1. An Introduction to Planning Graph
2019/6/2
An Introduction to Planning Graph
Chang, Han-Wen
A. Blum and M. Furst, "Fast Planning Through Planning Graph Analysis", Artificial Intelligence, 90: (1997)
AIMA textbook, Chap. 11, Section 11.4
March 29, 2007

2. 2019/6/2
Planning Problem
Planning is to come up with a sequence of actions that will achieve a goal from the initial state.

3. Representation states: conjunction of positive literals actions:
ground and function-free first-order literals
closed-world assumption
actions:
action name and parameter list
precondition
add-effect
delete-effect
no new object created

4. Rocket Example Cargo (A); Cargo (B); Rocket (R); Place (L);
Place (P); R A B
move (Rocket ?r, Place ?from, Place ?to)
Precond: At (?r, ?from) & HasFuel (?r)
Add: At (?r, ?to)
Delete: At (?r, ?from) & HasFuel (?r)
load (Rocket ?r, Place ?p, Cargo ?c)
Precond: At (?r, ?p) & At (?c, ?p)
Add: In (?c, ?r)
Delete: At (?c, ?p)
unload (Rocket ?r, Place ?p, Cargo ?c)
Precond: At (?r, ?p) & In (?c, ?r)
Add: At (?c, ?p)
Delete: In (?c, ?r)
Place L
Init:
At (A, L) & At (B, L) &
At (R, L) & HasFuel (R) B A
Goal:
At (A, P) & At (B, P)
Place P

5. Motivation
Search performance depends on branching factor, and constraints reduce the search space.
Independent actions can be done in any order.

6. Basic Idea
Construct a graph that encodes constraints on possible plans
Use this “planning graph” to constrain search for a valid plan:
If valid plan exists, it is a subgraph of the planning graph

7. Planning Graph Directed and Leveled Nodes Edges Proposition nodes
Action nodes
Edges
Precondition edges: from propositions to actions
Add edges: from actions to propositions
Del edges: from actions to propositions
No-op edges: from propositions to propositions

8. Graph Levels Alternate Levels
Proposition level: all propositions that could be true at time step t
Action level: all actions that could have their preconditions satisfied at time step t

9. Extending Planning Graph
Load R, L, B
Load R, L, A
At B, L
At B, L
At A, L
At A, L
In B, R
In A, R
At R, L
At R, L
Fuel R
Fuel R
At R, P
Move R, L, P

10. Rocket Example Precondition edges Add-effect edges Delete-effect edges
2019/6/2
Rocket Example
Add-effect edges
Delete-effect edges
Load R, L, B
Load R, L, A
Move R, L, P
Unload R, L, A
Unload R, L, B
Actions
Time 1
Load R, L, B
Load R, L, A
Move R, L, P
Unload R, L, A
Unload R, L, B
Unload R, P, A
Unload R, P, B
Actions
Time 2
No-op edges
Load R, L, B
Load R, L, A
Move R, L, P
Actions
Time 0
At B, L
At A, L
At R, L
Fuel R
Propositions
Time 0
At B, L
At A, L
At R, L
Fuel R
At R, P
In B, R
In A, R
Propositions
Time 1
At B, L
At A, L
At R, L
Fuel R
At R, P
In B, R
In A, R
Propositions
Time 2
At B, L
At A, L
At R, L
Fuel R
At R, P
In B, R
In A, R
At B, P
At A, P
Propositions
Time 3

11. Mutual Exclusions (mutex)
Inconsistent Effects
Interference
Competing Needs
Inconsistent Support

12. Inconsistent Effects (mutex)
The action deletes an add-effect of the other.
Load(R, L, A) deletes At(L, A) which is an add-effect of Unload(R, L, A), so the two actions are mutex.

13. Interference (mutex) The action deletes a precondition of the other.
Move(R, L, P) deletes At(R, L) which is an precondition of Load(R, L, A), so the two actions are mutex.

14. Competing Needs (mutex)
If there is a precondition p of action a and a precondition q of action b that are mutex in the previous proposition level, the two actions are mutex.
The precondition At(A, L) of action Load(R, A, L) and the precondition At(A, P) of action Load(R, A, P) are mutex, so the two actions are mutex.

15. Inconsistent Support (mutex)
If each action a having an add-effect of proposition p is marked as exclusive of each action b having an add-effect of proposition q, the two propositions are mutex.
The proposition At(A, L) and the proposition In(A, R) are mutex at time step t if the are mutex at time step t-1, and any action creates At(A, L) are mutex with any action creates In(A, R).

16. Spare Tire Example (AIMA)

17. GraphPlan Algorithm
function GRAPHPLAN(problem) return solution or failure
graph  INITIAL-PLANNING-GRAPH(problem)
goals  GOALS[problem]
loop do
if goals all non-mutex in last level of graph then do
solution  EXTRACT-SOLUTION(graph, goals, LENGTH(graph))
if solution  failure then return solution
else if NO-SOLUTION-POSSIBLE(graph) then return failure
graph  EXPAND-GRAPH(graph, problem)

18. Plan Extraction Valid plan Backward chaining
goals are satisfied
Non-mutex actions
Backward chaining
Achieve goals level by level
Non-mutex actions at level k
Preconditions as the goals for level k-1
No Non-mutex action found  backtrack

19. Features Literals increase monotonically
Actions increase monotonically
Mutexes decrease monotonically
Eventually level off
two consecutive levels are identical

20. Termination planning graph eventually leveled-off
If the graph is leveled-off and some literals of the goal do not appear or are marked as mutex in the latest proposition level, the problem is unsolvable.

21. Advanced Test
Let Sti be the collection of unachievable (sub)goal-sets stored for level i after trial at stage t
If graph leveled off at level n and St-1n = Stn at a stage t > n, then output “No Plan Exists”

22. Remarks on Planning Graph
2019/6/2
Remarks on Planning Graph
Polynomial space / graph creation time
p: |initial state|
n: #object
m: #operator
t: #level
l: max( #add-list )
k: max( #operator parameter )
#Max nodes action level: O(mnk)
#Max nodes proposition level: O(p+mlnk)

23. More Remarks Sound & complete Partially-ordered planning
Independent actions in the same level can be executed in any order

24. Pro and Con Cases with better performance
pairwise mutex relations capture important constraints
parallel actions reduce the depth of the graph

25. Thanks

26. Eat Cake Example (AIMA)
similar to drink water example

27. Spare Tire Example (AIMA)

28. Rocket Example Cargo (A); Cargo (B); Rocket (R); Place (L);
Place (P); R A B
move (Rocket ?r, Place ?from, Place ?to)
Precond: At (?r, ?from) & HasFuel (?r)
Add: At (?r, ?to)
Delete: At (?r, ?from) & HasFuel (?r)
load (Rocket ?r, Place ?p, Cargo ?c)
Precond: At (?r, ?p) & At (?c, ?p)
Add: In (?c, ?r)
Delete: At (?c, ?p)
unload (Rocket ?r, Place ?p, Cargo ?c)
Precond: At (?r, ?p) & In (?c, ?r)
Add: At (?c, ?p)
Delete: In (?c, ?r)
Place L
Init:
At (A, L) & At (B, L) &
At (R, L) & HasFuel (R) B A
Goal:
At (A, P) & At (B, P)
Place P

29. Rocket Example Precondition edges Add-effect edges Delete-effect edges
2019/6/2
Rocket Example
Add-effect edges
Delete-effect edges
Load R, L, B
Load R, L, A
Move R, L, P
Unload R, L, A
Unload R, L, B
Actions
Time 1
Load R, L, B
Load R, L, A
Move R, L, P
Unload R, L, A
Unload R, L, B
Unload R, P, A
Unload R, P, B
Actions
Time 2
No-op edges
Load R, L, B
Load R, L, A
Move R, L, P
Actions
Time 0
At B, L
At A, L
At R, L
Fuel R
Propositions
Time 0
At B, L
At A, L
At R, L
Fuel R
At R, P
In B, R
In A, R
Propositions
Time 1
At B, L
At A, L
At R, L
Fuel R
At R, P
In B, R
In A, R
Propositions
Time 2
At B, L
At A, L
At R, L
Fuel R
At R, P
In B, R
In A, R
At B, P
At A, P
Propositions
Time 3

30. Extending Planning Graph
Load R, L, B
Load R, L, A
At B, L
At B, L
At A, L
At A, L
In B, R
In A, R
At R, L
At R, L
Fuel R
Fuel R
At R, P
Move R, L, P

31. Inconsistent Effects (mutex)
The action deletes an add-effect of the other.
Load(R, L, A) deletes At(L, A) which is an add-effect of Unload(R, L, A), so the two actions are mutex.

32. Interference (mutex) The action deletes a precondition of the other.
Move(R, L, P) deletes At(R, L) which is an precondition of Load(R, L, A), so the two actions are mutex.

33. Competing Needs (mutex)
If there is a precondition p of action a and a precondition q of action b that are mutex in the previous proposition level, the two actions are mutex.
The precondition At(A, L) of action Load(R, A, L) and the precondition At(A, P) of action Load(R, A, P) are mutex, so the two actions are mutex.

34. Inconsistent Support (mutex)
If each action a having an add-effect of proposition p is marked as exclusive of each action b having an add-effect of proposition q, the two propositions are mutex.
The proposition At(A, L) and the proposition In(A, R) are mutex at time step t if the are mutex at time step t-1, and any action creates At(A, L) are mutex with any action creates In(A, R).

35. Drink Water Example literals: Initial Condition: Goal: EmptyCup
FullCup
Thirsty
NotThirsty
Initial Condition:
FullCup & Thirsty
Goal:
NotThirsty & FullCup

36. Drink Water Example -- Actions
FillCup
Preconds: EmptyCup
Add-effs: FullCup
Del-effs: EmptyCup
EmptyCupAction
Preconds: FullCup
Add-effs: EmptyCup
Del-effs: FullCup
Drink
Preconds:
FullCup & Thirsty
Add-effs:
EmptyCup & NotThirsty
Del-effs:

37. Drink Water Planning Graph
EmptyCupAction
EmptyCupAction
FillCup
EmptyCup
EmptyCup
FullCup
FullCup
FullCup
Thirsty
Thirsty
Thirsty
NotThirsty
NotThirsty
Drink
Drink
proposition
time 1
action
time 1
proposition
time 2
action
time 2
proposition
time 3

38. Inconsistent Effects (mutex)
The action deletes an add-effect of the other.
Drink deletes Full_Cup which is an add-effect of Fill_Cup, so the two actions are mutex.

39. Interference (mutex) The action deletes a precondition of the other.
EmptyCupAction deletes FullCup which is an precondition of Drink, so the two actions are mutex.

40. Competing Needs (mutex)
If there is a precondition p of action a and a precondition q of action b that are mutex in the previous proposition level, the two actions are mutex.
The precondition EmptyCup of action FillCup and the precondition FullCup of action Drink are mutex, so the two actions are mutex.

41. Inconsistent Support (mutex)
If each action a having an add-effect of proposition p is marked as exclusive of each action b having an add-effect of proposition q, the two propositions are mutex.
The proposition EmptyCup and the proposition FullCup are mutex at time step t if the are mutex at time step t-1, and any action creates EmptyCup are mutex with any action creates FullCup.