Consistency Methods for Temporal Reasoning

1. Consistency Methods for Temporal Reasoning Lin XU Constraint Systems Laboratory Advisor: Dr. B.Y. Choueiry April, 2003 Supported by a grant from NASA-Nebraska, CAREER Award #0133568, and a gift from Honeywell Laboratories.

2. Outline • Temporal Reasoning • motivation & background • Simple Temporal Problem (STP) & Temporal Constraint Satisfaction Problem (TCSP) • what are they & how to solve them • Contribution: • 3 research questions • their solutions • empirical evidence • Summary & future directions for research 2

3. Time, always time! • Tom wants to serve tea • Clear tea pot 3 min • Clear tea cups 10 min • Boil water 15 min With little reasoning, the task takes 18 min instead of 28 min 3

4. Temporal Reasoning in AI • Temporal Logic • Temporal Networks • Qualitative: interval algebra, point algebra • Before, after, during, etc. • Quantitative: temporal constraint networks • Metric: 10 min before, during 15 min, etc. • Simple TP (STP) & Temporal CSP (TCSP) 4

5. Temporal Network: example Tom has class at 8:00 a.m. Today, he gets up between 7:30 and 7:40 a.m. He prepares his breakfast (10-15 min). After breakfast (5-10 min), he goes to school by car (20-30 min). Will he be on time for class? 5

6. Simple Temporal Network (STP) • Variable: Time point for an event • Domain: A set of real numbers • Constraint: distance between time points ( [5, 10]  5Pb-Pa10 ) • Solution: A value for each variable such that all temporal constraints are satisfied 6

7. More complex example Tom has class at 8:00 a.m. Today, he gets up between 7:30 and 7:40 a.m. He either makes his breakfast himself (10-15 min), or gets something from a local store (less than 5 min). After breakfast (5-10 min), he goes to school either by car (20-30 min) or by bus (at least 45 min). 7

8. Possible questions • Can Tom arrive school in time for class? • Is it possible for Tom to take the bus? • If Tom wanted to save money by making breakfast for himself and taking the bus, when should he get up? 8

9. Temporal CSP • Constraint: a disjunction of intervals [10, 15]  [0, 5] • Rest, same as STP • Variable: Time point for an event • Domain: A set of real numbers • Solution: Each variable has a value that satisfies all temporal constraints 9

10. Temporal Networks: STP & TCSP • Simple temporal problem (STP) • One interval per constraint • Can be solved in polynomial time • Floyd-Warshall F-W algorithm (all-pairs shortest-paths) • Temporal Constraint Satisfaction Problem (TCSP) • A disjunction of intervals per constraint • is NP-hard • Solved with Backtrack search (BT-TCSP) [Dechter] 10

11. Solving the TCSP • Formulate TCSP as a meta-CSP: • Given • Variables: Edges in constraint network • Domains of variables: edge labels in constraint network • A unique global constraint ( checking consistency of an STP) • Find all solutions to the meta-CSP 11

12. BT search for meta-CSP <new tree> big 12

13. Solving the TCSP • Requires finding all solutions to the meta-CSP • Every node in the search tree is an STP to be solved An exponential number of STPs to be solved  13

14. Questions addressed • Is there a better algorithm for STP than F-W? • exploiting topology of the constraint graph • exploiting semantic properties of the temporal constraints • Is there a consistency filtering algorithm for reducing the size of TCSP? • Can we improve performance of BT-TCSP • By using a better STP solver? • By avoiding to check STP consistency at every node? • By exploiting the topology of the constraint graph?  again! • By finding a ‘good’ variable ordering heuristic? 14

15. Contributions • Two new algorithms for solving STP • Partial Path Consistency [adapted from Bliek & Sam-Haroud] • STP [Xu & Choueiry, TIME 03] • A new algorithm for filtering TCSP • AC[Xu & Choueiry, submitted] • Three heuristics to improve search • Articulation points (AP) [classical, never tested] • New cycle check (NewCyc) [Xu & Choueiry, submitted] • Edge ordering (EdgeOrd) [Xu & Choueiry, submitted]  Random generators: 2 for STP & 2 for TCSP 15

16. Contributions • Two new algorithms for solving STP • Partial Path Consistency [adapted from Bliek & Sam-Haroud] • STP [Xu & Choueiry, TIME 03] • A new algorithm for filtering TCSP • AC [Xu & Choueiry, submitted] • Three heuristics to improve search • Articulation points (AP) [classical, never tested] • New cycle check (NewCyc) [Xu & Choueiry, submitted] • Edge ordering (EdgeOrd) [Xu & Choueiry, submitted]  Random generators: 2 for STP & 2 for TCSP 16

17. Algorithms for solving the STP Our approach requires triangulation of the constraint graph 17

18. Partial Path Consistency(PPC) • Known features of PPC[Bliek & Sam-Haroud, 99] • Applicable to general CSPs • Triangulates the constraint graph • In general, resulting network is not minimal • For convex constraints, guarantees minimality (same as F-W, but much cheaper in practice) • Adaptation of PPC to STP [this thesis] • Constraints in STP are bounded difference, thus convex, PPC results in the minimal network 18

19. STP [TIME 03] STP considers the temporal graph as composed by triangles instead of edges PPC STP Temporal graph F-W 19

20. Advantages of STP • A finer version of PPC • Cheaper than PPC and F-W • Guarantees the minimal network • Automatically decomposes the graph into its bi-connected components • binds effort in size of largest component • allows parallellization • Best known algorithm for solving STP use it in BT-TCSP where it is applied an exponential number of times 20

21. Finding the minimal STP 21

22. Determining consistency of STP 22

23. Contributions • Two new algorithms for solving STP • Partial Path Consistency [adapted from Bliek & Sam-Haroud] • STP [Xu & Choueiry, TIME 03] • A new algorithm for filtering TCSP • AC[Xu & Choueiry, submitted] • Three heuristics to improve search • Articulation points (AP) [classical, never tested] • New cycle check (NewCyc) [Xu & Choueiry, submitted] • Edge ordering (EdgeOrd) [Xu & Choueiry, submitted]  Random generators: 2 for STP & 2 for TCSP 23

24. Filtering algorithm: AC Remove inconsistent intervals from the label of edge before search. Polynomial number of polynomial-size ternary constraints One global, exponential size constraint 24

25. AC reduces size of TCSP 25

26. Advantages of AC • It is powerful, especially under high density • It uses special, poly-size data structures • It is sound, effective, and cheap O (n |E |k3) • We show how to make it optimal [to be proved] • It uncovers a phase transition in TCSP 26

27. Contributions • Two new algorithms for solving STP • Partial Path Consistency [adapted from Bliek & Sam-Haroud] • STP [Xu & Choueiry, TIME 03] • A new algorithm for filtering TCSP • AC [Xu & Choueiry, submitted] • Three heuristics to improve search • Articulation points (AP) [classical, never tested] • New cycle check (NewCyc) [Xu & Choueiry, submitted] • Edge ordering (EdgeOrd) [Xu & Choueiry, submitted]  Random generators: 2 for STP & 2 for TCSP 27

28. Articulation points (AP) • Decompose the graph into bi-connected components • Solve each of them independently • Binds the total cost by the size of largest component • Classical solution, never implemented or tested 28

29. New cycle check (NewCyc) • Checks presence of new cycles O (|E |) • Checks consistency only if a new cycle is added 29

30. Advantages of NewCyc • Restricts effort to new bi-connected component • Reduces effort of consistency checking • Does not affect # of nodes visited in BT-TCSP 30

31. Edge Ordering in BT-TCSP • Repeat your graph 31

32. EdgeOrd Heuristic Order the edges using ‘triangle adjacency’ Priority list is a by-product of triangulation 32

33. Advantages of EdgeOrd • Localized backtracking • Automatic decomposition of the constraint graph  no need for AP 33

34. Experimental evaluations • With/without: AC, AP, NewCyc, EdgeOrd 34

35. Number of solutions 35

36. Nodes visited (without AC) 36

37. Nodes visited (after AC) 37

38. CC for DPC-TCSP (without AC) 38

39. CC for DPC-TCSP (after AC) 39

40. CC for PPC-A-TCSP (without AC) 40

41. CC for PPC-A-TCSP (after AC) 41

42. CC for STP-TCSP BEST 42

43. Random generators • STP generators • Implemented two new • Tested three • GenSTP-1 [Xu & Choueiry, submitted] • GenSTP-2 [Courtesy of Ioannis Tsamardinos] • SPRAND (sub-class of SPLIB) [Public domain] • TCSP generator • Implemented two new • Tested 1: GenTCSP-1 [Xu & Choueiry, submitted] 43

44. Output from thesis • 1 paper accepted in TIME-ICTL 2003 • 2 papers submitted to CP 2003 • 2 papers submitted to IJCAI 2003 workshop on Spatial & Temporal Reasoning 44

45. Answers to Question I • Is there a better algorithm for STP than F-W? • Exploiting topology: AP improves any STP solver • Constraint semantic: convexity STP is more efficient than F-W and PPC 45

46. Answer to Question II • Is there a consistency filtering algorithm for reducing the size of TCSP? • AC reduces the size of meta-CSP by eliminating intervals from the domain of edge • Effective, cheap, almost optimal 46

47. Answers to Question III • Can we improve the performance of BT-TCSP • by using a better STP solver? Yes, STP is better than DPC to reduce cost of BT • By avoiding to check STP consistency at every node? Yes, NewCyc avoids unnecessary checks & localizes updates • By exploiting the topology of the constraint graph? Yes, using articulation points • By finding a good variable ordering heuristic We propose EdgeOrd, significantly reduces cost of search 47

48. Future work • Improve AC, establish optimality • Integrate AC • with ULT (a closure algorithm) • with search, as in forward-checking • Exploit interchangeability in BT-TCSP, best method for finding all solution 48

49. The End • Thank you for your attention • Questions & comments are welcome 49