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Spatio -temporal Relational Constraint Calculi

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

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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 It all began within the Natural Language Processing. We often use temporal expression like, "Mauryadynasty in India was before or overlapped the period of QinDynasty 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. Motivation… • Mauryaemperors 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 Mauryarule and the Tang dynasty? • Answer: Maury “before” Tang AI Class

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

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

  6. 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

  7. 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

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

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

  10. 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

  11. 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

  12. Interval Composition Table https://www.ics.uci.edu/~alspaugh/cls/shr/allen.html

  13. 3. Cyclic-time… A B A overlaps B AI Class

  14. EQ North Northwest Northeast East West Eq Southwest Southeast South 4. Cardinal-directions Calculus Ligozat: AAAI 1998 • Space: R2 • Entity: Point • Atomic relations (9) AI Class

  15. 5. 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

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

  17. Star-calculus to Approximately represent tracks Track Representation by sequence 9, 6, 0, … AI Class

  18. Star-calc to Locate Organs Automatically • Need 3D Star-calc • We have atlas, a standard anatomical map • Motivation: Seed segmentation algorithm AI Class

  19. 6. 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

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

  21. 7. 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

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

  23. 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

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

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

  26. Reasoning/ Inferencing operators • Set Intersection B b |m o |d A b.o |m.o |b.d |m.d C  … |… |… |… … |… … |… D AI Class

  27. Reasoning/ Inferencing operators Set Intersection – leads to Path Consistency B b |m o |d A b.o |m.o |b.d |m.d C  {b,m,o} o o D AI Class

  28. Spatio-temporal Reasoning (STR) Problem • Satisfiability: Does there exist a satisfying assignment? (NP-hard, except point-calc) • Find one assignment: Singleton network (one constraint only on each arc) • All possible assignments / minimal network • Do: filter the network = constraint propagation • PC ≠> GC, but close (~90% in Interval Reasoning) AI Class

  29. 8. Quantitative Temporal Constraint Network (TCN) • Space: 1D Time-line (global clock) • Entity: Time-point • Constraints:(1) Domain constraint, (2) Sets of convex intervals AI Class

  30. Consistency issues • General case is NP-hard, • General case: not-necessarily convex, • e.g., [42,62] U [70-75] Dechter-Meiri-Pearl, AI Jnl. 1991 • Simple Temporal Problem or STP: Constraints as only single convex intervals • STP is tractable (≡ P-class) • Floyd-Warshall algorithm works for STP with some preprocessing AI Class

  31. Preprocessing [42,62] to t3 62 21 -42 -11 t4 t0 44 25 -5 -14 t2 AI Class

  32. Reasoning on STP by Floyd-Warshallfor All-pairs Shortest-path t3 62 21 -42 -11 t4 t0 44 25 -5 -14 t2 AI Class

  33. Improving Formal Verification with Timed-automata (TA) by using TCN-representation(with Dr. Bhattacharyya) AI Class

  34. Merci! Debasis Mitra dmitra@cs.fit.edu • Mitra D, and Launay F. (2010) "Explanation Generation over Temporal Interval Algebra." • ISP Book: Ed. Hazarika • Renz J, and Mitra D. (2004) “Qualitative Direction Calculi with Arbitrary Granularity.” • Pacific Rim Intl. Conf. on AI 2004 • Ligozat G, Mitra D, Condotta JF. (2004) “Spatial and Temporal Reasoning: Beyond Allen's Calculus.” • AI Communications AI Class

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