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Introduction to Spatio-temporal Qualitative Reasoning

Debasis MitraFlorida Institute of Technology

Introduction to Spatio-temporal Qualitative Reasoning

Debasis MitraFlorida Institute of Technology

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DEBASIS MITRA 

Associate Professor, Dept. of Computer Sciences, Florida Institute of Technology Ph.D., Computer Science, University of Louisiana at Lafayette, 1994Ph.D., Physics, Indian Institute of Technology, Kharagpur, India, 1984M.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.  

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  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.

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Time pointsTime 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.

Time point

a1< >

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 ReasoningPoint-based Reasoning

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Point-based ReasoningPoint-based Reasoning

Input 1:

(a1 < a2) and (a2 < a3) -> (a1 < a3)

A starting point of reasoning: Composition table

a2->a3:: < > =

a1->a2

< < < = > <<

> < = > > >>

= << >> ==

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Point-based ReasoningPoint-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 inverseinverse 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):

AB

AB

AB

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

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

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

A

B A

BA

BA A equals (eq) B

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

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

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

B

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Allen’s Interval AlgebraAllen’s Interval Algebra

Full Set is 2^{13 basic relations}

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

Interval Algebra

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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)

Pointisable Subalgebra

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 InstanceInput:

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]

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The Reasoning ProblemThe 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

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StdA

StdB

StdC

GSA-mt

(b | a) (a)(b)

(o)(o)

(b | m)

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]

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Current Focus of the STR Community 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

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Directional Interval Algebra (DIA)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

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Cardinal Algebra (Ligozat)Cardinal Algebra (Ligozat)

East

North

West

South

NortheastNorthwest

Southwest Southeast

Equal

Nine Basic relations in a 2D space

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Cyclic AlgebraCyclic Algebra

Sixteen basic relations between intervals/arcs on a directed circle

overlap

Partially-ordered Time

Four basic relations between points:

{<, >, =, ||}

||

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Region-conncetion Calculus-5Region-conncetion Calculus-5

Five basic relations between two sets:

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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

Current TrendsCurrent Trends

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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:

Our ContributionsOur Contributions

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Canonical representation of intervals(Ligozat’98)

Canonical representation of intervals(Ligozat’98)

Starting-pt

Ending-pt

Not allowed Not allowed regionregion

Not allowed Not allowed regionregion

(2, 5)

(-7, 4)overlap overlap regionregion

(-7, 2)

meet regionmeet region

45 degree-line

2

5

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Our Contributions: domain theoretic algorithms

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

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Generalized Framework

An extreme symmetry between different 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

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Examples of Qualitative space

2D Cardinal

Intervals

Northeast

meet

before

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Why study generalized framework?

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)

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Our Contributions: New ontologyOur Contributions: New ontology

Star Algebra - 2DStar Algebra - 2D

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)

Possible applications of interest: Ph.D. topic

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

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.

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

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

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

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: ….

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

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!Thanks!