Domain Independent Approaches for Finding Diverse Plans

1. Domain Independent Approaches for Finding Diverse Plans Biplav SrivastavaSubbarao Kambhampati IBM India Research Lab Arizona State University sbiplav@in.ibm.comrao@asu.edu Tuan A. Nguyen Minh Binh Do University of Natural Sciences Palo Alto Research Center natuan@fit.hcmuns.edu.vnminhdo@parc.com Alfonso Gerevini Ivan Serina University of Brescia University of Brescia gerevini@ing.unibs.itserina@ing.unibs.it IJCAI 2007, Hyderabad, India (6 Authors from 3 continents, 4 countries, 5 institutions) Domain Independent Approaches for Finding Diverse Plans

2. FPC FRE C={c1,c2,…c} X={x1,x2,…x} I={i1, i2,… i} Logical Composition Runtime Physical Composition S={S1,S2,…SK} W={W1,W2,…WL} Specifications RAW T={t1,t2,…t} RIW REW 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 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 Domain Independent Approaches for Finding Diverse Plans

4. 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? Domain Independent Approaches for Finding Diverse Plans

5. 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

6. 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) Domain Independent Approaches for Finding Diverse Plans

7. 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? Domain Independent Approaches for Finding Diverse Plans

8. Basis for Comparing Plans • Actions in the plan • States in the behavior of the plan • Causal support structures in the plan Domain Independent Approaches for Finding Diverse Plans

9. Quantifying Distances • Set-difference • Neighborhood based • Prefix-based • Suffix-based • … Domain Independent Approaches for Finding Diverse Plans

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

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

12. 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

13. 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 Domain Independent Approaches for Finding Diverse Plans

14. 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 Domain Independent Approaches for Finding Diverse Plans

15. GP-CSP Result: Using different distance bases (time) Solving for diversity guided by c or sis easier (givesmore results in the same time) than a Domain Independent Approaches for Finding Diverse Plans

16. GP-CSP Result: Using different distance bases (cross-validation on solution quality) 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 sfor each d, the average distance between k solutions measured by ais 0.485 * ds. 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) Domain Independent Approaches for Finding Diverse Plans

17. 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 domainscan be found relatively easily • Impact of  •  = 3 gives better results • Can randomization mechanisms in LPG givebetter result? • Distance measure needed to get diversity effectively Domain Independent Approaches for Finding Diverse Plans

18. 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 sec Avg. distance = 0.69 LPG-d solves 225 comb.Avg. time = 64.1 sec Avg. distance = 0.88 Domain Independent Approaches for Finding Diverse Plans

19. 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 Domain Independent Approaches for Finding Diverse Plans

20. 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 Domain Independent Approaches for Finding Diverse Plans

21. 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 Domain Independent Approaches for Finding Diverse Plans

22. Appendix Domain Independent Approaches for Finding Diverse Plans

23. 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