Performance of Coordinating Concurrent Hierarchical Planning Agents Using Summary Information

Performance of Coordinating Concurrent Hierarchical Planning

Agents Using Summary Information

Brad Clement and Ed Durfee

University of Michigan

Artificial Intelligence Laboratory

Overview

• Background– What is concurrent hierarchical plan coordination?

– What is summary information?

• Claims– Coordinating at abstract levels is much easier than coordinating at detailed

levels in finding some solution. (complexity analysis)

– Coordinating at abstract levels is better at finding optimal solutions.• search techniques and heuristics that leverage summary information• preliminary experimental results

• Other results– CHiP coordination algorithm is sound and complete.

– Resolving threats in a partial order plan is NP-complete.

Multi-level CoordinationA

B

DA

DB


B

DA

DB

A

B

DA

DB


B

DA

DB

A

B

DA

DB

A

B

DA

DB


B

DA

DB

A

B

DA

DB

A

B

DA

DB

A

B

DA

DB

Multi-level Coordination

A

B

DA

DB

A

B

DA

DB

A

B

DA

DB

A

B

DA

DB

A

B

DA

DB

Multi-level Coordination

A

B

DA

DB

A

B

DA

DB

A

B

DA

DB

A

B

DA

DB

A

B

DA

DB

temporalconstraints

blocked

Coordinating at Abstract Levels• Resolve conflicts at high level to minimize search time

• Better solutions may exist at lower levels

coordinationlevels

crispercrispersolutionssolutions

lowerlowercoordinationcoordination

costcost

flexibilityflexibility

Concurrent Hierarchical Plans (CHiPs)and Summary Information

• pre, in, & postconditions - sets of literals over a set of propositions

• summary information– external preconditions at(A, 0, 0)

– external postconditions at(A, 0, 4)

– internal conditions at(A, 1, 1)

– must, may, always, sometimes

at(A, 1, 2) must sometimes hold

at(A, 0, 1) may sometimes hold

havePower(A) must always hold

B - before

B B

B B B B B

A

B

DA

DB

0

1

2

0 1 2 3 4

Summary Information

• Summarize conditions of potential refinements at abstract levels

• Reason about abstract plan interactions among agents– resolve all conflicts at abstract level

– prune inconsistent refinement choices at abstract levels

– make refinement choices based on task interactions

Concurrent Hierarchical Plan Coordination

• Agents individually derive summary information for their plan hierarchies

• Coordinator requests summary information for expansions of agents’ hierarchies from the top down

• After each expansion, try to resolve threats by adding ordering constraints

• Algorithm shown to be sound and complete

Search for Coordinated Plan

• search state– set of expanded plans – set of blocked subplans– set of temporal constraints

• search operators– expand– block– constrain

blocked

blocked

temporal constraints

Reasoning at Abstract Levels Can Improve Performance

Total Cost

mid-level best

top-level best

primitive-level best

level computationtime

executiontime

top 4 60mid 159 40primitive 2375 35

A

B

DA

DB

Computation CostExecution Cost

Easier to Coordinate at Higher Levels

Number of summary conditions per plan grows exponentially up the hierarchyO(bd-ic)

b - branching factori - leveld - depthc - conditions per plan


Number of summary conditions per plan grows exponentially up the hierarchyO(bd-ic)

Number of plans per level grows exponentially down the hierarchyO(bi)



Complexity of identifying threats among plans is O(n2c´2) for n plan steps and c´ summary conditions per step orO(b2dc2)



The number of orderings to test grows doubly exponentially down the hierarchyO(bi!)




Resolving threats for a partial order plan is NP-complete (reduced from Hamiltonian Path)

Reasoning at Abstract Levels Can Improve Performance

Total Cost

mid-level best

top-level best

primitive-level best

level computationtime

executiontime

top 4 60mid 159 40primitive 2375 35

A

B

DA

DB

Computation CostExecution Cost

Search Techniques

• Prune inconsistent global plans • Branch & bound - abstract solutions

help prune space where cost is higher• “Expand most threats first” (EMTF)

– expand subplan involved in most threats

– focuses search on driving down to source of conflict

• “Fewest threats first” (FTF)– search plan states with fewest threats first

– or subplans involved in most threats are blocked first

NEO Domain Experiments

• Compare FAF’s and our strategies for ordering search states and ordering expansions

• 4 - 8 locations• 2 & 3 transports• no, partial, & complete overlap in locations visited

evacuateevacuate

noswitch

oneswitch

twoswitches

noswitch

oneswitch

twoswitches

cw ccw

go tofarthest

switch & goto farthest

go tosafe loc

move move movemove move movemove move

NEO Domain Experiments

Summary Information vs. FAF

1

10

100

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100000

1000000

10000000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Problems

CP

U T

ime

FAFSummary Information

CPU Time in units of 1/100 CPU sec.

FAF only found solutions for 6 problems

• FTF-EMTF found solutions for 23 problems, 14 optimal

• FTF-ExCon found solutions for 19 problems, 12 optimal

• FAF-FAF found solutions for 22 problems, 14 optimal

• DFS-ExCon found solutions for 6 problems, 3 optimal (not shown)

Future Work

• What properties of plan hierarchies benefit which heuristics?

• For different domains, how can the hierarchies be restructured to take advantage of different heuristics?

• How can greater numbers of agents be continually coordinated as they accomplish, change, or add plans/goals?

Contributions• Sound and complete concurrent hierarchical plan

coordination algorithm

• Complexity analysis showing that resolving conflicts at higher levels is much easier than at lower levels

• Search techniques including FTF and EMTF heuristics that take advantage of summary information

• Preliminary experiments showing that these techniques can greatly improve the search for optimal plans

Performance of Coordinating Concurrent Hierarchical Planning Agents Using Summary Information

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