Domain Independent Approaches for Finding Diverse Plans

Biplav Srivastava

IBM India Research Lab [email protected]

Subbarao Kambhampati

Arizona State University [email protected]

Tuan A. Nguyen

University of Natural Sciences [email protected]

Alfonso Gerevini

University of Brescia gerevini@ ing.unibs.it

Minh Binh Do

Palo Alto Research Center [email protected]

Ivan Serina

University of Brescia serina@ ing.unibs.it

IJCAI 2007, Hyderabad, India Jan 09, 2007 (6 Authors from 3 continents, 4 countries, 5 institutions) 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 Specifications Jan 09, 2007 C={c 1 ,c 2 ,…c  } I={i 1 , i 2 ,… i  } X={x 1 ,x 2 ,…x  } F PC F RE Logical Composition R AW S={S 1 ,S 2 ,…S K } Physical Composition Runtime R IW W={W 1 ,W 2 ,…W L } R EW Domain Independent Approaches for Finding Diverse Plans T= {t 1 ,t 2 ,…t  } 2

Jan 09, 2007

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 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:   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?

Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 4

Jan 09, 2007

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 Domain Independent Approaches for Finding Diverse Plans 5

Problem Formulation

 

dDISTANTkSET

  Given a distance measure d (.,.), and a parameter k , find k plans for solving the problem that have guaranteed minimum pair wise distance

d

among them in terms of d (.,.) Converse formulation for dCLOSEkSET

Variations on the formulations possible

 Related work – Multiple solutions for CSP problems (See Hebrard 2005, 2006) Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 6

Jan 09, 2007

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?

7 Domain Independent Approaches for Finding Diverse Plans

Basis for Comparing Plans

   Actions in the plan States in the behavior of the plan Causal support structures in the plan Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 8

Quantifying Distances

 Set-difference  Neighborhood based    Prefix-based Suffix-based … Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 9

Action A1 A2 A2’ A3 A3’ Plan S1-1, S1-2 S1-3 Goal Causal Chains g1 Ai-p1-A1-g1-Ag g2 g3 g1 g2 Preconditions Effect p1 p2 p2, g1 p3 p3, g2 g1 g2 g2 g3 g3 g3 Ai-p2-A2-g2-Ag Ai-p3-A3-g3-Ag Ai-p1-A1-g1-Ag Ai-p1-A1-g1 A2’,Ai-p2-A2’, A2’-g2-Ag Ai-p3 A3’, Ai-p1-A1-g1-A2’,Ai p2 A2’-g2-A3’, A3’-g3-Ag Initial State p1, p2, p3 A1 A2 Goal State g1, g2, g3 A3 Plan S1-1 p1, p2, p3 A1 A2 A3 Plan S1-2 g1, g2, g3 p1, p2, p3 A1 A2’ A3’ Plan S1-3 g1, g2, g3 Jan 09, 2007 10

Compute by Set-difference

•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 Initial State p1, p2, p3 A1 A2 Goal State g1, g2, g3 A3 Plan S1-1 p1, p2, p3 A1 A2 A3 Plan S1-2 g1, g2, g3 p1, p2, p3 A1 A2’ A3’ Plan S1-3 g1, g2, g3

Jan 09, 2007

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 Domain Independent Approaches for Finding Diverse Plans 12

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 13

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 14

GP-CSP Result: Using different distance bases (time)

Jan 09, 2007 Solving for diversity guided by d c or d s more results in the same time) than d a is easier (gives Domain Independent Approaches for Finding Diverse Plans 15

GP-CSP Result: Using different distance bases (cross-validation on solution quality)

Cell = 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: for d s < d s , d a > = 0.485 means that over 462 combinations of (d,k) solvable for each d, the average distance between k solutions measured by d a is 0.485 * d s .

The results indicate that when we enforce d for to d s (1.26* d a ) and d c (1.98* d a ) d a , we will likely find even more diverse solution sets according Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 16

Exploring with LPG

Jan 09, 2007 • 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 Domain Independent Approaches for Finding Diverse Plans 17

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

18

Jan 09, 2007

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 19 Domain Independent Approaches for Finding Diverse Plans

Jan 09, 2007

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 20 Domain Independent Approaches for Finding 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 Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 21

Appendix

Jan 09, 2007 Domain Independent Approaches for Finding Diverse Plans 22

Jan 09, 2007

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 Domain Independent Approaches for Finding Diverse Plans 23