1. Graphplan Joe Souto CSE 497: AI Planning Sources: Ch. 6 “Fast Planning through Planning Graph Analysis”, A. Blum & M. Furst

2. Classical Planning Every node is a partial plan Initial plan complete plan for goals

3. Neoclassical Planning Every node in search space is a set of several partial plans So not every action in a node appears in the solution

4. Planning Graph State-space: plan is sequence of actions Plan-space: plan is partially ordered set of actions  Planning graph: sequence of sets of parallel actions ex: ( {a1, a2}, {a3, a4}, {a5, a6, a7} )

5. Veloso’s Rocket Problem St. Louis San Francisco Seattle R1 R2 R3 C1 C2 C3 Solution can be generalized in 3 steps

6. Veloso’s Rocket Problem St. Louis San Francisco Seattle R1 R2 R3 C1 C2 C3 Step 1: Load all rockets

7. Veloso’s Rocket Problem St. Louis San Francisco Seattle Step 2: Move all rockets

8. Veloso’s Rocket Problem St. Louis San Francisco Seattle Step 3: Launch all rockets

9. What does Graphplan do? Explores the problem with a “planning graph” before trying to find a solution plan Uses STRIPS operators, except no negated literals allowed in preconditions or goals Plan-space used ‘least commitment’, but Graphplan uses ‘strong commitments’ Requires reachability analysis: can a state be reached from a given state? Requires disjunctive refinement: method of addressing flaws since multiple conflicting propositions can exist in each state We’ll start with the reachability concept

10. Reachability metric necessary since you have to know if a solution state can be reached from s 0 Can be computed w/ reachability graphs, but computing them is intractable Can be approximated w/ planning graph, but this is tractable Reachability

11. Reachability Trees Consider a simple Blocks World Domain C B A Move(x, y, z) Precond: On(y, x), Clear(x), Clear(z), etc. Effects: On(z, x), ~On(y, x), Clear(y), etc. S0:S0:

12. B CA BCA B C A Move(B,C,table) Move(A,table,B) B CA Move(A,B,table) A BC B CA Move(A,table,B)Move(B,table,A) C AB Move(C,table,A) etc… Reachability Trees S0:S0: Move(B,C,A) etc…

13. Reachability Trees Note that a reachability tree down to depth d solves all planning problems with s 0 and A, for every goal that is reachable in d or fewer actions This blows up into O(k d ) nodes where k = # valid actions, thus we move on to finding reachability with planning graphs Could be improved by making a graph rather than tree, but still intractable since #nodes = #states

14. Planning Graphs What if all the states reachable from s 0 were modeled as a single state? B CA BCA Move(B,C,table) B C A Move(A,table,B) Move(B,C,A) B CA

15. Planning Graph Idea B CA BCA B C A B CA Move(B,C,table) Move(A,table,B) Move(B,C,A)

16. Planning Graphs Planning graph considers an inclusive disjunction of actions from one node to next that contains all the effects of these actions Goal is considered reachable from s 0 only if it appears in some node of the planning graph Graph is of polynomial size and can be built in polynomial time in size of input Since some actions in a disjunction may interfere, we must keep track of incompatible propositions for each set of propositions and incompatible actions for each disjunction of actions

17. Planning Graphs Planning graph = directed layered graph with alternating levels of propositions (P) and actions (A) P 0 = initial state A n = set of actions whose preconditions are in P n P n = set of propositions that can be true after n actions have been performed ie: P n-1  effects + (A 1 )

18. Planning Graphs Precondition arcs go from preconditions in P n to associated actions in A n Add edges indicate positive effects of actions Delete edges mark negative effects of actions Also define a no-op operator  p : precond(  p ) = effects + (  p ) = p and effects - (  p ) =  Note that negative effects are not removed, just marked. P n-1  P n : “persistence principle” Precondition arcs Add edges Delete edges  b2

19. Move(B,C,table) Move(A,table,B) Move(B,C,A) Clear(B) On(B, C) Clear(A) On(A, table) On(C,table) P0P0 BCA B C A B CA B CA Clear(C) On(B, table) On(B, C) On(A, B) On(A,table) Clear(B) On(B, A) Clear(A) On(C,table) A1A1 P1P1

20. Definitions 1) Two actions(a,b) are independent iff: effects - (a)  [precond(b)  effects + (b)] =  effects - (b)  [precond(a)  effects + (a)] =  B CA B C A BCA Move(A,table,B) Move(B,C,table) Precond: clear(A), clear(B) Effects + : on(B,A) Effects - : clear(B) Precond: clear(B) Effects + : on(table,B), clear(C) Effects - : none

21. Definitions 2) A set of independent actions, , is applicable to a state iff precond(  )  s 3) A layered plan is a sequence of sets of actions. A valid plan,  =, is solution to problem iff: Each set  i   is independent  n is applicable to s n g   (…  (  (s 0,  1 ),  2 ) …  n )

22. Note Since planning graph explores results of all possible actions to level n: If a valid plan exists within n steps, that plan is a subgraph of the planning graph Allows you to find plan w/ min number of actions

23. Mutual Exclusion Can’t have 2 simultaneous actions in one level that are dependent Two actions at a given level in planning graph are mutually exclusive (“mutex”) if no valid plan can contain both, or no plan could make both true, ie: they are dependent or they have incompatible preconditions μA i = mutually exclusive actions in level i μP i = mutually exclusive propositions in level i

24. Finding Mutex relationships Two rules: 1. Interference: if one action deletes a precondition of another or deletes a positive effect 2. Competing Needs: if actions a and b have preconditions that are marked as mutex in previous proposition level

25. Mutex Example B CA B C A BCA Move(A,table,B) Move(B,C,table) Precond: clear(A), clear(B) Effects + : on(B,A) Effects - : clear(B) Precond: clear(B) Effects + : on(table,B), clear(C) Effects - : none Mutex by Interference

26. Mutex Example Mutex by Competing Needs St. Louis R1 R2 A) Load(R1, C2, St Louis) B) Load(R2, C2, Seattle)  Mutex because C2 cannot be in St Louis and Seattle at same time C2 Seattle

27. Break

28. Graphplan Algorithm Input: Proposition level P 0 containing initial conditions Output: valid plan or states no valid plan exists Algorithm: while (!done) { Expansion Phase: Expand planning graph to next action and proposition level; Search/Extraction Phase: Search graph for a valid plan; if (valid plan exists) return successful plan; else continue; }  Graphplan is sound and complete

29. Expanding Planning Graphs Create next Action level by iterating through each possible action for each possible instantiation given the preconditions in the previous proposition level, then insert no-ops and precondition edges Create next Proposition level from the Add- Effects of the actions just generated Associated with each action is a list of actions it is mutex with

30. Expansion Algorithm

31. Move(B,C,table) Move(A,table,B) Move(B,C,A) Clear(B) On(B, C) Clear(A) On(A, table) On(C,table) P0P0 BCA B C A B CA B CA Clear(C) On(B, table) On(B, C) On(A, B) On(A,table) Clear(B) On(B, A) Clear(A) On(C,table)  A1A1 P1P1 Mutex list for Move(B,C,table): -Move(A,table,B) -Move(B,C,A)     Mutex list for Move(A,table,B): -Move(B,C,table) -Move(B,C,A) Mutex list for Move(B,C,A): -Move(B,C,table) -Move(A,table,B)

32. Finding Graphplan Solution Solution found via backward chaining Select one goal at time t, find an action at t – 1 achieving this goal Continue recursively with next goal at time t Preconditions of actions in A t become the new goals Repeat above steps until reaching P 0 Performance improved w/ “forward checking”: after each action is considered, Graphplan checks that no goal becomes cut off by this action

33. Planning Graph Solution

34. Extraction Algorithm Optimization: Actions that failed to satisfy certain goals at certain levels are saved in “nogood” hash table ( ▼ ), indexed by level, so when you backtrack you can prevented wasting time examining actions that were not helpful earlier

35. Graphplan Algorithm

36. Algorithm Example Initial state: B CA D E BC A D E Goal state: On(A, table) On(B, A) On(D, B) Clear(D) On(E, table) On(C, E) Clear(C)

37. Move(B,C,table) Clear(B) On(B, C) Clear(A) On(A, table) On(C,table) On(E, table) On(D,E) Clear(D) P0P0 B CA Clear(A) On(B, A) Clear(C) Clear(D) On(C,table) On(B, table) On(B, C) On(E,table) On(A, B) On(A,table) Clear(B) On(D,E) Clear(E) On(D,table) A1A1 P1P1 D E Move(B,C,A) Move(D,E,table) A2A2 P2P2 Move(D,table,B) Move(C,table,E) On(E,table) On(B, A) Clear(C) On(C,E) On(C,table) On(B, table) On(B, C) Clear(D) On(A, B) On(A,table) Clear(B) On(D,E) Clear(E) On(D,table) On(D,B)        Move(B,C,D) Move(D,E,A) … … Solution: ({Move(B,C,A),Move(D,E,table)}, {(Move(C,table,E),Move(D,table,B)})

38. Monotonicity Property Recall persistence principle: Since negative effects are never removed, and for  : precond(  p ) = effects + (  p ) = p  P n-1  P n, propositions monotonically increase  Similarly, A n-1  A n, actions monotonically increase

39. Unsolvable problems Due to monotonic property of planning graphs, P n-1  P n, and A n-1  A n At some point, all possible propositions will have been explored, thus P n =P n+k for all k>0 Graph has “leveled off” (also called “Fixedpoint” in book) If you reach a proposition level that’s identical to the previous level, and all goal conditions are not present and non-mutex, problem is unsolvable Thus Graphplan is complete

40. Graphplan Planning System Two files required to specify a domain Facts file – describe objects in the problem, initial state, and goal state Operations file – describe valid operations in that domain

41. Sample Facts File (blockA OBJECT) (blockB OBJECT) (blockC OBJECT) (blockD OBJECT) (preconds (on-table blockA) (on blockB blockA) (on blockC blockB) (on blockD blockC) (clear blockD) (arm-empty)) (effects (on blockB blockA) (on blockC blockB) (on blockA blockD)) Things (operands) in the domain Initial state Goal State (variable_name variable_type) (…) (preconds (literal_name {variable_name1 variable_name2 …}) (…) ) (effects (literal_name {variable_name1 variable_name2 …}) (…) ) General Syntax

42. Sample Operations File (operator PICK-UP (params ( OBJECT)) (preconds (clear ) (on-table ) (arm-empty)) (effects (holding ))) (operator STACK (params ( OBJECT) ( OBJECT)) (preconds (clear ) (holding )) (effects (arm-empty) (clear ) (on ))) (operator Operator_name (params ( )) (preconds (literal { …}) (…) ) (effects (literal { …}) (…) ) General Syntax

43. More Samples: Rocket Facts (London PLACE) (Paris PLACE) (JFK PLACE) (r1 ROCKET) (r2 ROCKET) (alex CARGO) (jason CARGO) (pencil CARGO) (paper CARGO) (preconds (at r1 London) (at r2 London) (at alex London) (at jason London) (at pencil London) (at paper London) (has-fuel r1) (has-fuel r2)) (effects (at alex Paris) (at jason JFK) (at pencil Paris) (at paper JFK))

44. More Samples: Rocket Ops (operator LOAD (params ( CARGO) ( ROCKET) ( PLACE)) (preconds (at ) (at )) (effects (in ) (del at ))) (operator UNLOAD (params ( CARGO) ( ROCKET) ( PLACE)) (preconds (at ) (in )) (effects (at ) (del in ))) (operator MOVE (params ( ROCKET) ( PLACE) ( PLACE)) (preconds (has-fuel ) (at )) (effects (at ) (del has-fuel ) (del at )))

45. Important Graphplan has no concept of negation. Use propositions with equivalent meaning Ex:  inhand(B)  not-inhand(B) Cannot use _ in any token. Use – instead. Comments: Begin line with ; See README file for more details

46. Running Graphplan Access in my home directory on Suns: /home/jhs4/graphplan Contains executable and sample facts/operations files Execute with:./graphplan.sparc Program prompts for names of operations and fact files at runtime Source for Solaris and Linux in./solaris-src and./linux-src respectively

47. Graphplan System Live Demo

48. Contact Trouble running Graphplan? Email me: jhs4(at)lehigh.edu