Set I/O Architectures
Data diversity and storage access are dependent on how data is
physically represented on a computer.
Though data processing results are independent of computer representations,
the performance of processing operations vary dramatically.
Performance depends
on physical representations and organization of data.
An ideal storage management environment would support
many varieties of physical data representations.
Set I/O architectures are very different than
record I/O architectures.
Sets are mathematical objects.
Records are physical objects,
Mathematical objects
are formally defined abstract definitions that are independent
of any physical representation.
Physical objects
are arbitrarily defined concrete definitions that are dependent on
currently existing physical representations.
Systems processing data as mathematical objects
rely on the properties of the abstract definitions,
not on the current state of the physical representations.
Record I/O
requires knowledge of how data is physically represented in storage.
Set I/O
requires knowledge of how data is mathematically represented in storage.
Set I/O implementations were first introduced in
1968.^{[AFIP]}^{,[LLL]}
Though set accessing systems have been commercially available since
1971,^{[MREL]}
the performance advantages of set accessing I/O over record accessing I/O are little known.
The key I/O performance difference is that record accessing I/O
depends on physical representations of storage data,
while set accessing I/O depends on mathematical representations of storage data.
Storage independent representation of data is key to I/O performance.
■
Set I/O architectures use a formal foundation
for a mathematical representation
and manipulation of system data.
Changes to the physical representation
and organization of data can be made at any
time, as long as mathematical integrity is
maintained.
■
In 1965 ARPA initiated research to
explore the feasibility of
a mathematical foundation for representing
and manipulating
data on a computer.^{[IFIP]}
■
Since information
contained in data is independent of any representation,
and since mathematically welldefined objects
and operations on such objects
are also independent of representations,
ARPA directed the research to
provide applications with
machineindependent access to stored
data.^{[STDS]}
■
All the properties of Classical set theory,
except one, fit the criteria for modeling computer data
as mathematical objects.
ARPA research focused on
extending Classical set theory to include the property of
structure,
giving birth to the concept of
extended sets.^{[Ellis]}
■
Record I/O architectures specify physical
representations and organizations
of data that reflect specific
application processing requirements.
■
Set I/O architectures insulate applications from direct access
to storage by use of set operations.
Record I/O architectures bind applications to
storage by use of index structures.
■
Set I/O implementations have been commercially active since
1971.^{[MREL]}
Early implementations only supported data represented as labeled arrays.
XML documents became represented as extended sets in
2001.^{[XML]}
By 2011 extended set theory
provided a mathematical foundation
capable of modeling any computer representation of
data.^{[Blass]}^{,[FUN]}
■
Set I/O architectures provide applications global access to data,
while local platforms focus on performance issues.
Developers can use
set I/O for universal data access
while allowing local implementations freedom to provide
nearoptimal performance.^{[CDgm]}^{,[FUN]}
Data Access: Data access is a process.
Mathematically welldefined process descriptions
seldom exist to assist system developers.
Data access is an exception.
All data access on a digital computer can be
mathematically welldefined
in terms of XST.
Data Content
is an assertion about the relationship
of certain items of interest.
This assertion (data content) can be faithfully
represented as a set in XST,
Application Data
is a userfriendly representation of some
specific data content,
All application data representations can be faithfully
represented as a set in XST.
Storage Data
is a machinefriendly representation of
an application data representation.
All storage data representations can be faithfully
represented as a set in XST.
Data Access
is a process of exchanging data content
between application data representations
and storage representations.
Given the proper suite of XSP operations,
any application can access any data,
anywhere at any time.
Data as XSP Sets:
Twelve RDM tables
R1  R12 expressed by a single Labeled
set Ri, RDM.
A very simple XMLstructure expressed as a labeled
set, XML.
Three extended relations expressed as labeled
sets, Xrel1.
Two complex extended relations expressed as labeled
sets, Xrel2.
Conclusion
The fundamental result of ARPA's research
exploring the feasibility of a
machineindependent data model was the discovery that
data could be represented as a mathematical object.
A formal modeling notation was developed to
represent and manipulate all computer data as
extended sets.^{[Ellis]}
The evolution of this notation gave rise to XSP Technology.
The ability to represent and manipulate data as XSP sets is what
distinguishes Set I/O implementations from traditional
DBMS implementations.
For those interested in the formal description of
XST operations, a summary is presented
in [TGF].
For those interested in exploring the
formal modeling of systems behavior,
[FUN]
presents an XST representation of all 29 processes
available on a digital computer.
EPILOGUE
Data That Can't Be Accessed, Can't Be Processed.
If Data Can Be Accessed, It Has A Set Identity.
If Data Has A Set Identity, It Can Be Processed By Set Operations.
If Data Can Be Processed By Set Operations, Processes Are Limited Only By Imagination.
Though all XST operations are nonproprietary,
software and hardware
implementations of XST operations may be highlyproprietary.
The only criteria for a XST operation
implementation is that it has a welldefined XST definition.
All the necessary mathematical material for defining
any XST operation
is available below.
References

[IFIP]
Feasibility of a Settheoretic Data Structure:
IFIPS  1968

[AFIP]
Description of a Settheoretic Data Structure:
AFIPS  1968

[STDS]
CONCOMP Project Appendix D: Description of a SetTheoretic Data Structure  1970

[VLDB77]
Extended Set Theory:
A General Model For Very Large, Distributed, Backend Information Systems.

[VLDB84]
1984 VLDB Panel: Inexpensive Large Capacity Storage Will Revolutionize
The Design Of Database Management Systems.

[T&P]
XSP TECHNOLOGY: Theory & Practice Formal Modeling & Practical Implementation of XML & RDM Systems
♦
The RDM works in spite of set theory, not because of it.
XMLStructures and operations on these structures are settheoretically sound under XST.
In this paper the formal foundations of RDM and XML systems are examined in the light of XST to provide practical relevance of XSP Technology to the field of software systems development.

[MREL]
MICROSTDS RDBMS 19711998
(First commercially available Set I/O architecture.)

[LLL]
Lawrence Livermore Lab.:
Set Theoretic Data Structures (STDS): a tutorial
Technical Report, 1977.

[XML]
Champion, M.:
XSP: An Integration Technology for Systems Development and Evolution  2001

[SETS?]
Why Not Sets? All computer processing is settheoretic
in nature.  2010

[f(x)]
Mathematical Foundation
A multiple storage XOP commutative diagram supporting:
For all x in A, ri(hi(gi(x))) = f(x).

[DOA]
Adaptive Data Restructuring Functions Death Of A Dream.

[PILES]
Pebble Piles & Index Structures
Piles of pebbles or parchment with numbers?

[DAfUP]
Data Access for User Productivity

[CCDAS]
ContentContainer Data Access Strategies Content for Functionality  Containers for Performance  2016
♦
Since data represented for processing is not always ideal
for preservation, and since data represented for
preservation is not always ideal for processing,
accessing applicationfriendly data
from storagefriendly data
poses a genuine challenge.

[SPvRP]
Set Processing vs. Record Processing Performance:
Dynamic Data Restructuring vs. Prestructured Data Storage (1 page summary)
♦
Record processing systems and set processing systems are very different.
This paper attempts to clarify the major differences and demonstrate
the performance advantages of set processing.

[North]
North, K.:
Sets, Data Models and Data Independence  2010

[MMDR]
XSP: Extended Set Processing:
Mathematically Managing Data Representations (1 page summary)
♦
This paper presents a high level overview of why a mathematical model of
data representations is necessary, and how an extended set processing
model accomplishes the task.
 XST 

[Blass]
Blass, A., Childs, D L:
Axioms and Models for an Extended Set Theory  2011

[Ellis]
Ellis, T.:
Extended Set Theory: A Summary A clarification of extended set notation.  2015

[FUN]
Functions As Set Behavior Essential Concepts: Conceptual &
Formal modeling foundations.

[PFAC]
Processes, Functions, Applications & Composition
♦ Lambda application & category composition of processes.

[TGF]
XST Definitions, Operations, & Properties Tuples, Graphs, Functions 
♦ CST operations subsumed under XST operations.
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INTEGRATED INFORMATION SYSTEMS
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