Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 1
Domain Independent Approaches for Finding Diverse Plans
Biplav Srivastava Subbarao KambhampatiIBM India Research Lab Arizona State [email protected] [email protected]
Tuan A. Nguyen Minh Binh DoUniversity of Natural Sciences Palo Alto Research [email protected] [email protected]
Alfonso Gerevini Ivan SerinaUniversity of Brescia University of [email protected] [email protected]
IJCAI 2007, Hyderabad, India
(6 Authors from 3 continents, 4 countries, 5 institutions)
Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 2
Motivation
REW
S={S1,S2,…SK} W={W1,W2,…WL}
FPC FRE
RAW RIW
Logical Composition
PhysicalComposition
RuntimeSpecifications
C={c1,c2,…c} I={i1, i2,… i} X={x1,x2,…x}
T={t1,t2,…t}
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
Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 3
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
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
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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: Needs extensive domain modeling Not affordable for many types of applications
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 What bases for comparison are easier to enforce than others? How scalable are the algorithms?
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Outline
Motivation Problem Formulation (s) Distance Measures
Different bases for comparison Different bases for computation
Solution Approaches Constraint-satisfaction based Heuristic-search based
Results Related Work Conclusion Future Work
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Problem Formulation
dDISTANTkSET Given a distance measure (.,.), and a
parameter k, find k plans for solving the problem that have guaranteed minimum pair-wise distance d among them in terms of (.,.)
Converse formulation for dCLOSEkSET Variations on the formulations possible
Related work – Multiple solutions for CSP problems (See Hebrard 2005, 2006)
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Distance Measures
In what terms should we measure distances between two plans? The actions that are used in the plan? The behaviors exhibited by the plans? The roles played by the actions in the plan?
Choice may depend on 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?
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Basis for Comparing Plans
Actions in the plan States in the behavior of the plan Causal support structures in the plan
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Quantifying Distances
Set-difference
Neighborhood based Prefix-based Suffix-based …
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Action Preconditions Effect
A1 p1 g1
A2 p2 g2
A2’ p2, g1 g2
A3 p3 g3
A3’ p3, g2 g3
p1,p2,p3
g1,g2,g3
A1 A2
<p1,p2,p3>
A3
<g1,p2,p3><g1,g2,p3>
<g1,g2,g3>
Plan S1-2
p1,p2,p3
g1,g2,g3
A1
A2
A3<p1,p2,p3>
<g1,g2,g3>
Plan S1-1
Plan Goal Causal Chains
S1-1,
S1-2
g1 Ai-p1-A1-g1-Ag
g2 Ai-p2-A2-g2-Ag
g3 Ai-p3-A3-g3-Ag
S1-3 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’,Ai-p2-A2’-g2-A3’, A3’-g3-Ag
p1,p2,p3
g1,g2,g3
A1 A2’
<p1,p2,p3>
A3’
<g1,p2,p3><g1,g2,p3>
<g1,g2,g3>
Plan S1-3
Initial State Goal State
•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
Compute by Set-difference
p1,p2,p3
g1,g2,g3
A1 A2
<p1,p2,p3>
A3
<g1,p2,p3><g1,g2,p3>
<g1,g2,g3>
Plan S1-2
p1,p2,p3
g1,g2,g3
A1
A2
A3<p1,p2,p3>
<g1,g2,g3>
Plan S1-1
p1,p2,p3
g1,g2,g3
A1 A2’
<p1,p2,p3>
A3’
<g1,p2,p3><g1,g2,p3>
<g1,g2,g3>
Plan S1-3
Initial State Goal State
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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
Relative ease of enforcing diversity with different bases for distance functions
Heuristic-based LPG metric-temporal planner Scalability of proposed solutions
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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 ismore efficient than random search
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GP-CSP Result: Solution quality time with different bases
Solving for diversity guided by distance functions islikely to get better quality of results than random search
Comparison of the diversity in the solution sets returned by the random and distance function-guided greedy approaches
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GP-CSP Result: Using different distance bases (time)
Solving for diversity guided by c or s is easier (givesmore results in the same time) than a
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GP-CSP Result: Using different distance bases (cross-validation on solution quality)
The results indicate that when we enforce d for a, we will likely find even more diverse solution sets according to s (1.26* da) and c (1.98* da )
Cell <row, column> = ’, ” 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 ” and ’ respectively.
Example: <s ,a> = 0.485 means that over 462 combinations of (d,k) solvable for s for each d, the average distance between k solutions measured by a is 0.485 *
ds.
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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
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Experiments with LPG
LPG-d solves 109 comb.Avg. time = 162.8 secAvg. distance = 0.68Includes d<0.4,k=10; d=0.95,k=2
LPG-d solves 211 comb.Avg. time = 12.1 secAvg. distance = 0.69
LPG-d solves 225 comb.Avg. time = 64.1 secAvg. distance = 0.88
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Related Work
The problem of returning diverse relevant results is important in Information Retrieval Think “relevance” “solution ness”
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 The worst-case complexity results can be borrowed
for planning
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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
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
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Future Work
On the same thread Solution approaches for more problems Extensive experiments More suitable distance measures
Generalized problem Other action representations: Non-
deterministic, HTN actions, … Plans with different goals
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Appendix
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Purpose for Comparison and Characteristics of the Plan Distance Measure
Plans for visualization purpose Minimal and non-minimal plans should be
found similar. They achieve the goal, after all! Plans for different goals should be seen
different Plans for execution purpose
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