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Qualitative Spatial-Temporal Reasoning

Qualitative Spatial-Temporal Reasoning. Jason J. Li Advanced Topics in A.I. The Australian National University. Spatial-Temporal Reasoning. Space is ubiquitous in intelligent systems We wish to reason, make predictions, and plan for events in space

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Qualitative Spatial-Temporal Reasoning

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  1. Qualitative Spatial-Temporal Reasoning Jason J. Li Advanced Topics in A.I. The Australian National University

  2. Spatial-Temporal Reasoning • Space is ubiquitous in intelligent systems • We wish to reason, make predictions, and plan for events in space • Modelling space is similar to modelling time.

  3. Quantitative Approaches • Spatial-temporal configurations can be described by specifying coordinates: • At 10am object A is at position (1,0,1), at 11am it is at (1,2,2) • From 9am to 11am, object B is at (1,2,2) • At 11am object C is at (13,10,12), and at 1pm it is at (12,11,12)

  4. A Qualitative Perspective • Often, a qualitative description is more adequate • Object A collided with object B, then object C appeared • Object C was not near the collision between A and B when it took place

  5. Qualitative Representations • Uses a finite vocabulary • A finite set of relations • Efficient when precise information is not available or not necessary • Handles well with uncertainty • Uncertainty represented by disjunction of relations

  6. Qualitative vs. Fuzzy • Fuzzy representations take approximations of real values • Qualitative representations make only as much distinctions as necessary • This ensures the soundness of composition

  7. Qualitative Spatial-Temporal Reasoning • Represent space and time in a qualitative manner • Reasoning using a constraint calculus with infinite domains • Space and time is continuous

  8. Trinity of a Qualitative Calculus • Algebra of relations • Domain • Weak-Representation

  9. Algebra of Relations • Formally, it’s called Nonassociatve Algebra • Relation Algebra is a subset of such algebras that its composition is associative • It prescribes the constraints between elements in the domain by the relationship between them.

  10. Algebra of Relations • It usually has these operations: • Composition: • If A is related to B, B is related to C, what is A to C • Converse: • If A is related to B, what is B’s relation to A • Intersection/union: • Defined set-theoretically • Complement: • A is not related to B by Rel_A, then what is the relation?

  11. Example – Point Algebra • Points along a line • Composition of relations • {<} ; {=} = {<} • {<,=} ; {<} = {<} • {<,>} ; {<} = {<,=,>} • {<,=} ; {>,=} = {=}

  12. Example – RCC8

  13. Domain • The set of spatial-temporal objects we wish to reason • Example: • 2D Generic Regions • Points in time

  14. Weak-Representation • How the algebra is mapped to the domain (JEPD) • Jointly Exhaustive: everything is related to everything else • Pairwise Disjoint: any two entities in the domain is related by an atomic relation

  15. Mapping of Point Algebra • Domain: Real values • Between any two value there is a value • We say the weak representation is a representation • Any consistent network can be consistently extended • Domain: Discrete values (whole numbers) • Weak representation not representation

  16. Network of Relations • Always complete graphs (JEPD) • Set of vertices (VN) and label of edges (LN) • Vertice VN(i) denotes the ith spatial-temporal variable • Label LN(i,j) denote the possible relations between the two variables VN(i), VN(j) • A network M is a subnetwork of another network N iff all nodes and labels of M are in N

  17. Example of Networks • Greece is part of EU and on its boarder • Czech Republic is part of EU and not on its boarder • Russia is externally connected to EU and disconnected to Greece

  18. Example of Networks Czech NTPP U EC EU Russia U DC TPP Greece

  19. Path-Consistency • Any two variable assignment can be extended to three variables assignment • Forall 1 <= i, j, k <= n • Rij = Rij ∩ Rik ; Rkj

  20. Example of Path-Consistency Czech NTPP U EC EU Russia U DC TPP Greece

  21. Example of Path-Consistency Conv(NTPP) = NTPPi Czech NTPP EC ; NTPPi = DC DC EC EU Russia DC U TPP Greece

  22. Example of Path-Consistency Conv(DC) = DC Czech NTPP DC ; DC = U DC EC EU Russia U DC TPP Greece

  23. Example of Path-Consistency TPP ; NTPPi = {DC,EC,PO,TPPi, NTPPi} Conv(NTPP) = NTPPi Czech NTPP DC EC EU Russia DC TPP Greece DC,EC,PO,TPPi,NTPPi

  24. Example of Path-Consistency • From the information given, we were able to eliminate some possibilities of the relation between Czech and Greece

  25. Consistency • A network is consistent iff • There is an instantiation in the domain such that all constraints are satisfied.

  26. Consistency • A nice property of a calculus, would be that path-consistency entails consistency for CSPs with only atomic constraints. • If all the transitive constraints are satisfied, then it can be realized. • RCC8, Point Algebra all have this property • But many do not…

  27. Path-Consistency and Consistency • Path-consistency is different to (general) consistency • Consider 5 circular disks • All externally connected to each other • This is PC, but not Consistent!

  28. Important Problems in Qualitative Spatial-Temporal Reasoning • A very nice property of a qualitative calculus is that if path-consistency entails consistency • If the network is path-consistent, then you can get an instantiation in the domain • Usually, it requires a manual proof • Any way to do it automatically?

  29. Important Problems in Qualitative Spatial-Temporal Reasoning • Computational Complexity • What is the complexity for deciding consistency? • P? NP? NP-Hard? P-SPACE? EXP-SPACE?

  30. Important Problems in Qualitative Spatial-Temporal Reasoning • Unified theory of spatial-temporal reasoning • Many spatial-temporal calculi have been proposed • Point Algebra, Interval Algebra, RCC8, OPRA, STAR, etc. • How do we combine efficient reasoning calculi for more expressive queries.

  31. Important Problems in Qualitative Spatial-Temporal Reasoning • Unified theory of spatial-temporal reasoning • Some approaches combines two calculi to form a new calculi, with mixed results • IA (PA+PA), INDU (IA + Size), etc • BIG Calculus containing all information? • Meta-reasoning to switch calculi?

  32. Important Problems in Qualitative Spatial-Temporal Reasoning • Qualitative representations may have different levels of granularity • How coarse/fine you want to define the relations • Do you care PP vs. TPP? • What resolution do you want your representation? • What level of information do you want to use?

  33. Important Problems in Qualitative Spatial-Temporal Reasoning • Spatial Planning • Most automated planning problems ignore spatial aspects of the problem • Most real-life applications uses an ad-hoc representation for reasoning • How do we use make use of efficient reasoning algorithms to better plan for spatial-change

  34. Solving Complexity • If path-consistency decide consistency, the problem is polynomial • If not, then some complexity proof is required • Transform the problem to one of the known problems

  35. Solving Complexity • Show NP-Hardness, you need to show 1-1 transformation for a subset of the problems to a known NP-Complete Problem • Deciding consistency for some spatial-temporal networks • Deciding the Boolean satisfiability problem (3-SAT)

  36. Transforming Problem • Boolean satisfiability problem has • Variables • Literals • Constraints • Transform each component to spatial networks

  37. Transforming Problem • Show deciding consistency is same as deciding consistency for SAT problem, and vice versa • Program written to do this automatically (Renz & Li, KR’2008)

  38. Summary • Qualitative Spatial-Temporal Reasoning uses constraint networks of infinite domains • It reasons with relations between entities, and make only as few distinctions as necessary • It is useful for imprecise / uncertain information • Many open questions / problems in the field.

  39. Further Reading • A. G. Cohn and J. Renz, Qualitative Spatial Representation and Reasoning, in: F. van Hermelen, V. Lifschitz, B. Porter, eds., Handbook of Knowledge Representation, Elsevier, 551-596, 2008. • J. J. Li, T. Kowalski, J. Renz, and S. Li, Combining Binary Constraint Networks in Qualitative Reasoning, Proceedings of the 18th European Conference on Artificial Intelligence (ECAI'08), Patras, Greece, July 2008, 515-519. • G. Ligozat, J. Renz, What is a Qualitative Calculus? A General Framework, 8th Pacific Rim International Conference on Artificial Intelligence (PRICAI'04), Auckland, New Zealand, August 2004, 53-64 • J. Renz, Qualitative Spatial Reasoning with Topological Information, LNCS 2293, Springer-Verlag, Berlin, 2002. • The above can all be accessed at http://www.jochenrenz.info

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