1 introduction to spatio-temporal qualitative reasoning debasis mitra florida institute of...
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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!