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

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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,
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. Mitras 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/empirica
l 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 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 lt, , gt
• Three QUALITATIVE regions for a second point to
  be placed on the time line.

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Time point
lt
gt
a1
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Point-based Reasoning

• Input 1
• (a1 lt a2) and (a2 lt a3) (a1 lt a3)
• Input 2
• (a1 lt a2) and (a2 gt a3) (a1 ltgt a3)
• We need a relation not belonging to the set lt,
  gt,
• The full set needed for reasoning is lt, gt, ,
  lt, gt, ltgt, and also ltgt , null , the power set

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Point-based Reasoning

• Input 1
• (a1 lt a2) and (a2 lt a3) -gt (a1 lt a3)
• A starting point of reasoning Composition table
• a2-gta3 lt gt
• a1-gta2
• lt lt lt gt lt
• gt lt gt gt gt
• lt gt

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Point-based Reasoning

• We have already decided to allow disjunctions lt
  gt in the language
• Input 3
• (a1 lt a2) (a2 ltgt a3)
• (a1 lt.lt a3) (a1 lt.gt a3) (a1 .lt a3)
• (a1 .gt a3)
• A disjunctive composition scheme compose base
  relations and union the results

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Point Algebra

• We need composition operation and set union
  operation
• Input 4
• (a1 lt a2) (a2 lt a3) (a1 ltgt a3)
• (a1 lt a3) (a1 ltgt a3)
• (a1 lt a3)
• The last operation is set intersection

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Point Algebra

• The set lt, gt, , lt, gt, ltgt, lt gt, 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.
• lt, gt, does not form an algebra under
  composition.

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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
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Allens Interval Algebra
Full Set is 213 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 lt a2_start) (a1_end lt gt a2_start)
• (a1_start lt a2_end) (a1_end lt a2_end)

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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)
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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

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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

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StdB
(o)
(o)
StdC
StdA
(b m)
(b)
(a)
(b a)
GSA-mt
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Allens 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)
  

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Allens Algorithm

• Complexity O(N3) for N nodes in the graph.
• Reasoning with Interval Algebra A is NP-hard!
• Allens relaxation algorithm works fine for
  tractable cases e.g., point algebra, pointisable
  interval algebra
• Allens 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
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)
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)
Nine Basic relations in a 2D space
North
Northeast
Northwest
Equal
East
West
Southwest
Southeast
South
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Cyclic Algebra
Sixteen basic relations between intervals/arcs on
a directed circle

overlap
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Partially-ordered Time
Four basic relations between points lt, gt, ,


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Region-conncetion Calculus-5

Five basic relations between two sets
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Current Trends
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
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Our Contributions
Domain-theoretic approach as opposed to
relational algebraic approach Relational-algebrai
c approach constrain labels on arcs (set of
symbols/ basic-relations), e.g. Allens
algorithm Domain-theoretic approach create a
qualitative space and place each object there.
Example
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Canonical representation of intervals(Ligozat98)
Ending-pt
45 degree-line
(2, 5)
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overlap region
Not allowed region
(-7, 4)
meet region
2
(-7, 2)
Starting-pt
Not allowed region
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Our Contributions domain theoretic algorithms
Reworking 1D (point) case for a better
understanding (new result solution for
incremental adding a point is contiguous) Study
ing 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
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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
Northeast
2D Cardinal
Intervals
meet
before
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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 ontology
Star Algebra - 2D
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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)

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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
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

• 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???

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.)
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Pointers

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