Graphplan

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
Velosos Rocket Problem
C1
R1
C2
San Francisco
R2
C3
R3
St. Louis
Seattle

• Solution can be generalized in 3 steps

6
Velosos Rocket Problem
C1
R1
C2
San Francisco
R2
C3
R3
St. Louis
Seattle

• Step 1 Load all rockets

7
Velosos Rocket Problem
San Francisco
St. Louis
Seattle

• Step 2 Move all rockets

8
Velosos Rocket Problem
San Francisco
St. Louis
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
• Well start with the reachability concept

10
Reachability

• Reachability metric necessary since you have to
  know if a solution state can be reached from s0
• Can be computed w/ reachability graphs, but
  computing them is intractable
• Can be approximated w/ planning graph, but this
  is tractable

11
Reachability Trees

• Consider a simple Blocks World Domain

S0
B
C
A

• Move(x, y, z)
• Precond On(y, x), Clear(x), Clear(z), etc.
• Effects On(z, x), On(y, x), Clear(y), etc.

12
Reachability Trees
B
S0
C
A
Move(B,C,A)
Move(A,table,B)
Move(B,C,table)
etc
A
B
C
A
B
etc
C
Move(A,table,B)
Move(B,table,A)
Move(C,table,A)
Move(A,B,table)
A
B
C
B
B
C
C
A
A
B
C
A
etc
13
Reachability Trees

• Note that a reachability tree down to depth d
  solves all planning problems with s0 and A, for
  every goal that is reachable in d or fewer
  actions
• This blows up into O(kd) 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 s0 were
  modeled as a single state?

B
C
A
Move(A,table,B)
Move(B,C,table)
Move(B,C,A)
A
B
C
A
B
B
C
A
C
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Planning Graph Idea
B
C
A
Move(B,C,table)
A
B
Move(A,table,B)
B
C
A
C
Move(B,C,A)
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 s0 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)
• P0 initial state
• An set of actions whose preconditions are in Pn
• Pn set of propositions that can be true after n
  actions have been performed ie Pn-1 ?
  effects(A1)

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Planning Graphs
Delete edges

• Precondition arcs go from preconditions in Pn to
  associated actions in An
• Add edges indicate positive effects of actions
• Delete edges mark negative effects of actions
• Also define a no-op operator ?pprecond(?p)
  effects(?p) p and effects-(?p) ?
• Note that negative effects are not removed, just
  marked.
• Pn-1 ? Pn persistence principle

?b2
Precondition arcs
Add edges
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P0
A1
P1
B
B
C
A
C
A
Clear(C) On(B, table) On(B, C) On(A,
B) On(A,table) Clear(B) On(B, A) Clear(A) On(C,t
able)
Move(B,C,table)
A
Clear(B) On(B, C) Clear(A) On(A,
table) On(C,table)
B
C
Move(A,table,B)
Move(B,C,A)
B
C
A
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Definitions

• 1) Two actions(a,b) are independent iff
• effects-(a) ? precond(b) ? effects(b) ?
• effects-(b) ? precond(a) ? effects(a) ?

Precond clear(A), clear(B) Effects
on(B,A) Effects- clear(B)
A
Move(A,table,B)
B
B
C
C
A
Precond clear(B) Effects on(table,B),
clear(C) Effects- none
Move(B,C,table)
B
C
A
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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, ? lt?1, , ?ngt, is
  solution to problem iff
• Each set ?i ? ? is independent
• ?n is applicable to sn
• g ? ?(?(?(s0, ?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

• Cant 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
• µAi mutually exclusive actions in level i
• µPi mutually exclusive propositions in level i

24
Finding Mutex relationships

• Two rules
• Interference if one action deletes a
  precondition of another or deletes a positive
  effect
• Competing Needs if actions a and b have
  preconditions that are marked as mutex in
  previous proposition level

25
Mutex Example

• Mutex by Interference

Precond clear(A), clear(B) Effects
on(B,A) Effects- clear(B)
A
Move(A,table,B)
B
B
C
C
A
Precond clear(B) Effects on(table,B),
clear(C) Effects- none
Move(B,C,table)
B
C
A
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Mutex Example

• Mutex by Competing Needs

A) Load(R1, C2, St Louis) B) Load(R2, C2,
Seattle) ?Mutex because C2 cannot be in St Louis
and Seattle at same time
R1
R2
C2
Seattle
St. Louis
27

• Break

28
Graphplan Algorithm

• Input Proposition level P0 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

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Expansion Algorithm
31
P0
A1
P1
B
B
C
A
Mutex list for Move(B,C,table) -Move(A,table,B) -
Move(B,C,A)
C
A
Clear(C) On(B, table) On(B, C) On(A,
B) On(A,table) Clear(B) On(B, A) Clear(A) On(C,t
able)
Move(B,C,table)
A
Clear(B) On(B, C) Clear(A) On(A,
table) On(C,table)
Mutex list for Move(A,table,B) -Move(B,C,table) -
Move(B,C,A)
B
?
C
Move(A,table,B)
Mutex list for Move(B,C,A) -Move(B,C,table) -Move
(A,table,B)
?
?
?
?
Move(B,C,A)
B
C
A
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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 At become the new
  goals
• Repeat above steps until reaching P0
• 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
D
C
A
E

• Goal state

On(A, table) On(B, A) On(D, B) Clear(D) On(E,
table) On(C, E) Clear(C)
D
B
C
A
E
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P0
P1
A1
A2
P2
Solution (Move(B,C,A),Move(D,E,table),
(Move(C,table,E),Move(D,table,B))
B
D
C
A
E
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,tabl
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,tabl
e) On(D,B)
?
Move(B,C,D)
Clear(B) On(B, C) Clear(A) On(A,
table) On(C,table) On(E, table) On(D,E) Clear(D)
Move(C,table,E)
Move(B,C,A)
?
?
?
?
?
Move(B,C,table)
?
Move(D,table,B)
Move(D,E,table)

Move(D,E,A)

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Monotonicity Property

• Recall persistence principle Since negative
  effects are never removed, and for ?
  precond(?p) effects(?p) p? Pn-1 ? Pn,
  propositions monotonically increase
• ? Similarly, An-1 ? An, actions monotonically
  increase

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Unsolvable problems

• Due to monotonic property of planning graphs,
  Pn-1 ? Pn, and An-1 ? An
• At some point, all possible propositions will
  have been explored, thus PnPnk for all kgt0
• Graph has leveled off (also called Fixedpoint
  in book)
• If you reach a proposition level thats 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
General Syntax

• (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))

(variable_name variable_type) () (preconds
(literal_name variable_name1 variable_name2
) () ) (effects (literal_name
variable_name1 variable_name2 ) () )
Things (operands) in the domain
Initial state
Goal State
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Sample Operations File

• (operator
• PICK-UP
• (params (ltob1gt OBJECT))
• (preconds
• (clear ltob1gt) (on-table ltob1gt) (arm-empty))
• (effects
• (holding ltob1gt)))
• (operator
• STACK
• (params (ltobgt OBJECT) (ltunderobgt OBJECT))
• (preconds
• (clear ltunderobgt) (holding ltobgt))
• (effects
• (arm-empty) (clear ltobgt) (on ltobgt
  ltunderobgt)))

(operator Operator_name (params (ltop1gt
ltop_typegt)) (preconds (literal ltop1gt ltop2gt
) () ) (effects (literal ltop1gt ltop2gt
) () ) )
General Syntax
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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))
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More Samples Rocket Ops

• (operator
• LOAD
• (params
• (ltobjectgt CARGO) (ltrocketgt ROCKET) (ltplacegt
  PLACE))
• (preconds
• (at ltrocketgt ltplacegt) (at ltobjectgt ltplacegt))
• (effects
• (in ltobjectgt ltrocketgt) (del at ltobjectgt
  ltplacegt)))
• (operator
• UNLOAD
• (params
• (ltobjectgt CARGO) (ltrocketgt ROCKET) (ltplacegt
  PLACE))
• (preconds
• (at ltrocketgt ltplacegt) (in ltobjectgt ltrocketgt))
• (effects
• (at ltobjectgt ltplacegt) (del in ltobjectgt
  ltrocketgt)))

(operator MOVE (params (ltrocketgt ROCKET)
(ltfromgt PLACE) (lttogt PLACE)) (preconds
(has-fuel ltrocketgt) (at ltrocketgt ltfromgt))
(effects (at ltrocketgt lttogt) (del has-fuel
ltrocketgt) (del at ltrocketgt ltfromgt)))
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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/gr
  aphplan
• 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

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• Graphplan System Live Demo

48
Contact

• Trouble running Graphplan? Email me
• jhs4(at)lehigh.edu