programming languages 25

26 Computation Problems Although the situation calculus is a very powerful representation. It is not always efficient enough to use to compute sequences of actions. The problem of computing a sequence of actions to achieve a goal is planning Next we will study some less rich representations which support more efficient planning. 26

27 CWA Classical Planning. No incomplete or uncertain knowledge. Use the Closed World Assumption in our knowledge representation and reasoning. The Knowledge base used to represent a state of the world is a list of positive ground atomic facts. CWA is the assumption that a) if a ground atomic fact is not in our list of known facts, its negation must be true. b) the constants mentioned in KB are all the domain objects. 27

28 CWA CWA makes our knowledge base much like a database: if employed(john,cibc) is not in the database, we conclude that employed(john, CIBC) is true. 28

29 CWA Example KB = {handempty clear(c), clear(b), on(c,a), ontable(a), ontable(b)} C A B 1. clear(c) clear(b)? 2. on(b,c)? 3. on(a,c) on(b,c)? 4. X.on(X,c)? (D = {a,b,c}) 5. X.ontable(X) X = a X = b? 29

30 Querying a Closed World KB With the CWA, we can evaluate the truth or falsity of arbitrarily complex first-order formulas. This process is very similar to query evaluation in databases. Just as databases are useful, so are CW KB s. 30

31 Querying A CW KB Query(F, KB) /*return whether or not KB = F */ if F is atomic return(f KB) 31

32 Querying A CW KB if F = F 1 F 2 return(query(f 1 ) && Query(F 2 )) if F = F 1 F 2 return(query(f 1 ) Query(F 2 )) if F = F 1 return(! Query(F 1 )) if F = F 1 F 2 return(!query(f 1 ) Query(F 2 )) 32

33 Querying A CW KB if F = X.F 1 for each constant c KB if (Query(F 1 {X=c})) return(true) return(false). if F = X.F 1 for each constant c KB if (!Query(F 1 {X=c})) return(false) return(true). 33

34 Querying A CW KB Guarded quantification (for efficiency). if F = X.F 1 for each constant c KB if (!Query(F 1 {X=c})) return(false) return(true). E.g., consider checking X. apple(x) sweet(x) we already know that the formula is true for all nonapples 34

35 Querying A CW KB Guarded quantification (for efficiency). X:[p(X)] F 1 X: p(x) F 1 for each constant c s.t. p(c) if (!Query(F 1 {X=c})) return(false) return(true). X:[p(X)]F 1 X: p(x) F 1 for each constant c s.t. p(c) if (Query(F 1 {X=c})) return(true) return(false). 35

36 STRIPS representation. STRIPS (Stanford Research Institute Problem Solver.) is a way of representing actions. Actions are modeled as ways of modifying the world. since the world is represented as a CW-KB, a STRIPS action represents a way of updating the CW-KB. Now actions yield new KB s, describing the new world the world as it is once the action has been executed. 36

37 Sequences of Worlds In the situation calculus where in one logical sentence we could refer to two different situations at the same time. on(a,b,s 0 ) on(a,b,s 1 ) In STRIPS, we would have two separate CW- KB s. One representing the initial state, and another one representing the next state (much like search where each state was represented in a separate data structure). 37

38 STRIPS Actions STRIPS represents actions using 3 lists. 1. A list of action preconditions. 2. A list of action add effects. 3. A list of action delete effects. These lists contain variables, so that we can represent a whole class of actions with one specification. Each ground instantiation of the variables yields a specific action. 38

39 STRIPS Actions: Example pickup(x): Pre: {handempty, clear(x), ontable(x)} Adds: {holding(x)} Dels: {handempty, clear(x), ontable(x)} pickup(x) is called a STRIPS operator. a particular instance e.g. pickup(a) is called an action. 39

40 Operation of a STRIPS action. For a particular STRIPS action (ground instance) to be applicable to a state (a CW-KB) every fact in its precondition list must be true in KB. This amounts to testing membership since we have only atomic facts in the precondition list. If the action is applicable, the new state is generated by removing all facts in Dels from KB, then adding all facts in Adds to KB. 40

41 Operation of a Strips Action: Example C A B pickup(b) C A B pre = {handmpty, clear(b), ontable(b)} add = {holding(b)} del = {handmpty, clear(b), ontable(b)} KB = {handempty clear(c), clear(b), on(c,a), ontable(a), ontable(b)} KB = { holding(b), clear(c), on(c,a), ontable(a)} 41

42 STRIPS Blocks World Operators. pickup(x) Pre: {clear(x), ontable(x), handempty} Add: {holding(x)} Del: {clear(x), ontable(x), handempty} putdown(x) Pre: {holding(x)} Add: {clear(x), ontable(x), handempty} Del: {holding(x)} 42

43 STRIPS Blocks World Operators. unstack(x,y) Pre: {clear(x), on(x,y), handempty} Add: {holding(x), clear(y)} Del: {clear(x), on(x,y), handempty} stack(x,y) Pre: {holding(x),clear(y)} Add: {on(x,y), handempty, clear(x)} Del: {holding(x),clear(y)} 43

44 STRIPS has no Conditional Effects putdown(x) Pre: {holding(x)} Add: {clear(x), ontable(x), handempty} Del: {holding(x)} stack(x,y) Pre: {holding(x),clear(y)} Add: {on(x,y), handempty, clear(x)} Del: {holding(x),clear(y)} The table has infinite space, so it is always clear. If we stack(x,y) if Y=Table we cannot delete clear(table), but if Y is an ordinary block c we must delete clear(c). 44

45 Conditional Effects Since STRIPS has no conditional effects, we must sometimes utilize extra actions: one for each type of condition. We embed the condition in the precondition, and then alter the effects accordingly. 45

46 ADL Operators. ADL operators add a number of features to STRIPS. 1. Their preconditions can be arbitrary formulas, not just a conjunction of facts. 2. They can have conditional and universal effects. 3. Open world assumption: 1. States can have negative literals 2. The effect (P Q) means add P and Q but delete P and Q. But they must still specify atomic changes to the knowledge base (add or delete ground atomic facts). 46

47 ADL Operators Examples. move(x,y,z) Pre: on(x,y) clear(z) Effs: ADD[on(X,Z)] DEL[on(X,Y)] Z table DEL[clear(Z)] Y table ADD[clear(Y)] 47

48 ADL Operators, example C A B move(c,a,b) A C B move(c,a,b) Pre: on(c,a) clear(b) Effs: ADD[on(c,b)] DEL[on(c,a)] b table DEL[clear(b)] a table ADD[clear(a)] KB = { clear(c), clear(b), on(c,a), on(a,table), on(b,table)} KB = { on(c,b) clear(c), clear(a) on(a,table), on(b,table)} 48

49 ADL Operators Examples. cleartable() Pre: Effs: X. on(x,table) DEL[on(X,table)] 49

50 ADL Operators. 1. Arbitrary formulas as preconditions. in a CW-KB we can evaluate whether or not the preconditions hold for an arbitrary precondition. 2. They can have conditional and universal effects. Similarly we can evaluate the condition to see if the effect should be applied, and find all bindings for which it should be applied. Specify atomic changes to the knowledge base. CW-KB can be updated just as before. 50

51 Other Example Domains 8 Puzzle as a planning problem A constant representing each position, P1,,P9 P1 P4 P7 P2 P5 P8 P3 P6 P9 A constant for each tile. B,T1,, T8. 51

52 8-Puzzle at(t,p) tile T is at position P at(t1,p1), at(t2,p2), at(t5,p3), adjacent(p1,p2) P1 is next to P2 (i.e., we can slide the blank from P1 to P2 in one move. adjacent(p5,p2), adjacent(p5,p8), 52

53 8-Puzzle slide(t,x,y) Pre: {at(t,x), at(b,y), adjacent(x,y)} Add: {at(b,x), at(t,y)} Del: {at(t,x), at(b,y)} at(t1,p1), at(t5,p3), at(t8,p5), at(b,p6),, at(t1,p1), at(t5,p3), at(b,p5), at(t8,p6),, slide(t8,p5,p6)

54 Elevator Control 54

55 Elevator Control Schindler Lifts. Central panel to enter your elevator request. Your request is scheduled and an elevator is assigned to you. You can t travel with someone going to a secure floor, emergency travel has priority, etc. Modeled as a planning problem and fielded in one of Schindler s high end elevators. 55

56 Solving Planning Problems Given a CW-KB representing the initial state, a set of STRIPS or ADL (Action Description Language) operators, and a goal condition we want to achieve (specified either as a conjunction of facts, or as a formula) The planning problem is determine a sequence of actions that when applied to the initial CW-KB yield an updated CW-KB which satisfies the goal. 56

57 Solving Planning Problems We cover two approaches: GraphPlan: building a graph to represent the planning problem and finding the solution using the graph. Planning as Search: Casting the problem as a standard heuristic search (using reachablity analysis) The performance of GraphPlan is not at good as heuristic search, but it does perform better than partial-order planning and it can be generalized to other types of planning, e.g., finding optimal plans, planning with sensing, etc. 57

58 Graphplan Operates in two phases. Phase I. Guess a concurrent plan length k, then build a leveled graph with k alternating layers. Phase II. Search this leveled graph for a plan. If no plan is found, return to phase I and build a bigger leveled graph with k+1 alternating layers. The final plan, if found, consists of a sequence of sets of actions {a 1 1,a 2 1, } {a 1 2, a 2 2, } {a 1 3, a 2 3, } The plan is concurrent in the sense that at stage I, all actions in the i-th set are executed in parallel. 58

59 Graphplan The leveled graph alternates between levels containing propositional nodes and levels containing action nodes. (Similar to the reachability graph). Three types of edges: precondition-edges, add-edges, and delete-edges. 59

60 GraphPlan Level Graph ontable(a) ontable(b) clear(a) clear(b) handempty pickup(a) pickup(b) no-op( op(ontable ontable(a)) no-op( op(ontable ontable(b)) no-op(clear(a)) op(clear(a)) no-op(clear(b)) op(clear(b)) no-op( op(handempty handempty) holding(a) holding(b) ontable(a) ontable(b) clear(a) clear(b) handempty Initial state Only the propositions true in the initial state. Possible actions Only the actions whose preconditions are in the previous level. Also have no-ops for capturing nonchanges. All propositions added by actions in previous level Precondition Delete Add 60

61 GraphPlan Level Graph ontable(a) ontable(b) clear(a) clear(b) handempty pickup(a) pickup(b) no-op( op(ontable ontable(a)) no-op( op(ontable ontable(b)) no-op(clear(a)) op(clear(a)) no-op(clear(b)) op(clear(b)) no-op( op(handempty handempty) holding(a) holding(b) ontable(a) ontable(b) clear(a) clear(b) handempty Level S 0 contains all facts true in the initial state. Level A 0 contains all actions whose preconditions are true in S 0. Included in the set of actions are no-ops. One no-op for every ground atomic fact. The precondition of the no-op is its fact, its add effect is its fact. Level S i contains all facts that are added by actions at level A i-1 Level A i contains all actions whose preconditions are true in S i 61

62 GraphPlan Mutexes. ontable(a) ontable(b) clear(a) clear(b) handempty pickup(a) pickup(b) no-op( op(ontable ontable(a)) no-op( op(ontable ontable(b)) no-op(clear(a)) op(clear(a)) no-op(clear(b)) op(clear(b)) no-op( op(handempty handempty) holding(a) holding(b) ontable(a) ontable(b) clear(a) clear(b) handempty In addition to the facts/actions. GraphPlan also computes and adds mutexes to the graph. Mutexes are edges between two labels, indicating that these two labels cannot be true at the same time. Mutexes are added as we construct each layer, and in fact alter the set of labels the eventually appear in a layer. 62

63 Mutexes A mutex between two actions a 1 and a 2 in the same layer A i, means that a 1 and a 2 cannot be executed simultaneously (in parallel) at the i th step of a concurrent plan. A mutex between two facts F 1 and F 2 in the same state layer S i, means that F 1 and F 2 cannot be be simultaneously true after i stages of parallel action execution. 63

64 Mutexes It is not possible to compute all mutexes. This is as hard as solving the planning problem, and we want to perform mutex computation as a precursor to solving a planning instance. However, we can quickly compute a subset of the set of all mutexes. Although incomplete these mutexes are still very useful. This is what GraphPlan does. 64

65 Mutexes ontable(a) ontable(b) clear(a) clear(b) handempty pickup(a) pickup(b) no-op( op(ontable ontable(a)) no-op( op(ontable ontable(b)) no-op(clear(a)) op(clear(a)) no-op(clear(b)) op(clear(b)) no-op( op(handempty handempty) holding(a) holding(b) ontable(a) ontable(b) clear(a) clear(b) handempty Two actions are mutex if either action deletes a precondition or add-effect of another. Note no-ops participate in mutexes. Intuitively these actions have to be sequenced they can t be executed in parallel 65

66 Mutexes ontable(a) ontable(b) clear(a) clear(b) handempty pickup(a) pickup(b) no-op( op(ontable ontable(a)) no-op( op(ontable ontable(b)) no-op(clear(a)) op(clear(a)) no-op(clear(b)) op(clear(b)) no-op( op(handempty handempty) holding(a) holding(b) ontable(a) ontable(b) clear(a) clear(b) handempty Two propositions p and q are mutex if all actions adding p are mutex of all actions adding q. Must look at all pairs of actions that add p and q. Intuitively, can t achieve p and q together at this stage because we can t concurrently execute achieving actions for them at the previous stage. 66

67 Mutexes holding(a) holding(b) ontable(a) ontable(b) clear(a) clear(b) handempty putdown(a) putdown(b) no-op( op(ontable ontable(a)) no-op( op(ontable ontable(b)) no-op(clear(a)) op(clear(a)) no-op(clear(b)) op(clear(b)) no-op( op(handempty handempty) Two actions are mutex if two of their preconditions are mutex. Intuitively, we can t execute these two actions concurrently at this stage because their preconditions can t simultaneously hold at the previous stage. 67

68 How Mutexes affect the level graph. 1. Two actions are mutex if either action deletes a precondition or add effect of another 2. Two propositions p and q are mutex if all actions adding p are mutex of all actions adding q 3. Two actions are mutex if two of their preconditions are mutex We compute mutexes as we add levels. S 0 is set of facts true in initial state. (Contains no mutexes). A 0 is set of actions whose preconditions are true in S 0. Mark as mutex any action pair where one deletes a precondition or add effect of the other. S 1 is set of facts added by actions at level A 0. Mark as mutex any pair of facts p and q if all actions adding p are mutex with all actions adding q. A 1 is set of actions whose preconditions are not mutex at S 1. Mark as mutex any action pair with preconditions that are mutex in S 1, or where one deleted a precondition or add effect of the other. 68

69 How Mutexes affect the level graph. 1. Two actions are mutex if either action deletes a precondition or add effect of another 2. Two propositions p and q are mutex if all actions adding p are mutex of all actions adding q 3. Two actions are mutex if two of their preconditions are mutex S i is set of facts added by actions in level A i-1 Mark as mutex all facts satisfying 2 (where we look at the action mutexes of A i-1 is set of facts true in initial state. (Contains no mutexes). A i is set of actions whose preconditions are true and nonmutex at S i. Mark as mutex any action pair satisfying 1 or 2. 69

70 How Mutexes affect the level graph. Hence, mutexes will prune actions and facts from levels of the graph. They also record useful information about impossible combinations. 70

71 Example NoOp-on(a,b) on(a,b), on(b,c), ontable(c), ontable(b), clear(a), clear(c), handempty unstack(a,b) pickup(c) on(a,b), on(b,c), ontable(c), ontable(d), clear(a), clear(c), handempty, holding(a), clear(b), holding(c) unstack(a,b) deletes the add effect of NoOpon(a,b), so these actions are mutex as well precondition add effect del effect pickup deletes handempty, one of unstack(a,b) s preconditions. 71

72 Example NoOp-on(a,b) on(a,b), on(b,c), ontable(c), ontable(b), clear(a), clear(c), handempty unstack(a,b) pickup(c) on(a,b), on(b,c), ontable(c), ontable(d), clear(a), clear(c), handempty, holding(a), clear(b), holding(c) unstack(a,b) is the only action that adds clear(b), and this is mutex with pickup(c), which is the only way of adding holding(c). precondition add effect del effect 72

73 Example NoOp-on(a,b) on(a,b), on(b,c), ontable(c), ontable(b), clear(a), clear(c), handempty unstack(a,b) pickup(c) on(a,b), on(b,c), ontable(c), ontable(d), clear(a), clear(c), handempty, holding(a), clear(b), holding(c) These two are mutex for the same reason. precondition add effect del effect 73

74 Example NoOp-on(a,b) on(a,b), on(b,c), ontable(c), ontable(b), clear(a), clear(c), handempty unstack(a,b) pickup(c) on(a,b), on(b,c), ontable(c), ontable(d), clear(a), clear(c), handempty, holding(a), clear(b), holding(c) unstack(a,b) is also mutex with the NoOpon(a,b). So these two facts are mutex (NoOp is the only way on(a,b) can be created). precondition add effect del effect 74

75 Phase II. Searching the Graphplan on(a,b) on(b,c) ontable(c) ontable(b) clear(a) clear(b) handempty K Build the graph to level k, such that every member of the goal is present at level k, and no two are mutex. Why? 75

76 Searching the Graphplan stack(a,b) stack(b,c) no-op(on(b,c)) op(on(b,c))... on(a,b) on(b,c) ontable(c) ontable(b) clear(a) clear(b) handempty K Find a non-mutex collection of actions that add all of the facts in the goal. 76

77 Searching the Graphplan holding(a) clear(b) on(b,c) K-1 stack(a,b) stack(b,c) no-op(on(b,c)) op(on(b,c))... on(a,b) on(b,c) ontable(c) ontable(b) clear(a) clear(b) handempty K The preconditions of these actions at level K-1 become the new goal at level K-1. Recursively try to solve this new goal. If this fails, backtrack and try a different set of actions for solving the goal at level k. 77

78 Phase II-Search Solve(G,K) forall sets of actions A={a i } such that no pair (a i, a j ) A is mutex the actions in A suffice to add all facts in G Let P = union of preconditions of actions in A If Solve(P,K-1) Report PLAN FOUND At end of forall. Exhausted all possible action sets A Report NOPLAN This is a depth first search. 78

79 Graph Plan Algorithm Phase I. build leveled graph. Phase II. Search leveled graph. Phase I: While last state level does not contain all goal facts with no pair being mutex add new state/action level to graph if last state/action level = previous state/action level (including all MUTEXES) graph has leveled off) report NO PLAN. Phase II: Starting at last state level search backwards in graph for plan. Try all ways of moving goal back to initial state. If successful report PLAN FOUND. Else goto Phase I. 79

80 Dinner Date Example Initial State {dirty, cleanhands, quiet} Goal {dinner, present, clean} Actions Cook: Wrap: Pre: {cleanhands} Add: {dinner} Pre: {quiet} Add: {present} Tidy: Pre: {} Add: {clean} Del: {cleanhands, dirty} Vac: Pre: {} Add: {clean} Del: {quite, dirty} 80

81 Dinner example: rule1 action mutex Actions (including all No-OP actions) Cook: Pre: {H} Add: {D} Del: {} Wrap: Pre: {Q} Add: {P} Del: {} Tidy: Pre: {} Add: {C} Del: {H,R} Vac: Pre: {} Add: {C} Del: {Q, R} NO(C): Pre: {C} Add: {C} Del: {} NO(D): Pre: {D} Add: {D} Del: {} NO(H): Pre: {H} Add: {H} Del: {} NO(P): Pre: {P} Add: {P} Del: {} NO(Q): Pre: {Q} Add: {Q} Del: {} NO(R): Pre: {R} Add: {R} Del: {} Legend: NO:No-Op, C:clean, D: Dinner, H: cleanhands, P:Present, Q:Quiet, R: dirty Look at those with non-empty Del, and find others that have these Del in their Pre or Add: So, Rule 1 action mutex are as follows (these are fixed): (Tidy,Cook), (Tidy, NO(H)), (Tidy, NO(R)), (Vac, Wrap), (Vac,NO(Q)), (Vac, NO(R)) Rule 3 action mutex depend on state layer and you have to build the graph. 81

82 Dinner Example: Legend: Arrows: Blue: pre, Green: add, Red: Del, Black: Mutex D: Dinner, C:clean, H: cleanhands, Q:quiet, P:Present, R: dirty Init={R,H,Q} Goal={D,P,C} Cook Wrap D P Note: At layer S1 all goals are present and no pair forms a mutex Tidy c So, go to phase II and search the graph: R H Q Vac NO(R) NO(H) NO(Q) R H Q i.e. Find a set of non-mutex actions that adds all goals {D,P,C}: x{cook, Wrap, Tidy} mutex Tidy&Cook x{cook, Wrap, Vac} mutex Vac&Wrap S0 A0 S1 No such set exists, nothing to backtrack, so goto phase I and add one more action and state layers 82

83 Dinner Example: R H Q Arrows: Blue: pre, Green: add, Red: Del, Black: Mutex D: Dinner, C:clean, H: cleanhands, Q:quiet, P:Present, R: dirty Init={R,H,Q} Goal={D,P,C} Cook Wrap Tidy Vac NO(R) NO(H) NO(Q) D P c R H Q Cook Wrap Tidy Vac NO(D) NO(P) NO(C) NO(R) NO(H) NO(Q) S0 A0 S1 A1 S2 D P c R H Q At layer A1, first draw rule1 action mutex,, then find rule3 action mutex (for this only look at mutex fact at level S1). At S2, apply rule 2 for fact mutex.. At layer S2 all goals are present and no pair forms a mutex,, so PhaseII phase II: Find a set of non- mutex actions that adds all goals {D,P,C}: x{cook, Wrap, Tidy} x{cook, Wrap, Vac} {Cook, Wrap, NO(C)} S1 Cannot find any non-mutex action set in A0 Backtrack to S2, try another action set {Cook, NO(P), Vac} 83

84 Dinner Example: R H Q Arrows: Blue: pre, Green: add, Red: Del, Black: Mutex D: Dinner, C:clean, H: cleanhands, Q:quiet, P:Present, R: dirty Init={R,H,Q} Goal={D,P,C} Cook Wrap Tidy Vac NO(R) NO(H) NO(Q) D P c R H Q Cook Wrap Tidy Vac NO(D) NO(P) NO(C) NO(R) NO(H) NO(Q) S0 A0 S1 A1 S2 D P c R H Q so try {Cook, NO(P), Vac} S1 Find a nonmutex set of act in A0 to get NewG: {NO(H),Wrap} NewG S0 Done! So, the actions are Wrap, {Cook, Vac}: Wrap,Cook,Vac Wrap,Vac,Cook Note that we could still backtrack to S2, try remaining action sets! 84

85 Second Approach: Planning As Search Planning can be treated as a search problem. The initial CW-KB is the initial state. The actions are operators mapping a state (i.e. a CW-KB) to a new state (an updated CW-KB). The goal is satisfied by any state (CW-KB) that satisfies the goal. 85

86 Example. move(b,c) B C A C A B move(c,b) A C B move(c,table) A B C move(a,b) B A C 86

87 Problems Search tree is generally quite large randomly reconfiguring 9 blocks takes thousands of CPU seconds. The representation suggests some structure. Each action only affects a small set of facts, actions depend on each other via their preconditions. Planning algorithms are designed to take advantage of the special nature of the representation. 87

88 Planning as Heuristic Search We look at heuristics computed from Relaxed Plans : The idea is to consider what happens if we ignore the delete lists of actions. This is yields a relaxed problem that can produce a useful heuristic estimate 88

89 Finding Relaxed Plans In the relaxed problem actions add new facts, but never delete facts. Given the relaxed planning problem we use GraphPlan to build a plan for solving the relaxed problem. As we will see, this will be very easy. We then take the number of steps in the GraphPlan plan as a heuristic estimate of the distance from the current state to the goal This is the method used by the Fast Forward System (FF) you are employing in your assignment. It turns out that this heuristic is not admissible (will see). It is possible to compute admissible heuristics, but in general they don t perform as well: they force the planner to find optimal plans. 89

90 GraphPlan with no delete lists Recall: 1. Two actions are mutex if either action deletes a precondition or add effect of another 2. Two propositions p and q are mutex if all actions adding p are mutex of all actions adding q 3. Two actions are mutex if two of their preconditions are mutex With no deletes in the actions, rule #1 can never apply. With no action mutex, rule #2 cannot apply, and thus rule #3 can never apply. So: with no deletes we never need to propagate mutexes. We also do not need to keep track of noop actions (as these were only used to create mutexes). 90

91 GraphPlan with no delete lists Graph Plan procedure on relaxed plans is then simplified to the following: 1. We start with the initial state S We alternate between state and action layers. 3. We find all actions whose preconditions are contained in S 0. These actions comprise the first action layer A The next state layer consists of all of S 0 as well as the adds of all of the actions in A In general A i is the set of actions whose preconditions are contained in S i. S i+1 is S i union the add lists of all of the actions in A i. 91

92 Example a b c d a b c a d d on(a,b), on(b,c), ontable(c), ontable(d), clear(a), clear(d), handempty unstack(a,b) pickup(d) S 0 A 0 on(a,b), on(b,c), ontable(c), ontable(d), clear(a), handempty, clear(d), holding(a), clear(b), holding(d) S 1 Note: this is not a state! 92

93 Example on(a,b), on(b,c), ontable(c), ontable(d), clear(a), clear(d), handempty, holding(a), clear(b), holding(d) S 1 putdown(a), putdown(d), stack(a,b), stack(a,a), stack(d,b), stack(d,a), pickup(d), unstack(b,c) A 1 93

94 Reachability We continue until the goal G is contained in the state layer, or until the state layer no longer changes. Intuitively, the actions at level A i are the actions that could be executed at the i-th step of some plan, and the facts in level S i are the facts that could be made true after some i-1 step plan. Some of the actions/facts have this property. But not all! This process corresponds to doing a reachablity analysis. 94

95 Reachability a b c on(a,b), on(b,c), ontable(c), ontable(b), clear(a), clear(c), handempty unstack(a,b) pickup(c) S 0 A 0 on(a,b), on(b,c), ontable(c), ontable(b), clear(a), clear(c), handempty, holding(a), clear(b), holding(c) S 1 stack(c,b) and on(c,b) needs 4 actions A 1 on(c,b), but stack(c,b) cannot be executed after one step 95

96 Heuristics from Reachability Analysis Grow the levels until the goal is contained in the final state level S[K]. If the state level stops changing and the goal is not present. The goal is unachievable. (The goal is a set of positive facts, and in STRIPS all preconditions are positive facts). Now, we search the graph to extract a plan (i.e. Phase II of the Graph Plan process). 96

97 Recall Phase II of GraphPlan Solve(G,K): forall sets of actions A={a i } such that no pair (a i, a j ) A is mutex and the actions in A suffice to add all facts in G Let P = union of preconditions of actions in A If Solve(P,K-1) Report PLAN FOUND At end of forall. Exhausted all possible action sets A Report NOPLAN Relaxed plan graph contains no mutexes. So the recursive call to Solve(P,K-1) will never fail. Additionally since we don t have no-ops in the graph, we need to change the test suffice to add all facts in G (some facts were previously created at earlier levels and added by no-ops. Without the no-ops we have to delay these facts. 97

98 Heuristics from Reachability Analysis CountActions(G,S K ): /* Compute the number of actions contained in a relaxed plan achieving the goal. */ Split G into facts in S K-1 and elements in S K only. These sets are the previously achieved and just achieved parts of G. Find a minimal set of actions A whose add-effects cover the just achieved part of G. (The set contains no redundant actions, but it might not be the minimum sized set.) Replace the just achieved part of G with the preconditions of A, call this updated G, NewG. Now return CountAction(NewG,S K-1 ) + number of actions needed to cover the just achieved part of G. 98

99 Example legend: [pre]act[add] S 0 = {f 1, f 2, f 3 } A 0 = {[f 1 ]a 1 [f 4 ], [f 2 ]a 2 [f 5 ]} S 1 = {f 1,f 2,f 3,f 4,f 5 } A 1 = {[f 2,f 4,f 5 ]a 3 [f 6 ]} S 2 ={f 1,f 2,f 3,f 4,f 5,f 6 } G = {f 6,f 5, f 1 } We split G into G P and G N : CountActs(G,S 2 ) G P ={f 5, f } //already in S1 1 G N = {f 6 } //New in S2 A = {a 3 } //adds all in G N //the new goal: G P Pre(A) G 1 = {f 5,f 1,f 2,f 4 } Return 1 + CountActs(G 1,S 1 ) 99

100 Example Now, we are at level S1 S 0 = {f 1, f 2, f 3 } A 0 = {[f 1 ]a 1 [f 4 ], [f 2 ]a 2 [f 5 ]} S 1 = {f 1,f 2,f 3,f 4,f 5 } A 1 = {[f 2,f 4,f 5 ]a 3 [f 6 ]} S 2 ={f 1,f 2,f 3,f 4,f 5,f 6 } G 1 = {f 5,f 1,f 2,f 4 } We split G1 into G P and G N : CountActs(G 1,S 1 ) G P ={f 1,f 2 } //already in S0 G N = {f 4,f 5 } //New in S1 A = {a 1,a 2 } //adds all in G N //the new goal: G P Pre(A) G 2 = {f 1,f } 2 Return 2 + CountActs(G 2,S 0 ) = So, in total CountActs(G,S2)=1+2=3 100

101 Using the Heuristic 1. To use CountActions as a heuristic, we build a layered structure from a state S that reaches the goal. 2. Then we CountActions to see how many actions are required in a relaxed plan. 3. We use this count as our heuristic estimate of the distance of S to the goal. 4. This heuristic tends to work better as a best-first search, i.e., when the cost of getting to the current state is ignored. 101

102 Admissibility An optimal length plan in the relaxed problem (actions have no deletes) will be a lower bound on the optimal length of a plan in the real problem. However, CountActions does NOT compute the length of the optimal relaxed plan. The choice of which action set to use to achieve G P ( just achieved part of G ) is not necessarily optimal. In fact it is NP-Hard to compute the optimal length plan even in the relaxed plan space. So CountActions will not be admissible. 102

103 Empirically However, empirically refinements of CountActions performs very well on a number of sample planning domains. 103