Artificial Intelligence. The Planning Problem. Typical Assumptions. Planning. Given: Find

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1 rtificial Intelligence Planning 1 The Planning Problem Given: set of oerators n initial state descrition goal state descrition or redicate Find sequence of oerator instances such that erforming them in order from the initial state will modify the state to achieve the goal. 2 Tyical ssumtions Each action is indivisible atomic. No concurrent actions allowed. ctions are deterministic i.e. no uncertainty in outcome. gent is sole cause of change to world. gent is omniscient. losed World ssumtion Everything known to be true is included in state descrition. Otherwise it s false. 3 1

2 Possible roaches Situation alculus: ugment FOL to reason about actions in time. dd situation variables. Define redicates as a function of situation. dd new function resultas that returns new situation given action and current situation. Examle: Definition of action agent-walks-to-location. x y s atgentxs traedgents atgentyresultwalkys 4 Possible roaches State-Sace Search: We already need an initial state. Goal test checks to see if goal state is achieved. Successor function based on set of oerators. Once goal is found the lan is simly the sequence of oerators on ath from initial state to goal state. Unfortunately this aroach relies totally on algorithm and ignores information inherent in state es. goal and oerator descritions. 5 Oening States/Oerators State/situation reresentation is a conjunction of ground literals i.e. facts with no variables. Goal is a state where all literals are ositive i.e. all true. STRIPS Oerators have three arts: Oerator/action name Preconditions: what needs to be true Effects: what is now true ND false 6 2

holdingx ontablex clearx handemty 7 ction Descrition Language lternative method for reresenting actions.

3 Examle Oerator Descrition Suose we want to describe an oerator that icks u an object. Name: ickux Preconditions: ontablex clearx handemty dd List: holdingx Delete List: ontablex clearx handemty lternatively effects could be Effect List: holdingx ontablex clearx handemty 7 ction Descrition Language lternative method for reresenting actions. States include both ositive and negative literals oen world assumtion. Quantifiers and disjunction now allowed in goals. Equality redicate built in. Variables can be tyed reduces search sace. 8 Examle: Moving a Piano 1 Goal: Move the iano through the window. 9 3

ction c In a c a irort Plane c argo a a c a c Load ction SFO JFK Goal SFO irort JFK irort P Plane P Plane argo argo JFK P SFO P JFK SFO Init 12 Examle: Sare Tire EFFET : PREOND : EFFET : PREOND

4 4 10 Examle: Moving a Piano 2 Goal: Move the iano through the window. 11 Examle: ir argo EFFET : PREOND : EFFET : PREOND : EFFET : PREOND : to from to irort from irort Plane from from to Fly ction c In a c a irort Plane c argo a c In a c Unload ction c In a c a irort Plane c argo a a c a c Load ction SFO JFK Goal SFO irort JFK irort P Plane P Plane argo argo JFK P SFO P JFK SFO Init 12 Examle: Sare Tire EFFET : PREOND : EFFET : PREOND : EFFET : PREOND : EFFET : PREOND : xle Flat Ground Flat Trunk Sare xle Sare Ground Sare LeaveOvernight ction xle Sare Ground Sare xle Flat Ground Sare xle Sare PutOn ction Ground Flat xle Flat xle Flat xle Flat Remove ction Ground Sare Trunk Sare Trunk Sare Trunk Sare Remove ction xle Sare Goal Trunk Sare xle Flat Init

5 Planning as Search Generally lanning is regarded as a search roblem due to the natural descrition. Two main aroaches suggest themselves: Situation-Sace Search: Search sace is sace of all ossible states. Plan is sequence of oerators on ath from start to goal. Plan-Sace Search: Search sace is sace of all ossible lans or artial lans. 13 Situation-Sace Planning Two aroaches Progression Planning Forward chaining Use standard search techniques FS * etc. State-sace search excet with STRIPS oerators Regression Planning ackward chaining Historically More efficient that rogression lanning Must consider re-conditions and add-list of effects ll search methods now benefit from heuristics. 14 STRIPS Uses a goal stack instead of the regression algorithm. Maintains current state throughout. roach: Pick an order for achieving each of the goals. When a goal is oed from stack ush oerator that adds goal followed by its reconditions. Reeat until all reconditions solved. 15 5

6 STRIPS lgorithm rocedure rocedurestripsi G 1 1 G n n OPS OPS ushachieveg ushachieveg 1 1 G n n GoalStack GoalStack do do if ifemtygoalstack then then success success N ogoalstack ogoalstack if ifn of of form form achieveg achieveg 1 1 gg m m then then if ifn true true in ini I continue continue if if N reviously reviously tried tried fail fail Mark Mark N attemted attemted ushn ushn GoalStack GoalStack order order N s N s goals goals gg i i ushachieveg ushachieveg i i GoalStack GoalStack in in order order else else if if N of of form form achieveg achieveg then then if if N true true in in I I continue continue choose choose oerator oerator O from from OPS OPS that that may may add add gg if if none none fail fail choose choose unifyo unifyo I I so so O adds adds gg ushalyo ushalyo GoalStack GoalStack ushachievereconditionso GoalStack GoalStack else else if if N of of form form alyo alyo then theni I alyo alyo I I 16 Examle Oerators ickux P: ontablex clearx handemty E: holdingx ontablex clearx handemty utdownx P: holdingx E: ontablex clearx handemty holdingx stackxy P: holdingx cleary E: onxy clearx handemty holdingx cleary unstackxy P: clearx onxy handemty E: holdingx cleary clearx onxy handemty 17 Examle Initial State: clear clear on ontable ontable handemty 18 6

7 Examle Goal State: on on 19 State: clear clear on ontable ontable handemty Plan: Ste 1 GoalStack: achieveonon 20 State: clear clear on ontable ontable handemty Plan: Ste 2 GoalStack: achieveon achieveon achieveonon Subgoals 21 7

8 8 22 Ste 3 State: clear clear on ontable ontable handemty GoalStack: achieveclearholding alystack achieveon achieveonon oerator reconditions Plan: 23 Ste 4 State: clear clear on ontable ontable handemty GoalStack: achieveholding achieveclear achieveclearholding alystack achieveon achieveonon Subgoals Plan: 24 Ste 5 State: clear clear on ontable ontable handemty GoalStack: achievehandemtyclearony alyunstacky achieveclear achieveclearholding alystack achieveon achieveonon Plan:

9 9 25 Ste 6 State: clear ontable ontable holding clear GoalStack: achieveclear achieveclearholding alystack achieveon achieveonon Plan: Unstack 26 Ste 7 State: ontable ontable clear on clear handemty GoalStack: achieveon achieveonon Plan: Unstack Stack 27 Ste 8 State: ontable ontable clear on clear handemty GoalStack: achieveclearholding alystack achieveonon Plan: Unstack Stack

10 10 28 Ste 9 State: ontable ontable clear on clear handemty GoalStack: achieveholding achieveclear achieveclearholding alystack achieveonon Plan: Unstack Stack 29 Ste 10 State: ontable ontable clear on clear handemty GoalStack: achieveontableclearhandemty alyicku achieveclear achieveclearholding alystack achieveonon Plan: Unstack Stack 30 Ste 11 State: ontable on clear holding GoalStack: achieveclear achieveclearholding alystack achieveonon Plan: Unstack Stack Picku

11 State: Plan: ontable Unstack on Stack on Picku clear Stack handemty Ste 12 GoalStack: achieveonon 31 State: ontable on on clear handemty Plan: Unstack Stack Picku Stack GoalStack: Ste 13 DONE 32 Progression Planning Initial State: The initial state for lanning corresonds to the initial search state. ctions: lication actions are those whose reconditions are met by the current state. Successor state results from alying EFFETS list. Goal Test: hecks whether goal state is achieved. Ste ost: Tyically unit cost but any cost function can be used. ny comlete search algorithm can be used to solve

12 Irrelevant ctions Forward rogression lanning considers all alicable actions from a state. Many of these actions are likely to be irrelevant for the roblem. Potentially high branching factor can bog down search. an imrove the situation with a good heuristic as in any informed search. 34 ranching Factor onsider task get milk bananas and cordless drill. Progression lanning tends to have trouble. Start Pet Store School Market Parrot Dog lass Slee Read Tuna Milk Finish Sit Read 35 Regression Planning Hels with the irrelevant action roblem. Only relevant actions considered based on definition: n action is relevant to a conjunctive goal if it achieves one of the conjuncts of the goal. Regression involves matching EFFETS in rules satisfying goals thus making relevant. First identify states from which alying an action will yield goal or goal conjunct. dd new actions reconditions to goal descrition. Delete ositive effects from goal descrition. onsistent actions do not undo any desired literal

13 Goal Interaction Most lanners assume goals to be achieved are indeendent. Sometimes this is not true. Resulting interaction of goals can result in infinite loos or the inability to create a lan. 37 Sussman nomaly Initial State Goal State Solving on will be undone when solving on Solving on will be undone when solving on 38 Heuristics oth rogression and regression lanning benefit greatly from good heuristics. Neither rogression nor regression lanning are efficient without heuristics. In fact lanning is PSPE-comlete unless limited to only ositive reconditions and single effects. Recent advances in defining heuristics have enabled many ractical lanning roblems to be solved

14 Relaxation Note that we have exlicit reresentations of both reconditions and effects. We will modify these to derive a simler lanning roblem. One aroach remove all reconditions from actions i.e. make all actions alicable at all times. hn = # unsatisfied subgoals? Not quite. Negative interactions one action canceling out another. Single actions may achieve multile goals. 40 Subgoal Indeendence ssume ure divide-and-conquer will work Estimate cost of solving as sum of cost of solving each subgoal indeendently. Note that this can be either otimistic or essimistic. Otimistic when there are negative interactions between sublans that must be corrected. Pessimistic when sublans contain redundant actions e.g. actions from two sublans could be relaced with another single action. 41 Minimal Set over First relax the roblem by ignoring reconditions. Then further relax the roblem by removing negative effects. Finally count the minimum number of actions required such that the union of the actions ositive effects satisfies the goal. Goal ctionxeffet: P ctionyeffet: Q ctionzeffet: P Q Minimal set cover given by {X Y} =

15 Emty Delete List an also relax roblems by removing negative effects without removing reconditions. Examle: If action has effect it just has effect in relaxed roblem. Negative interactions no longer matter since no action can delete literals achieved by another action. Requires solving the new lanning roblem. Heuristic is just number of stes to solve the roblem. In ractice this aroach is usually worthwhile. 43 Total Order vs Partial Order Total-Order Planner: a lanner that maintains a artial solution as a totallyordered list of stes found so far. lso known as a linear lanner. Partial-Order Planner: a lanner that only reresents artial-order constraints on stes. lso known as a non-linear lanner. STRIPS is a total-order lanner. 44 Princile of Least ommitment Never make a choice unless required to do so. In lanning never order the stes unless the order is required. Thus ordering constraints define order of stes in a lan. Leads to artial-order lanning

16 Reresenting Partial-Order Plans Reresented as a grah. Each node is a ste in the lan. Each arc reresents a temoral constraint between two stes. S1 S2 S5 S3 S4 46 Examle S1 S2 S5 S3 S4 Reresents three total-order lans: 1. [S1 S2 S3 S4 S5] 2. [S1 S3 S2 S4 S5] 3. [S1 S3 S4 S2 S5] 47 Examle 2 Partial Order Plan On On Start Finish On On 48 16

17 Examle 2 Total Order Plans Start Finish Start Finish Start Finish Start Finish Start Finish Start Finish 49 omonents Each artial order lan has the following comonents: set of actions that make u the lan. set of ordering constraints of the form before where and are actions. Note this does not require to be immediately before. set of causal links of the form achieves for where is a redicate that is now satisfied by erforming action. Note that is also a recondition for that is now satisfied. Note: Once is satisfied a new action cannot be added that conflicts with this causal link. set of oen reconditions i.e. reconditions not achieved by some action in the lan. 50 roach Search in lan-sace. Start node in lan-sace consists of a lan with two connected seudo-nodes. Start P: None E: all ositive literals defining initial state Finish P: literals defining the conjunctive goal E: None 51 17

18 Searching in Plan-Sace There are two main reasons why a lan may not be a solution. Unsatisfied goal There is a goal or subgoal that is not satisfied by the current lan stes. Possible threat lan ste could cause the undoing of a needed goal if that ste is done at the wrong time. Define lan modification oerators to detect and fix these roblems. 52 Examle Setting the Table Goal: {ontablecloth outglasses outplates outsilver} Initial State: cleartable Oerators lay-tablecloth P: cleartable E: ontablecloth cleartable ut-outx P: None E: outx cleartable 53 Ste 1 Start Finish 54 18

19 Ste 2 Insert the 4 unsolved goals in Finish Start S1: lay-tablecloth Finish S2: ut-outglasses S3: ut-outplates S4: ut-outsilver 55 Threat Precondition to S1 is cleartable. S2 S3 and S4 do not have this as a recondition. Performing S2 S3 or S4 before S1 would revent S1 from ever being erformed. Need to add a temoral constraint to revent this from haening. 56 Ste 3 dd temoral constraints Start S1: lay-tablecloth Finish S2: ut-outglasses S3: ut-outplates S4: ut-outsilver ny total order satisfying above is legal; therefore this is a comlete lan

20 Partial-Order Planning lgorithm function POPinitial goal goal oerators returns lan lan lan lan MakeMinimalPlaninitial goal goal do do if if Solution?lan return lan lan S need need re-s SelectSubgoallan hooseolan oerators S need need re-s ResolveThreatslan end_do 58 Partial-Order Planning lgorithm function SelectSubgoallan returns S need need re-s ick icka lan lan ste ste S need from need from Steslan with with a recondition re-s not not yet yet achieved return S need need re-s 59 Partial-Order Planning lgorithm rocedure hooseolan oerators S need need re-s choose ste ste S add from add from oerators or or Steslan that that has has re-s as as an an effect. if if no no such such ste ste fail fail add add link link S add -re-s-> add S need to need to Linkslan add add constraint S add < add S need to need to Orderingslan if if S add is add is newly added ste ste from from oerators add add S add to add to Steslan add add Start Start < S add < add Finish to to Orderingslan end_if 60 20

21 Partial-Order Planning lgorithm rocedure ResolveThreatslan for for each each S threat threatening threat link link S i -re-s-> i S j j in in Linkslan do do choose either Demote: dd dd S threat < threat S i to i to Orderingslan Promote: dd dd S j < j S threat to threat to Orderingslan if if not not onsistentlan fail fail end_do 61 hanging a Flat Tire InitFlatxle SareTrunk GoalSarexle ctionremovesaretrunk PREOND: SareTrunk EFFET: SareTrunk SareGround ctionremoveflatxle PREOND: Flatxle EFFET: Flatxle FlatGround ctionputonsarexle PREOND: SareGround Flatxle EFFET: SareGround Sarexle ctionleaveovernight PREOND: EFFET: SareGround Sarexle SareTrunk FlatGround Flatxle 62 hanging a Flat Tire Start with the initial lan containing a Start action effects: SareTrunk Flatxle and a Finish action recondition: Sarexle. First ick the only oen recondition namely Sarexle. The only alicable action is PutOnSarexle. Next ick the SareGround recondition of the PutOn action. gain there is only one alicable action RemoveSareTrunk. SareTrunk RemoveSareTrunk Sarexle SareTrunk SareGround Start PutOnSarexle Finish Flatxle Flatxle 63 21

22 hanging a Flat Tire Now ick Flatxle recondition of the PutOn action. Suose we choose LeaveOvernight as the action. The effect includes SareGround but this conflicts with the causal link Sare Ground Remove Sare Trunk PutOn Sare xle Now we add ordering constraint LeaveOvernight before RemoveSareTrunk. 64 hanging a Flat Tire SareTrunk RemoveSareTrunk Sarexle SareTrunk SareGround Start PutOnSarexle Finish Flatxle Flatxle LeaveOvernight Flatxle FlatGround Sarexle SareGround SareTrunk 65 hanging a Flat Tire Only one oen recondition remains SareTrunk which is a recondition of RemoveSareTrunk. Only one action can achieve this the Start action. Unfortunately we have a conflict between the causal link between Start and Remove and the SareTrunk effect of LeaveOvernight. Note we cannot demote the LeaveOvernight action to recede Start and we cannot romote it to follow Remove. This leads to a backtrack to remove LeaveOvernight from the lan altogether. Following in this manner yields the final lan

23 hanging a Flat Tire SareTrunk RemoveSareTrunk Sarexle SareTrunk SareGround Start PutOnSarexle Finish Flatxle Flatxle Flatxle RemoveFlatxle 67 Planning Grahs s we see lanning is a search roblem. We would like to be able to aly search heuristics to imrove the rocess. One aroach is to construct a grah of the search sace and use this to guide search. This leads to the concet of a lanning grah. 68 Planning Grahs Planning grahs are constructed as a sequence of levels. Each level contains a set of literals and a set of actions. The literals are those that could be true at that time ste. The actions are those that could have their reconditions satisfied at that time ste. Planning grahs work only for roositional lanning roblems

24 onstructing Planning Grahs Start with state level S 0 reresenting the roblem s initial state. Insert action level 0 and lace all actions whose reconditions are satisfied in the revious level. onnect each action to its reconditions in S 0 and to its effects in S 1. This introduces new literals into S 1 not found in S 0. ontinue until the grah is filled out. 70 Persistent ctions We need to reresent ersistence like frame axioms. Persistent actions are actions that have the same reconditions and effects. Thus ersistent action has reconditions and effects. These are reresented in lanning grahs with small boxes. 71 Mutual Exclusion articular level shows all actions that could occur in the corresonding state. The level also records conflicts between actions that could revent them from occurring together. These conflicts are reresented through mutual exclusion mutex links. These are shown as gray links in the lanning grah

25 Mutex Links Mutex links exist between two actions at a given level if any of the following holds: Inconsistent effects: One action negates an effect of another action. Interference: One of the effects of an action is the negation of a recondition of another action. ometing needs: One of the reconditions of an action is mutually exclusive with the recondition of another action. Mutex links exist between two literals if one is the negation of the other or if each ossible air of actions that could achieve the literals is mutually exclusive inconsistent suort. 73 GRPHPLN Given a lanning grah it is ossible to extract a lan directly from the grah. The rocess follows in two main stes that alternate in a loo: heck whether all the goal literals are resent in the current level with no mutex links between any air of them. If this holds attemt to extract the lan; otherwise exand the grah by adding the actions for the current level and the state literals for the next level. 74 GRPHPLN function GRPHPLNroblem returns solution or failure grah INITIL-PLNNING-GRPHroblem goals GOLS[roblem] loo do if goals all non-mutex in last level of grah then do solution EXTRT-SOLUTIONgrah goals LENGTHgrah if solution failure then return solution else if NO-SOLUTION-POSSILEgrah then return failure grah EXPND-GRPHgrah roblem 75 25

26 Sare Tire Planning Grah 76 26

CRIKEY - A Temporal Planner Looking at the Integration of Scheduling and Planning

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