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New Mexico State UniversitySon Thanh ToTran Cao Son
Enrico Pontelli
Contingent Planning
as And/Or forward Search
with Disjunctive Representation
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• Contingent planning• Approach overview
Employed DNF representationPrAO: a new And/Or forward search with novel pruning techniques
• DNF representation: review & extension• PrAO And/Or forward search algorithm• Experimental evaluation• Conclusion and future work
Outline
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Contingent planningAn example & formulationAnd/Or search for solutions
• Approach overviewEmployed DNF representationPrAO: an And/Or forward search
• DNF representation: review & extension• And/Or forward search algorithm: PrAO• Experimental evaluation• Conclusion and future work
Outline
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Contingent Planning: An Example
at-same-room
at-same-room
bug-is-dead
bug-is-dead
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• Problem P = F, A, , I, G Fluents: F = {bug-is-dead, at-same-room} Actions: A = {move, kill-bug}
◊ pre(move) = move: at-same-room at-same-roommove: at-same-room at-same-room
◊ pre(kill-bug) = at-same-roomkill-bug: bug-is-dead
Sensing: = {sense-bug}, pre(sense-bug) = ◊ l(sense-bug) = at-same-room
Initial State: I = bug-is-dead (? at-same-room) Goal: G = bug-is-dead
• The initial Belief State: set of states satisfying I:BS(I) = { {bug-is-dead, at-same-room} ,
{bug-is-dead, at-same-room} }
Contingent Planning Problem
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• Contingent planningFormulationAnd/Or search for solutions
• Approach overviewEmployed DNF representationPrAO: an And/Or forward search
• DNF representation: review & extension• And/Or forward search algorithm: PrAO• Experimental evaluation• Conclusion and future work
Outline
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bug-is-dead
at-same-room
N2
bug-is-dead
at-same-room
N1
bug-is-dead
at-same-room
N3 (goal)
bug-is-dead
?at-same-room
N0 (initial)
sense-bug sense-bug
move
kill-bug
move
move
Start node
(initial belief state)
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• Extend DNF representation for conformant planning (ICAPS-2009) to handle Non-deterministic actions Sensing actions
• Develop PrAO: an And/Or forward search with Novel pruning techniques The remaining search graph when a solution
is detected is also the solution
Overview of Our Approach
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Contingent planningApproach overviewDNF representationBrief review (conformant planning)Extending for contingent planing
• And/Or forward search algorithm: PrAO• Experimental evaluation• Conclusion and future work
Outline
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Contingent planning Approach overview•DNF representationReview (conformant planning)Extending for contingent planning
• And/Or forward search algorithm: PrAO• Experimental evaluation• Conclusion and future work
Outline
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• A non-deterministic action a contains a set of outcomes: o1, …, on. (a is deterministic if n = 1)
• oi is a set of conditional effects • ai : the deterministic action with set of effects oi
DNF(a, )
non-det. action: a
Extending to Non-deterministic action
?
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• A non-deterministic action a contains a set of outcomes: o1, …, on. (a is deterministic if n = 1)
• oi is a set of conditional effects • ai : the deterministic action with set of effects oi
DNF(a, )
DNF(a1, )det. action: a1
non-det. action: a
Extending to Non-deterministic action
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• A non-deterministic action a contains a set of outcomes: o1, …, on. (a is deterministic if n = 1)
• oi is a set of conditional effects • ai : the deterministic action with set of effects oi
DNF(a, )
DNF(a1, )
DNF(an, )
…
det. action: a1
det. action: an
non-det. action: a
Extending to Non-deterministic action
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Union of
DNF(a1, ), ...
DNF(an, )
• A non-deterministic action a contains a set of outcomes: o1, …, on. (a is deterministic if n = 1)
• oi is a set of conditional effects • ai : the deterministic action with set of effects oi
DNF(a, )
DNF(a1, )
DNF(an, )
…
det. action: a1
det. action: an
non-det. action: a
Extending to Non-deterministic action
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Union of
DNF(a1, ), ...
DNF(an, )
• A non-deterministic action a contains a set of outcomes: o1, …, on. (a is deterministic if n = 1)
• oi is a set of conditional effects • ai : the deterministic action with set of effects oi
DNF(a, )
DNF(a1, )
DNF(an, )
…
det. action: a1
det. action: an
Remove super
partial states
non-det. action: a
Extending to Non-deterministic action
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• Sensing action : (, S) (S+, S-)
◊ S+ = {s | s S, s l()}◊ S- = {s | s S, s l()}
(, ) (+, -): + S+ - S-
(+ l() - l())
Initially: + = - = For every :
◊ If l() then: + = + { }◊ Else if l() then: - = - { }◊ Else:
+ = + { +} + = { l() }- = - { -} - = {l()}
Intuitively: + - + S+ - S-
Successor states for Sensing Actions
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Contingent planning
Approach overview
DNF representation: review & extension
PrAO: a new And/Or forward search: Search space is a directed graph Novel pruning techniques
• Experimental evaluation
• Conclusion and future work
Outline
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• Node N is goal reachable (GR) if:− Goal node (satisfies the goal), or− Has a goal-reachable or-child, or − Both dual and-children are goal-reachable
Goal Reachability
GR
N
GR
GR
N
GR GR
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bug-is-dead
at-same-room
N2
bug-is-dead
at-same-room
N1
bug-is-dead
at-same-room
N3 (goal)
bug-is-dead
?at-same-room
N0 (initial)
sense-bug sense-bug
move
kill-bug
move
move
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bug-is-dead
at-same-room
N2
bug-is-dead
at-same-room
N1
bug-is-dead
at-same-room
N3 (GR)
bug-is-dead
?at-same-room
N0 (initial)
sense-bug sense-bug
move
kill-bug
move
move
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bug-is-dead
at-same-room
N2
bug-is-dead
at-same-room
N1 (GR)
bug-is-dead
at-same-room
N3 (GR)
bug-is-dead
?at-same-room
N0 (initial)
sense-bug sense-bug
move
kill-bug
move
move
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bug-is-dead
at-same-room
N2 (GR)
bug-is-dead
at-same-room
N1 (GR)
bug-is-dead
at-same-room
N3 (GR)
bug-is-dead
?at-same-room
N0 (initial)
sense-bug sense-bug
move
kill-bug
move
move
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bug-is-dead
at-same-room
N2 (GR)
bug-is-dead
at-same-room
N1 (GR)
bug-is-dead
at-same-room
N3 (GR)
bug-is-dead
?at-same-room
N0 (GR)
sense-bug sense-bug
move
kill-bug
move
move
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• Node N is dead if N goal and either:− N has no outgoing edges; or− Every or-child is dead, and for every pair of
dual and-children, at least one is dead
Dead Node Propagation
dead
N
dead dead
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• Observation 1:− Some edges may become “useless” due to
goal-reachable path
Pruning
GR
N
GR
N’
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PrAO Algorithm: Pruning Technique
GR
N
GR
N’
“useless” edges
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PrAO Algorithm: Pruning Technique
GR
N
GR
N’
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PrAO Algorithm: Pruning Technique
GR
N
GR
N1
GR
N2
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PrAO Algorithm: Pruning Technique
GR
N
GR
N1
GR
N2
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PrAO Algorithm: Pruning Technique
GR
N
GR
N1
GR
N2
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• Observation 2:− Edges can be removed due to a dead-node
PrAO Algorithm: Pruning Technique
dead
N
N1
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• Observation 2:− Edges can be removed by dead-nodes
PrAO Algorithm: Pruning Technique
dead
N
N1
N2
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• Observation 2:− Edges can be removed by dead-nodes
PrAO Algorithm: Pruning Technique
dead
N
N1
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• Active nodes: can be reached from the start node
• Some nodes (and their descendants if exist) may become disabled (inactive) after removing useless edges
• Expand only active nodes
• A disabled node N may become active again if it is a child of being expanded node. If N is active then so are its descendants (reactivate-propagation).
PrAO Algorithm: Pruning Technique
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PrAO: Reactivate Disabled Nodes
N1
active
N3
disabled
N1
active
N2
disabled
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PrAO: Reactivate Disabled Nodes
N1
active
N3
disabled
N1
active
N2
disabled
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PrAO: Reactivate Disabled Nodes
N1
active
N3
disabled
N1
active
N2
active
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PrAO: Reactivate Disabled Nodes
N1
active
N3
active
N1
active
N2
active
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1. Initialize the front queue Q with the start node n0 (initial belief state).
2. Iteratively perform the following steps:3. Pick an active node n from Q with the best heuristic4. If no such node n exists, terminate the search5. Expand node n6. If n is goal-reachable, then
7. Execute goal-propagation(n)8. If n0 is goal-reachable then extract and return the
solution.9. Else if n is dead, then
10. Execute dead-end-propagation(n)11. If n0 is a dead node then terminate the search
10. Else for every disabled child n’ of n, activate n’ and execute reactivate-propagation(n’)
PrAO Algorithm
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• Incorporate the pruning techniques in: goal-propagation dead-end-propagation
• Expand only active unexpanded nodes: avoid redundant expansion and reducenumbers of expanded/generated nodes
• PrAO is complete due to reactivate-propagation
PrAO Algorithm
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PrAO Algorithm
N0
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PrAO Algorithm
N0
N2N1 N3
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PrAO Algorithm
N0
N5N4
N2N1 N3
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PrAO Algorithm
N0
N5
N4
dead
N2N1 N3
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PrAO Algorithm
N0
N5
N4
dead
N2N1 N3
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PrAO Algorithm
N0
N5
disabled
N4
dead
N2
N1
deadN3
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PrAO Algorithm
N0
N5
disabled
N4
dead
N2
N1
deadN3
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PrAO Algorithm
N0
N5
disabled
N4
dead
N2
N1
deadN3
N7 N8
N6
goal
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PrAO Algorithm
N0
N5
disabled
N4
dead
N2
N1
dead
N3
GR
N7 N8
N6
goal
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PrAO Algorithm
N0
N5
disabled
N4
dead
N2
N1
dead
N7
disabled
N8
disabled
N3
GR
N6
goal
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PrAO Algorithm
N0
N5
disabled
N4
dead
N2
N1
dead
N7
disabled
N8
disabled
N3
GR
N6
goal
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PrAO Algorithm
N0
N5
N4
dead
N2
N1
dead
N7
disabled
N8
disabled
N3
GR
N6
goal
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PrAO Algorithm
N0
N5
N4
dead
N2
N1
dead
N8
disabled
N7
disabled
N3
GR
N6
goal
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PrAO Algorithm
N0
N4
dead
N2
N1
dead
N8
disabled
N7
disabled
N5
GR
N3
GR
N6
goal
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PrAO Algorithm
N0
N4
dead
N1
dead
N8
disabled
N7
disabled
N2
GR
N5
GR
N3
GR
N6
goal
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PrAO Algorithm
N0
GR
N4
dead
N1
dead
N8
disabled
N7
disabled
N2
GR
N5
GR
N3
GR
N6
goal
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Contingent planning Approach overviewDNF representation: review & extensionAnd/Or forward search: PrAO algorithm Experimental evaluation
•Experimental setupPerformance comparison
• Conclusion and future work
Outline
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• Planners: compare DNFct with most state-of the-art: CLG (Albore, Palacios, and Geffner IJCAI-2009)
Contingent-FF (Hoffmann and Brafman ICAPS-2005)
POND (Bryce, Kambhampati, and Smith JAIR-2006)
• Benchmarks:From Pond, Contingent-FF, and CLG distribution. Several obtained by our modification from conformant domains. Total: 64 instances out of 17 domains.
Experimental Setup
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Performance Comparisonon 64 Instances of 17 Domains
0
10
20
30
40
50
60
70
DNFct CLG cont-FF Pond
# of instances theplanner solvesfastest
# of instances theplanner can solve
# of the largestinstances of adomain the plannersolves fastest
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Contingent planning Approach overviewDNF representation: review & extensionAnd/Or forward search: PrAO algorithm Experimental evaluationConclusion and future work
Outline
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• Extended DNF Representation to handle non-det & sensing actions in contingent planning: Compact Fast state computation Complete
• Developed a new And/Or search algorithm PrAO with novel pruning techniques: Avoid redundant expansion Less expanded/generated nodes Complete
Conclusion
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• DNF Representation Not good when the size of DNF formulae
representing belief states is too large.
• PrAO: And/Or forward search In several problem instances, application
of pruning techniques results in more generated/explored nodes.
Future Work
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Thank you!
Question?