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,^_[M-REL] 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 well-defined objects and operations on such objects are also independent of representations, ARPA directed the research to provide applications with machine-independent 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.^_[M-REL] 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 near-optimal performance.^{_[CDgm],[FUN]}

Data Access: Data access is a process. Mathematically well-defined process descriptions seldom exist to assist system developers. Data access is an exception. All data access on a digital computer can be mathematically well-defined 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 user-friendly representation of some specific data content,
All application data representations can be faithfully represented as a set in XST.

Storage Data is a machine-friendly 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 XML-structure 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 machine-independent 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 non-proprietary, software and hardware implementations of XST operations may be highly-proprietary. The only criteria for a XST operation implementation is that it has a well-defined XST definition. All the necessary mathematical material for defining any XST operation is available below.

References