jan 09, 2007 domain independent approaches for finding diverse plans 1 biplav srivastavasubbarao...

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)

mailto:[email protected]









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


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


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?


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


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)


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?


Basis for Comparing Plans

Actions in the plan States in the behavior of the plan Causal support structures in the plan


Quantifying Distances

Set-difference

Neighborhood based Prefix-based Suffix-based …

Jan 09, 2007 10

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


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


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


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


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


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.


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


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


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


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


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


Appendix


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

jan 09, 2007 domain independent approaches for finding diverse plans 1 biplav srivastavasubbarao...

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