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Spatio -temporal Relational Constraint Calculi

Spatio -temporal Relational Constraint Calculi. Debasis Mitra Florida Institute of Technology Melbourne, USA. Abstract. Spatio-temporal Relational Constraint Calculi Debasis Mitra, Florida Inst. Tech.

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Spatio -temporal Relational Constraint Calculi

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  1. Spatio-temporal Relational Constraint Calculi Debasis Mitra Florida Institute of Technology Melbourne, USA

  2. Abstract Spatio-temporal Relational Constraint Calculi Debasis Mitra, Florida Inst. Tech It all began within the Natural Language Processing. We often use temporal expression like, "Maurya dynasty in India was before or overlapped the period of Qin Dynasty in China." Only qualitative relation, no numbers, are involved, and the relation is disjunctive: “before or overlapped”. Take another example, “Heart is located above and left of liver.” Formalizing such notions over both space and time led to a set of beautiful Spatio-temporal Qualitative Calculii. A few operators involved in reasoning with such qualitative spatio-temporal relations generated a class of relational algebras. In this talk I will briefly introduce some of these algebras and delve into a few projects that we have had an opportunity to contribute. I will also briefly raise some questions for future and mention some possibilities on how to utilize these results. AI Class

  3. Acknowledgement • FlorentLaunay, Student • Gerard Ligozat, LIMSI, Paris • Jochen Renz, ANU • National Science Foundation AI Class

  4. Motivation… • Maurya emperors ruled “before” or “overlapped” Qin dynasty • Tang dynasty was “after” Qin dynasty • Huen-Tsang lived “during” Tang dynasty • Huen-Tsang visited India “during” or “overlapping” king Harsha’s rule • What is the relationship between Maurya rule and the Tang dynasty? • Answer: Maury “before” Tang AI Class

  5. Constraint Network before | overlap Maurya Qin after Harsha Tang during | overlap during Huen-Tsang AI Class

  6. Constraint Network • Network may be Inconsistent, and “local consistency” not enough • Reasoning = Implied information • Reasoning involve operations: Converse, Compose, Intersect, Union before | overlap Maurya Qin ? (implied) after Harsha Tang after (implied) during | overlap during Huen-Tsang AI Class

  7. Overview • Qualitative Spatio-temporal calculi: definition, sort of … • Introduction to some calculi • Temporal calculi • Spatial calculi • Definition of the Reasoning problem • Some algebraic issues • Algorithms • Quantitative temporal constraint problem AI Class

  8. Basics of the Calculi • (S, E, B, CT) • S: Continuous, and Dense Space • E: Basic entity • B: Atomic relations • CT: Composition table of basic relations AI Class

  9. 1. Point-based Calculus Vilain and Kautz, AAAI 1986 Space: One-dimensional time-line (R) Entity: Time point in the space Atomic/ Basic relations: { <, =, > } (B < A) B A AI Class

  10. Composition table of Point-calculus = > B<C A<B = > AI Class

  11. 2. Interval Calculus • Space: Time-line (R) • Entity: Interval on R (Ordered 2-tuple of points) • Atomic Relations: Allen, CACM 1983 AI Class

  12. A A A A A A A A A A A A A B B B B B B B B B B B B B AI Class

  13. Interval Relations: a graphical model Ligozat: AAAI 1996 End-point Start-point AI Class

  14. Interval Relations: a graphical model Ligozat: AAAI 1996 End-point During inverse Start inverse Before inverse Overlap inverse Finish inverse Equal Finish During Overlap Start Start-point Meet AI Class Before

  15. Interval Composition Table • A 13 x 13 table • Sample: ( A overlaps B ) & (B overlaps C)  (A before | meets | overlaps C) before meet overlap A B C AI Class

  16. 3. Point-interval hybrid calculus • Space: Time-line (R) • Atomic Entities: Point and Interval • Basic Relations: 3 + 13 + 5*2 = 26 • Point to Point: 3 • Interval to Interval: 13 • Point to Interval: 5 • Interval to Point: 5 AI Class

  17. Point-interval hybrid calculus • Basic Relations: • Between Interval and Point: 5 AI Class

  18. 4. Generalized Interval Calculi Ligozat: AAAI 1991 • Space: R • Entity: n-interval (ordered n-tuple) • The points are semantically linked AI Class

  19. Generalized Interval Calculi • Atomic relations: • n-interval to 2-interval: (2n+1) + 2*(2n+1-2) + 2*(2n+1-4) +… + 2*1 • n-interval to m-interval: left as an exercise! AI Class

  20. 5. Multi-dimensional Calculi • Space: Rn X2 X1 AI Class

  21. 6. Partially-ordered time Broxwall and Jonosson: AAAI 2000 Space: Arbitrarily branching time-lines Entity: Time point or interval in the space Atomic point-relations: { <, =, >, ||} (A > B) (A || C) A B C AI Class

  22. 7. Interval-calculus on Cyclic time Osmani: LNCS-1611, 1999 • Space: Cyclic ordered set (C), or Directed-Circumference • Entity: Interval or Directed-Arc • Atomic relations: 16 • 13 linear intervals - {before/after} + {double-overlap, double-meet, met_by-overlap, overlapped_by-meet} AI Class

  23. Cyclic-time… A B A overlaps B AI Class

  24. EQ North Northwest Northeast East West Eq Southwest Southeast South 8. Cardinal-directions Calculus Ligozat: AAAI 1998 • Space: R2 • Entity: Point • Atomic relations (9) AI Class

  25. 9. Star-calculi Renz and Mitra: PRICAI 2004 • Space: R2 • (4n +1) atomic relations generated by n concurrent infinite lines 0 22 2 1 23 20 4 3 21 5 19 6 eq 18 7 17 9 15 8 16 11 13 14 10 AI Class 12

  26. Star-calc May be used for approximate shape representation, e.g. a 3D protein’s backbone Cα‘s 20 16 0 AI Class

  27. 10. Region-connection Calculi (RCC-x) Randell, Cui, Kohn: KR 1992 • Space: R2 • Atomic relations of RCC-5: Five basic relations between 2 closed sets A disjoint B A overlaps B A contains B A inside B A equal B AI Class

  28. 11. Region-connection Calculi (RCC-8) Renz: LNCS-2293 • Atomic relations: 8, by splitting RCC-5 relations • Distinguish between “inside” and “boundary” of a set • Problem with limits! What is “boundary”? • Disjoint Disjoint, & Touches-from-outside • Inside Inside, & Touches-from-inside • Contains Contains, & Contains-and-touches AI Class

  29. RCC-8 A contains BA contains-and-touches B AI Class

  30. Reasoning b.o |m.o |b.d |m.d AI Class

  31. Spatio-temporal Reasoning (STR) Problem • Input: Constraint network (V, E) • V: set of entities • E: labeled binary relations between entities (v1, v2, R12) • R12: disjunctive set of relations  B • Output / Inference: Satisfiable/ Unsatisfiable • Output 2: In case of satisfiability: a “solution” • [Output 3: In case of unsatisfiability, “culprit” constraints] AI Class

  32. Reasoning operators • Disjunctive composition B b |m o |d C A b.o |m.o |b.d |m.d AI Class

  33. Reasoning operators • Converse B B b |m  b~ |m~ A A AI Class

  34. Reasoning/ Inferencing operators • Set Intersection B b |m o |d A 1) b.o |m.o |b.d |m.d C 2) … |… |… |… … |… … |… D AI Class

  35. Spatio-temporal Reasoning (STR) Problem • Satisfiability: Does there exist a satisfying assignment? • Find one assignment: Singleton network (one constraint only on each arc) • Results: linear search to find assignment on each node • All possible assignments / minimal network (bound constraints, because all solutions=infinite) • Do: filter the network = constraint propagation • PC ≠> GC, but close (~90%) • Satisfiability question is NP-hard, in most calculi (except in linear point calc.) AI Class

  36. Spatio-temporal Constraint Algebra(Algebra with Inferencing “operators”) • Power set P(B) of atomic relations B is closed under: • Composition • Converse • Set intersection • Required for reasoning or constraint propagation • Example: Interval algebra (linear time), |P(B)| = 213 AI Class

  37. Special elements of the algebra • Power set P(B) includes: • Tautology: {B}, no constraint • Null: Φ, inconsistent AI Class

  38. Sub-algebra • Closure may exist for some θP(B) under: composition, converse, intersection • The sub-problem STR(θ) AI Class

  39. Maximal tractable subalgebras • STR is NP-hard over most of the known calculi • except for the point-algebra on linear time • STR in some θ-subalgebra’s are tractable AI Class

  40. Pointisable Subalgebra:Tractable Van Beek, AI Jnl 1992 • (A before | meet | overlap B)  (A- < B-) & (A- < B+) & (A+>|=|< B-) & (A+< B+) [implicit here: & (A- < A+) & (B- < B+)] A- before meet overlap A A+ B AI Class

  41. Pointisable Subalgebra • (A before | after B)  ? cannot be expressed as a Conjunctive normal form on end-points: hence, notpointisable AI Class

  42. Pointisable Subalgebra • We look for Maximal tractable sub-algebras (MTS): no superset is tractable • Pointisable subalgebra is not an MTS of Interval Algebra (IA) • Reminder: Algebra is over a closed subset of P(B) AI Class

  43. ORD-Horn ALJ[1] ALJ[2] ... ALJ[17] Non-tractable 90% 10% Tractability of the full Interval Algebra Krokhin, Jeavons, Jonsson. JACM 2003 Full Algebra AI Class

  44. Explanation • On a temporal constraint network, inconsistencies are identified as strong connected components • We then suggest a minimal set of constraints to relax to ‘roll back’ to a consistent system AI Class

  45. 12. Quantitative Temporal Constraint NetworkTCSP = Temporal Constraint Satisfaction • Space: Time-line • Entity: Time-point • Constraints:(1) Domains, (2) Sets of convex intervals AI Class

  46. Consistency issues of TCSP • General case is NP-hard, Dechter-Meiri-Pearl, AI Jnl. 1991 • Simple Temporal Problem or STP: Constraints as only single convex intervals • STP is tractable • Floyd-Warshall works for STP with some preprocessing AI Class

  47. Preprocessing t3 62 21 -42 -11 t4 t0 44 25 -5 -14 t2 AI Class

  48. Explanations on TCN • FW detects inconsistency by achieving a negative value on the distance matrix • Culprits as inconsistent cycles • Explanation on general case (non- STP)??? --Yours truly, unpublished AI Class

  49. Conclusion… • Spatio-temporal relational algebras investigated so far: ~10-15 • Tractable sub-algebras or other structures have been identified for some of them • No General theory exists for tractability! • Some calculi even do not form relational algebras, e.g., Star-odd!! AI Class

  50. Conclusion… • Some hybrid algebras have been studied: • Point + Interval • RCC + Size constraints • Not enough works on hybridization – may be useful for practical purposes., e.g., quantitaive+relational information! AI Class

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