Domain Independent Approaches
for Finding Diverse Plans
Biplav Srivastava
IBM India Research Lab
sbiplav@in.ibm.com
Subbarao Kambhampati
Arizona State University
rao@asu.edu
Tuan A. Nguyen
University of Natural Sciences
natuan@fit.hcmuns.edu.vn
Minh Binh Do
Palo Alto Research Center
minhdo@parc.com
Alfonso Gerevini
University of Brescia
gerevini@ing.unibs.it
Ivan Serina
University of Brescia
serina@ing.unibs.it
IJCAI 2007, Hyderabad, India
(6 Authors from 3 continents, 4 countries, 5 institutions)
Jan 09, 2007
Domain Independent Approaches for Finding Diverse Plans
1
Motivation


Traditionally, Planning has been seen as a
problem of finding a single plan for going
from an initial to a goal state
Often, we need a set of inter-related plans
instead of a single plan
C={c1,c2,…c}
X={x1,x2,…x}
I={i1, i2,… i}
FPC
Specifications
Logical
Physical
Composition
Composition
RAW
S={S1,S2,…SK}
Jan 09, 2007
FRE
Runtime
RIW
REW
T=
{t1,t2,…t}
W={W1,W2,…WL}
Domain Independent Approaches for Finding Diverse Plans
2
Motivation


Traditionally, Planning has been seen as a
problem of finding a single plan for going
from an initial to a goal state
Often, we need a set of inter-related plans
instead of a single plan

Diverse plans





Jan 09, 2007
A set of web service compositions that can
cover as much of the runtime failure
circumstances as possible
Or a set of intrusion plans that are
qualitatively different
Similar plans: plan stability (Fox et al ICAPS
06); a set of query plans so that partial results
of time-out queries can be used
First diverse, then similar; etc …
We explore domain-independent approaches
for finding diverse plans
Domain Independent Approaches for Finding Diverse Plans
3
Finding Diverse plans


How do we formulate and solve this problem?
Naïve idea: Let the planner just continue to search for
more plans


It is not enough for the planner to just produce multiple
plans. We want the plans to have some guaranteed diversity
Domain-dependent approach


Have a meta-theory of the domain in terms of predefined
attributes and their possible values covering roles, features
and measures. Use these attributes to compare plans [Myers
ICAPS 2006]
Issue:



We are interested in domain-independent approach. Need
to:


Formalize notions of diversity (distance measures)
Need to develop (or adapt existing) planning algorithms to
search for diverse plans


Jan 09, 2007
Needs extensive domain modeling
Not affordable for many types of applications
What bases for comparison are easier to enforce than others?
How scalable are the algorithms?
Domain Independent Approaches for Finding Diverse Plans
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Outline



Motivation
Problem Formulation (s)
Distance Measures



Solution Approaches






Jan 09, 2007
Different bases for comparison
Different bases for computation
Constraint-satisfaction based
Heuristic-search based
Results
Related Work
Conclusion
Future Work
Domain Independent Approaches for Finding Diverse Plans
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Problem Formulation

dDISTANTkSET



Variations on the formulations possible

Jan 09, 2007
Given a distance measure d(.,.), and a
parameter k, find k plans for solving the
problem that have guaranteed minimum pairwise distance d among them in terms of d(.,.)
Converse formulation for dCLOSEkSET
Related work – Multiple solutions for CSP
problems (See Hebrard 2005, 2006)
Domain Independent Approaches for Finding Diverse Plans
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Distance Measures

In what terms should we measure
distances between two plans?




Choice may depend on


Jan 09, 2007
The actions that are used in the plan?
The behaviors exhibited by the plans?
The roles played by the actions in the plan?
The ultimate use of the plans
 E.g. Should a plan P and a non-minimal
variant of P be considered similar or different?
What is the source of plans and how much is
accessible?
 E.g. do we have access to domain theory or
just action names?
Domain Independent Approaches for Finding Diverse Plans
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Basis for Comparing Plans



Jan 09, 2007
Actions in the plan
States in the behavior of the plan
Causal support structures in the plan
Domain Independent Approaches for Finding Diverse Plans
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Quantifying Distances

Set-difference

Neighborhood based



Jan 09, 2007
Prefix-based
Suffix-based
…
Domain Independent Approaches for Finding Diverse Plans
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Goal State
Initial State
Action
Preconditions
Effect
A1
p1
g1
A2
p2
g2
A2’
p2, g1
g2
A1
<g1,g2,g3>
p1,
p2,
p3
A3
<p1,p2,p3>
A3
p3
g3
A3’
p3, g2
g3
Plan
Plan S1-1
Goal
Causal Chains
p1,
p2,
p3
A1
<g1,p2,p3>
S1-1,
S1-2
S1-3
g1,
g2,
g3
A2
g1
Ai-p1-A1-g1-Ag
g2
Ai-p2-A2-g2-Ag
g3
Ai-p3-A3-g3-Ag
g1
Ai-p1-A1-g1-Ag
g2
Ai-p1-A1-g1-A2’,Ai-p2-A2’,
A2’-g2-Ag
g3
Ai-p3-A3’, Ai-p1-A1-g1-A2’,Aip2-A2’-g2-A3’, A3’-g3-Ag
A2
A3
<g1,g2,p3>
g1,
g2,
g3
<g1,g2,g3>
<p1,p2,p3>
Plan S1-2
p1,
p2,
p3
A1
<g1,p2,p3>
A2’
A3’
<g1,g2,p3>
g1,
g2,
g3
<g1,g2,g3>
<p1,p2,p3>
Jan 09, 2007
Plan S1-3
10
Compute by Set-difference
Goal State
Initial State
A1
<g1,g2,g3>
•Action-based
comparison: S1-1, S1-2
are similar, both
dissimilar to S1-3; with
another basis for
computation, all can be
seen as different
•State-based comparison:
S1-1 different from S1-2
and S1-3; S1-2 and S1-3
are similar
•Causal-link comparison:
S1-1 and S1-2 are
similar, both diverse from
S1-3
p1,
p2,
p3
g1,
g2,
g3
A2
A3
<p1,p2,p3>
Plan S1-1
p1,
p2,
p3
A1
<g1,p2,p3>
A2
A3
<g1,g2,p3>
g1,
g2,
g3
<g1,g2,g3>
<p1,p2,p3>
Plan S1-2
p1,
p2,
p3
A1
<g1,p2,p3>
A2’
A3’
<g1,g2,p3>
<g1,g2,g3>
<p1,p2,p3>
Plan S1-3
g1,
g2,
g3
Solution Approaches

Possible approaches



[Parallel] Search simultaneously for k solutions
which are bounded by given distance d
[Greedy] Search solutions one after another with
each solution constraining subsequent search
Explored in

CSP-based GP-CSP classical planner


Heuristic-based LPG metric-temporal planner

Jan 09, 2007
Relative ease of enforcing diversity with different
bases for distance functions
Scalability of proposed solutions
Domain Independent Approaches for Finding Diverse Plans
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GP-CSP Result: Solving time with
different bases
Average solving time (in seconds) to find a plan using greedy (first 3
rows) and by random (last row) approaches
Solving for diversity guided by distance functions is
more efficient than random search
Jan 09, 2007
Domain Independent Approaches for Finding Diverse Plans
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GP-CSP Result: Solution quality time
with different bases
Comparison of the diversity in the solution sets returned by the random
and distance function-guided greedy approaches
Solving for diversity guided by distance functions is
likely to get better quality of results than random search
Jan 09, 2007
Domain Independent Approaches for Finding Diverse Plans
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GP-CSP Result: Using different distance bases (time)
Solving for diversity guided by dc or ds is easier (gives
more results in the same time) than da
Jan 09, 2007
Domain Independent Approaches for Finding Diverse Plans
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GP-CSP Result: Using different distance bases
(cross-validation on solution quality)
Cell <row, column> = d’, d” indicates that over all combinations of (d,k) solved
for distance d, the average value d”/d’ where d” and d’ are distance measured
according to d” and d’ respectively.
Example: <ds ,da> = 0.485 means that over 462 combinations of (d,k) solvable
for ds for each d, the average distance between k solutions measured by da is
0.485 * ds.
The results indicate that when we enforce d for da, we
will likely find even more diverse solution sets according
to ds (1.26* da) and dc (1.98* da )
Jan 09, 2007
Domain Independent Approaches for Finding Diverse Plans
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Exploring with LPG
• Details of changes to LPG in the paper
• Looking for:
• How large a problem can be solved easily
• Large sets of diverse plans in complex domains
can be found relatively easily
• Impact of 
•  = 3 gives better results
• Can randomization mechanisms in LPG give
better result?
• Distance measure needed to get diversity
effectively
Jan 09, 2007
Domain Independent Approaches for Finding Diverse Plans
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Experiments with LPG
LPG-d solves 109 comb.
Avg. time = 162.8 sec
Avg. distance = 0.68
Includes d<0.4,k=10; d=0.95,k=2
Jan 09, 2007
LPG-d solves 211 comb.
Avg. time = 12.1 sec
Avg. distance = 0.69
Domain Independent Approaches for Finding Diverse Plans
LPG-d solves 225 comb.
Avg. time = 64.1 sec
Avg. distance = 0.88
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Related Work

The problem of returning diverse relevant results
is important in Information Retrieval


The problem of finding “similar” plans has been
investigated in Replanning and Plan Reuse.



But limited notions of distance measures
Myers 2006 gives a meta-theoretic basis for plan
comparison
For CSPs, Hebrard et al 2005 have formulated
the problem and proposed solutions

Jan 09, 2007
Think “relevance” “solution ness”
The worst-case complexity results can be borrowed
for planning
Domain Independent Approaches for Finding Diverse Plans
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Conclusion

Contributions


Formalize notions of bases for plan distance
measures
Proposed adaptation to existing representative,
state-of-the-art, planning algorithms to search for
diverse plans



Jan 09, 2007
Showed that using action-based distance results in
plans that are likely to be also diverse with respect
to behavior and causal structure
LPG can scale-up well to large problems with the
proposed changes
The approach and results are representative of how
other planners may be modified to find diverse
plans
Domain Independent Approaches for Finding Diverse Plans
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Future Work

On the same thread




Generalized problem


Jan 09, 2007
Solution approaches for more problems
Extensive experiments
More suitable distance measures
Other action representations: Nondeterministic, HTN actions, …
Plans with different goals
Domain Independent Approaches for Finding Diverse Plans
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Appendix
Jan 09, 2007
Domain Independent Approaches for Finding Diverse Plans
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Purpose for Comparison and Characteristics
of the Plan Distance Measure

Plans for visualization purpose



Plans for execution purpose


Jan 09, 2007
Minimal and non-minimal plans should be
found similar. They achieve the goal, after all!
Plans for different goals should be seen
different
Minimal and non-minimal plans should be
found different.
Plans with similar execution trace should be
seen similar even if they are for different goals
Domain Independent Approaches for Finding Diverse Plans
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