1. Theory for Coordinating Hierarchical Planning Agents Using Summary Information
Brad Clement and Ed Durfee
Artificial Intelligence Laboratory
University of Michigan
{bradc,

2. Multi-level Coordination & PlanningDA DA A B DB B DB DA DA A A B DB B DB
blocked
temporal constraints

3. Hierarchical Planning and Concurrent Execution
Hierarchical plans (e.g. HTNs)
abstract plans as promising classes of long-term activity
incremental refinement to specific actions
conditional (contingency) plans
robust execution systems (PRS, RAPS, JAM)
Concurrent execution
plan merging/coordination
temporal reasoning and planning
Need to avoid costly, irreversible decisions that result in failure or backtracking
Need to reason about concurrent interactions of abstract plans without dealing with entire decompositions

4. Interleaving Planning, Execution, and Coordination
Reason about plans at abstract levels in order to make safe decisions and avoid backtracking
summarize conditions for potential decompositions
develop mechanisms for determining how plans can or might interact
coordinate hierarchical plans at abstract levels
commit to safe abstract plans, refine, and execute

5. Improving Coordination & Planning
Complete at high level using summary information to gain flexibility in execution
Better solutions may exist at lower levels
Summary information aids in pruning subplans to resolve threats
New theory deals with concurrent execution
crisper
solutions
lower coordination cost
coordination
levels
more flexibility

6. Reasoning at Abstract Levels Can Improve PerformanceDA DB
Total Cost
top-level best
BFS algorithm
mid-level best
primitive-level best
Computation Cost Execution Cost

7. Concurrent Hierarchical Plans (CHiPs)
pre, in, & postconditions - sets of literals for a set of propositions
type - and, or, primitive
subplans - execute all for and, one for or; empty for primitive
order - conjunction of point or interval relations
B - before DA A B B B DB B B B B B

8. CHiP Executions, Histories, and Runs
Plan execution e = < ts, tf, d >
preconds hold at ts, postconds at tf, and inconds within (ts, tf)
decomposition d is a set of subplan executions constrained by order
History h = < E, sI >
E - set of plan executions
sI - initial state before any execution
Run r : H    S
r(h, 0) = sI
postconds asserted at tf
inconds asserted just after ts B B
E = { , , , } sI
E = { , , , }

9. Asserting, Clobbering, Achieving, Undoing
e asserts in/postcondition l at t
e achieves precondition l of e'
e clobbers pre/in/postcondition l of e'
e undoes postcondition l of e'
must - for all executions in all histories
may - for some execution in some history l l e e' l l e e' l l e e' e e' l l e e' l l l l e' e l l

10. Summary Information must, may always, sometimes first, last
pre: at(A,1,3) in: at(A,1,3), at(B,1,3), at(B,0,3), at(A,0,3), at(B,0,4), at(A,1,3) post: at(A,1,3), at(B,1,3), at(B,0,3), at(A,0,4), at(A,0,3), at(B,0,3), at(B,0,4)
1,3->0,4HI
must, may
always, sometimes
first, last
external preconditions
external postconditions
1,3->0,3
0,3->0,4
pre: at(A,1,3) in: at(A,1,3), at(B,1,3), at(B,0,3) post: at(A,0,3), at(A,1,3), at(B,1,3), at(B,0,3)
pre: at(A,0,3) in: at(A,0,3), at(B,0,3), at(B,0,4) post: at(A,0,4), at(A,0,3), at(B,0,3), at(B,0,4) 1 2 3 4
pre: at(A,1,3) in: at(A,1,3), at(B,1,3), at(B,0,3), at(B,1,4), at(A,0,3), at(A,1,4), at(B,0,4), at(A,1,3) post: at(A,1,3), at(B,1,3), at(B,0,3), at(A,0,4), at(A,0,3), at(A,1,4), at(B,0,3), at(B,1,4), at(B,0,4)
1,3->0,4 DA A 1
1,3->0,4HI
1,3->0,4LO DB 2 B

11. Deriving Summary Information
Recursive procedure bottoming out at primitives
Derived from those of immediate subplans
O(n2c2) for n plans in hierarchy and c conditions in each set of pre, in, and postconditions
Proven procedures for determining must/may - achieve/undo/clobber
Properties of summary conditions are proven based on procedure

12. Determining Temporal Relations
CanAnyWay(relation, psum, qsum) - relation can hold for any way p and q can be executed
MightSomeWay(relation, psum, qsum) - relation might hold for some way p and q can be executed A B DA DB
B - before
O - overlaps
CanAnyWay(before, psum, qsum)
ØCanAnyWay(overlaps, psum, qsum)
MightSomeWay(overlaps, psum, qsum)

13. Determining Temporal Relations among Abstract Plans
CanAnyWay(order, Psum) - all executions succeed for all plans with summary info Psum for all histories MightSomeWay(order, Psum) - all executions succeed for some set of plans with summary info Psum for some history
Psum is a set of summary information for a group of plans
order is a conjunction of point/interval relation constraints
CAW and MSW are proven sound and complete
Proof takes advantage of must-clobber and may-clobber
Rules identify threats to resolve (resources in contention)

14. Top-Down Search search state search operators set of expanded plans
set of temporal constraints
set of blocked subplans
search operators
expand, block, constrain
CAW used to identify solutions
MSW used to identify failure
CAW and MSW improve search
CAW and MSW
synchronize, look deeper, or backtrack
MSW identifies threats to resolve
blocked
temporal constraints
blocked

15. Summary of Contributions
General model of concurrent interaction of hierarchical plans
Formalization of planning concepts
Proven properties of summary information
Sound and complete rules for determining temporal relations
Toolbox for building sound and complete planning and coordination mechanisms

16. Work in Progress & Future Work
Constructing a concurrent hierarchical planner
Coordinating overlapping goals
Adopting other agents’ plans/goals
Dealing with global plan failure during execution
Summarizing plan information differently (probabilistic)
Handling more expressive plan representations
Scaling number of agents and difficulty of coordination
Interleaving planning, execution, and coordination

17. Complexity of Deriving Summary Information
# plans n = O(bd)
at level d-1, bd-1 · b2c2 compares at level d-2, bd-2 · b2(c+bc)2 compares
O((bd-i · b2(c+bi-1c)2)) =
O(b2c2 · (bd-i · b2i-2)) =
O(b2c2 · (bd+i-2)) =
O(b2c2 · b2d-2)) =
O(b2dc2) =
O(n2c2)
. . . d
. . .
i = 1
. . .
depth d
# conditions c
. . .
branching factor b

18. Difficulty of CAW and MSW Rule Specification1 A
MSW(overlaps, psum, qsum)
false if postconditions conflict
unsound
false if conflicts with pin and qpre  qin or ppost and qin where
pin is set of must, always inconds of p not achieved by inconds of q
ppost is set of postconds of p
qpre is set of must preconds of q not achieved by in- or postconds of p
qin is set of must, always inconds of p not achieved by inconds of p
O - overlaps 1 B 2
pre: at(A,0,0) in: at(A,0,0), at(A,0,0), at(A,0,1), at(A,0,1), at(A,1,1), at(B,0,0), at(B,0,1), at(B,1,1), at(B,1,0) post: at(A,0,0), at(A,0,1), at(A,1,1), at(A,1,0), at(B,0,0), at(B,0,1), at(B,1,1), at(B,1,0)
pre: at(B,2,0) in: at(B,2,0), at(B,2,0), at(B,2,1), at(B,2,1), at(B,1,1), at(A,2,0), at(A,2,1), at(A,1,1), at(A,1,0) post: at(B,2,0), at(B,2,1), at(B,1,1), at(B,1,0), at(A,2,0), at(A,2,1), at(A,1,1), at(A,1,0)