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Introduction to Spatio-temporal Qualitative Reasoning Debasis Mitra Florida Institute of Technology

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Introduction to Spatio-temporal Qualitative Reasoning Debasis MitraFlorida Institute of Technology

Associate Professor, Dept. of Computer Sciences, Florida Institute of Technology

Ph.D., Computer Science, University of Louisiana at Lafayette, 1994

Ph.D., Physics, Indian Institute of Technology, Kharagpur, India, 1984

M.Sc., Physics, Indian Institute of Technology, Kharagpur, India, 1977

Dr. Mitra joined Florida Tech in the Fall semester of 2001 as an Associate Professor. Before that he was a faculty member at Jackson State University in Jackson, Mississippi since fall of 1994. He worked as an exploration geophysicist for some time in between his two graduate studies on Physics and Computer Science. Dr. Mitra’s current research interest is on reasoning with space and time, particularly with incomplete and qualitative information. This area broadly falls under the Knowledge Representation branch within the Artificial Intelligence (AI). The primary methodology deployed in this type of research is similar as in the Constraint Propagation. Apart from doing theoretical/empirical works in the area Dr. Mitra is also interested in applying spatio-temporal reasoning to other fields of computation outside the AI.

An introduction to spatio-temporal qualitative reasoning ABSTRACT

Space and time are two of the most important entities dealt with in our lives. Although computer programs routinely manage them using some quantitative measures (e.g., clock), from a human-centric angle it is also necessary to develop a qualitative framework for them. By qualitative framework we mean handling terms like "overlap," "during," "Southeast," etc. Such terms appear not only in the natural language context, but also in many other systems like databases (e.g., Geographical Information Systems). Systems managing these types of qualitative notions of time and space can behave more intelligently than the traditional ones. Fortunately, these qualitative frameworks form perfect relational algebras and so, can be handled normally within the context of computation. In this talk I will introduce a few such algebras as examples, describe the graph theoretical techniques deployed in representing and reasoning with them, some open problems in the area, and mention my current works on this project. I will also briefly touch upon some other projects that I am involved with or is planning to get involved with in the near future.

Time points

- Linear time (like many other domains) is mappable to real numbers.
- Put a point (event) in a time-line:
- The“space” gets divided into three equivalent regions with respect to that point {<, =, >}
- Three QUALITATIVE regions for a second point to be placed on the time line.

Point-based Reasoning

- Input 1:
- (a1 < a2) and (a2 < a3) :: (a1 < a3)
- Input 2:
- (a1 < a2) and (a2 > a3) :: (a1 <|=|> a3)
- We need a relation not belonging to the set {<, >, =}
- The full set needed for reasoning is {<, >, =, <=, >=, <>, and also <=> , null }, the power set

Point-based Reasoning

- Input 1:
- (a1 < a2) and (a2 < a3) -> (a1 < a3)
- A starting point of reasoning: Composition table
- a2->a3:: < > =
- a1->a2
- < < < = > <
- > < = > > >
- = < > =

Point-based Reasoning

- We have already decided to allow disjunctions {< | = | >} in the language
- Input 3:
- (a1 <|= a2) & (a2 <|> a3) ::
- (a1 <.< a3) | (a1 <.> a3) | (a1 =.< a3) |
- (a1 =.> a3)
- A disjunctive composition scheme: compose base relations and union the results

Point Algebra

- We need composition operation and set union operation
- Input 4:
- (a1 <|= a2) & (a2 <|= a3) &(a1 <|> a3) ::
- (a1 <|= a3) &(a1 <|> a3) ::
- (a1 < a3)
- The last operation is set intersection

Point Algebra

- The set {<, >, =, <=, >=, <>, < = >, null} is closed under composition, union, intersection, and inverse operations
- This is POINT ALGEBRA
- This is a type of Relational Algebra
- Nice things about an algebra is that you can reason without getting outside the set.
- {<, >, =} does not form an algebra under composition.

Time Interval Relations

- Basic Relations (13):

A before (b) B B after (a) A

A

B

A

A meets (m) B B met-by (mi) A

B

A

A overlaps (o) B B overlapped-by (oi) A

B

A

A starts (s) B B started-by (si) A

B

A during (d) B B during-inverse (di) A

A

B

A finishes (f) B B finished-by (fi) A

A

B

A

A equals (eq) B

B

Allen’s Interval Algebra

Full Set is 2^{13 basic relations}

Forms algebra A under composition, union, intersection, and inverse operations:

Interval Algebra

A Subalgebra of Interval Algebra

- A subset of A: relations expressible as conjunction of end-points of two intervals
- a1 (before | meet | overlap) a2 ::
- a1------ ------------ --------
- --------------- a2
- (a1_start < a2_start) & (a1_end < = > a2_start)
- & (a1_start < a2_end) & (a1_end < a2_end)

Set of interval relations which are expressible as conjunction of point relations between their end points form Pointisable Subalgebra (~150 relations) A

{before | after} is not a pointisable relation: try it!

You can stick with only pointisable relations and reason within the set (need for having algebra)

A Reasoning Problem Instance

- Input:
- GSA_meeting should be {b | a} StdA office hour
- GSA_meeting should be {a} StdB office hour
- GSA_meeting should be {b} StdC office hour
- StdA should have office hour {overlap} that of StdB
- StdB should have office hour {overlap} that of StdC
- StdA should have office hour {b | m} that of StdC
- [Note NOT all of 4C2 possible inputs need to be present in input]
- Question 1: Is the information consistent? (decision problem)
- Question 2: Develop a scenario, if it is consistent
- Solution 1: No![2, 3, and 5 contradicts]

The Reasoning Problem

- Given a set of objects (points, intervals, …) and some binary relations between some of them answer Question 1 and 2 as above.
- Typical methodology: In a graph the objects are nodes and the binary relations are labels on directed edges between the nodes, algorithms are typically graph theoretical

Allen’s Algorithm

- Initialize a queue Q with all constrained edges
- Do until Q is empty
- e = pop (Q)
- for all triangles (e, e1, e2) formed by e do
- update e1 using (e and e2)
- update e2 using (e and e1)
- if ei becomes null return INCONSISTENCY
- else if ei gets further constrained push(ei, Q)

Allen’s Algorithm

- Complexity: O(N3) for N nodes in the graph.
- Reasoning with Interval Algebra A is NP-hard!
- Allen’s relaxation algorithm works fine for tractable cases e.g., point algebra, pointisable interval algebra
- Allen’s algorithm does not return correct answer for full Interval Algebra: not all inconsistencies are detected [Approximate algorithm]

Current Focus of the STR Community

Finding tractable subalgebras

Maximal Tractable subalgebras: no proper superset (other than the whole) forms a subalgebra. Note a subset or superset of any subalgebra is not necessarily closed under the said operations)

Hope: somebody would need such a subalgebra in a real application

Finding subalgebras is interesting theoretically

Directional Interval Algebra (DIA)

Direction of an interval could be opposite to the line-direction: e.g., a car on a road

Twenty-six basic relations, e.g.,

---------- ------------

------------ ---------------

Renz (IJCAI-2001) proposed it and found some max-tractable subalgebras of it

Cardinal Algebra (Ligozat)

Nine Basic relations in a 2D space

North

Northeast

Northwest

Equal

West

East

Southwest

Southeast

South

Region-conncetion Calculus-5

Five basic relations between two sets:

Come up with new ontology / algebra

Prove NP-hardness (most of them are), and find maximal tractable subalgebras

Develop data-structures and algorithms for efficient reasoning

Find applications

Domain-theoretic approach as opposed to relational algebraic approach

Relational-algebraic approach: constrain labels on arcs (set of symbols/ basic-relations), e.g. Allen’s algorithm

Domain-theoretic approach: create a qualitative space and place each object there. Example:

Canonical representation of intervals(Ligozat’98)

Ending-pt

45 degree-line

(2, 5)

5

overlap region

(-7, 4)

Not allowed region

meet region

2

(-7, 2)

Starting-pt

Not allowed region

Our Contributions: domain theoretic algorithms

Reworking 1D (point) case for a better understanding

(new result: solution for incremental adding a point is “contiguous”)

Studying and developing algorithms for 2D and nD Cardinal-algebra cases

Developing a generalized framework for “all” ontology /algebra - based on a domain-theoretic approach

An extreme symmetry betweendifferent algebra (note canonical rep of Interval Algebra vs 2D-Cardinal Algebra): not studied traditionally

Max-tractable algebras (across different ontology) seem to be have strong similarity

Understand these issues by studying a generalized framework rather than working on each ontology separately

Generalized Framework: Two approaches

- Relational algebraic approach: study the underlying algebra from an ontology independent fashion
- Domain theoretic approach: study the underlying geometry of a qualitative space and topology of relations

Why study generalized framework?

A very clear theoretical direction is suggested from current max-tractability results: we just need to understand it!!!

Some new directions are bound to come up, e.g., new tractable subsets (may not be subalgebras)

Applications would benefit from this deeper understanding

New ontology are better understood (PO time, the least understood area)

Our Contributions: New ontology

Star Algebra - 2D

Possible applications of interest: Ph.D. topic

- Bio-informatics: Two 1D chromosome, proteins have folding angles:: what type of ontology? (Merging different labs’ data as a CSP)
- Graphics / Visualization: Does “Qualiataive space” make any sense in modeling / information-storage?
- Robotics: Spatio-temporal modeling of the world, pattern matching, e.g. DIA in traffic management by autonomous traffic helicop (WITAS project)

Other future directions in the project: Ph.D. topic

Add certainty information to the incompleteness/disjunctions currently handled: e.g. Analysis of Intelligence Information

Study spatio-temporal reasoning needs in tactical deployment (involve databases): emergency management, battle entities, etc.

Other projects under development (or dormant): MS Thesis/Project

AI Planning: application in component-oriented program development (with Dr. Bond)

Empirical studies: of hard problems, and their phase transition

Multi-dimensional Datamodeling: for scientific databases

Other projects under development (or dormant): MS Thesis/Project

- Studying some search algorithms: a new heuristic for “island-based” search technique (for computer games??)
- Studying some CSP problem: new heuristics for N-queens problem that may have fundamental implications
- Quantum Computing: ….

Too much theory: how can one find employment???

Research methodology: (1) Mathematics, (2) algorithmics and programming, (3) deeper understanding of space and time, (4) interests in specific applications are welcome

Skills on information systems development: design your own research product (e.g. GUI, backend database, etc.)

Pointers

- My web page: www.cs.fit.edu/~dmitra
- Bibliography linked from there
- My publications list in my resume
- Thanks!

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