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MANAGING DATA MATHEMATICALLY. MANAGING DATA MATHEMATICALLY. DATA AS A MATHEMATICAL OBJECT. MANAGING DATA MATHEMATICALLY. - THEORETICAL FOUNDATIONS. - PRACTICAL APPLICATIONS. PREVIEW. THEORETICAL FOUNDATIONS EXTENTIONS TO SET THEORY DATA AS A MATHEMATICAL OBJECT - PowerPoint PPT Presentation
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MANAGINGDATA
MATHEMATICALLY
Copyright © 2006 IIS.
MANAGINGDATA
MATHEMATICALLY
DATA AS A MATHEMATICAL OBJECT
Copyright © 2006 IIS.
MANAGINGDATA
MATHEMATICALLY - THEORETICAL FOUNDATIONS
- PRACTICAL APPLICATIONS
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PREVIEW
• THEORETICAL FOUNDATIONS– EXTENTIONS TO SET THEORY
–DATA AS A MATHEMATICAL OBJECT–TRANSACTIONS AS SET OPERATIONS
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PREVIEW
• PRACTICAL APPLICATIONS
– ADAPT TRANSACTION TO DATA
– ADAPT DATA TO TRANSACTION
– LIVE DEMONSTRATION
• ADAPTIVE DATA RESTRUCTURING
– ON 1, 2, 4, 8, & 10 GB OF RAW DATA
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ARPA - 1965
CONCOMP RESEARCH PROJECT
• MATHEMATICALLY SOUND SYSTEMS
DATA MANAGEABLE MATHEMATICALLY?
– MATHEMATICAL IDENTITY FOR DATA
– MATHEMATICAL EXPRESSION FOR DATA BEHAVIOR
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Identifying Characteristics of Data
• Content– Represented Relationships
• Structure– Form of Representation
• Behavior– Response to Manipulation
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Copyright © 2006 IIS.
Copyright © 2006 IIS.
Copyright © 2006 IIS.
Identifying Characteristics of Data
• Content– Represented Relationships
• Structure– Form of Representation
• Behavior– Response to Manipulation
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Two Remarks on Set Theory[Th. Skolem, Math. Scand. 5 (1957), 40-46]
2. The ordered n-tuples as sets
“ But in literature I have found no answer to the general question how to define the ordered n-tuple as a set.”
In conclusion: “I shall not pursue these considerations here, but only emphasize that it is still a problem how the ordered n-typle can be defined in the most suitable way.”
n-tuples behave badly, e.g., <a, b> <a, c> = <a, a>
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XST Definition of n-tuple
<a, b, c> = { a1, b2, c3}
<x, b, y> = { x1, b2, y3}
<a, b, c> <x, b, y> = {b2}
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Copyright © 2006 IIS.
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Working Definitions
• Data A Representation of Relationships
• Data Transaction Any Transformation of Data from One State to Another
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• Logical Data– Representations friendly to humans– Used to specify enterprise transactions
• Physical Data– Representations friendly to machines– Used to support execution of enterprise
transactions
• Transaction Types– Logical Data to Logical Data– Logical Data to Physical Data– Physical Data to Physical Data– Physical Data to Logical Data
L
P
L
L
L
PP
P
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Visual Summary
E-DATA
M-DATA
Specification
Execution
Brains
Bytes
Logical Enterprise
MachinePhysical
Disjoint Environments
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Mathematical View
P1
L1L2
P2
Specification
Execution
Brains
Bytes
f
F
a b
Logical Enterprise
MachinePhysical
XST: f (L1) = b ( F ( a ( L1) ) ) = L2
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Information Access Strategies
• Adapt Query to Data
• Adapt Data to Query
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Copyright © 2006 IIS.
Copyright © 2006 IIS.
Copyright © 2006 IIS.
Information Access Strategies
• Adapt Query to Data
• Adapt Data to Query
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