Introducing: A Complete Algebra of Data

Grab some
coffee and
enjoy the
pre-show
banter before
the top of the
hour!

It All Adds Up
Ø  Math is the cornerstone of engineering
Ø  It helped build the pyramids
Ø  It led to skyscrapers
Ø  It put a man on the moon
Ø  Maybe, it will finally solve data management!

Guests
Robin Bloor
Chief Analyst, The Bloor Group
@robinbloor robin.bloor@bloorgroup.com
Eric Kavanagh
CEO, The Bloor Group
@eric_kavanagh eric.kavanagh@bloorgroup.com

The
Algebra
of
Data
Robin Bloor
Chief Analyst, The Bloor Group

Mathematics
There has never previously been an
algebra of data

The “Relational Algebra” Diversion
§  “Relational Algebra” was
introduced as a
mathematical approach to
data management
§  It began with the best
intentions
§  But it never provided a
mathematical basis for
data, and subsequently it
went off the rails…

“Relational Algebra”
§  Not established axiomatically
§  Apparently based on set
theory, but:
§  No definition of the basic
element
§  No definition of the genesis
set or ground set
§  Abuse of the mathematical
term “relation”

SQL and RDBMS
§  SQL killed the possibility of
a data algebra
§  The NULL –- something that
exists and yet doesn’t –- is
mathematical nonsense
§  As a consequence, RDBMS
were constrained to tabular
data structures
§  And thus a wide variety of
alternative “database”
types gradually emerged

Data Algebra
There is NO NEW MATHEMATICS in
data algebra.
It is all garden variety SET THEORY.

The Algebra of Data: Background
§  Algebraix Data Corporation financed research into data algebra
§  It was conceived and formally defined by Professor Gary
Sherman, who was engaged for that specific purpose
§  The algebra has been tested on implementations of a “relational
database” and an RDF database, and proven against almost all
data structures
§  It has multiple potential areas of application, but has been
demonstrated for:
§  Data management
§  Query optimization
§  Data integration

Data Algebra – The Basics
§  Based on several interrelated
algebras:
§  Couplets (G)
§  Relations (G)
§  Clans (G)
§  They sit one within the other
like Russian nesting dolls
The algebra of couplets - Couplets (G)
ab
R = {ab
, ac
, bd
, cd
, ee
}
C = {{ab
, aa
, bd
, cd
, ee
},
{ab
, bd
, bc
, cd
, ed
}}
The algebra of relations - Relations (G)
The algebra of clans - Clans (G)
operators: composition %, transposition ).
additional operators: cross-union▼, cross-intersection▲,
cross-superstriction _, cross superstriction ^.
additional operators: unionj, intersection+,
complementation’, superstriction q, substriction p.

The JOIN
AAB := (Aq[A%{{xx
}}])UFyang
(Bq[B%{{xx
}}])
where xd(yang(A)+yang(B))
and Fyang := {Rd P(G x G): R is yang-functional}
A Natural Join

Data Algebra - Extensions
Because data algebra is based on
(finite) set theory, it brings ALL of
finite mathematics with it: numerical
analysis, group theory, matrix algebra,
vector algebra, topology, etc.

The Book
§  Written for: software developers,
data professionals &
mathematicians
§  Requires a basic knowledge of
set theory and mathematics
§  Goal: To demonstrate that data
algebra can be applied to all data
structures
§  Takes a step by step approach to
explain the algebra
§  Definitely not your grandfather’s
math text

The Power of Mathematics
Accounting: Double Entry Bookkeeping, Balance Sheet,
Trial Balance, etc. Enabling banking, the mercantile economy
and capitalist economy
Log Tables and slide rules became calculation technology and
the prime tool for a whole range of professions, including
scientists and engineers of every kind
Aside from gaming theory, provided basis for statistics and
the whole of the insurance sector. Also has applications
throughout science e.g. quantum mechanics.
The use of calculus is ubiquitous in all fields of engineering
in architecture, physics, operations research, statistics,
commerce and, notably in space exploration
Used in modern particle physics and molecular physics,
music, crystallography and materials science, cryptography
and error-correction in computing.
The computer and hence whole of the computer industry is
based on the work done on computer architecture, including
electronic calculators and code-breaking technology
Applications include large integer multiplication, efficient
matrix vector multiplication, filtering algorithms, solving
difference equations, etc.
(Proven application in data integration,
data management and query optimization.)
James Cooley, John Tukey:
Fast Fourier Transform
1965
Blaise Pascal, Pierre de Fermat:
Probability Theory
1654
Isaac Newton, Gottfried Leibniz:
Calculus
1684
Évariste Galois
Group Theory
1830
Alan Turing, John Von Neumann
Computer Architecture
1937
John Napier
Logarithms
1614
Luca Pacioli
Bookkeeping
1494
Gary Sherman
Data Algebra
2015
THE GROWTH OF
MATHEMATICAL
KNOWLEDGE
INCREASING VALUE
(SCIENTIFIC OR
COMMERCIAL)
WHO AND WHAT WHAT IT ENABLESTIME
LINE

Data Stores
§  File systems
§  XML files
§  Databases of all
varieties
§  Document stores
Algebraically, these
are NOT distinct
Likely Impact -1
Data algebra provides a COMMON LANGUAGE
for ALL data
It will ease data integration problems

Data Retrieval
§  File navigation
§  SQL, JSONiq, XQuery,
SPARQL
§  Search
Algebraically, these
are the SAME problem
Likely Impact - 2
It will HARMONIZE data retrieval

Optimization: The Algebraix Platform

In time we expect standards to be developed for:
§  DATA DECLARATION – data description
(think XML, except algebraic)
§  DATA RETRIEVAL (think UQL – Universal
Query Language)
Standards

The Bloor Group is building a training course for
companies who would like to train their staff in
data algebra.
Contact: MaryJo.Nott@Bloorgroup.com
Data Algebra Training

There is an Algebraix library built in Python
It instantiates the mathematics in Python and can be
downloaded along with a PDF of the book
Programming with Data Algebra
The URL is http://algebraixlib.readthedocs.org/en/latest/

An invasion of armies can be
resisted, but not an idea whose time
has come.
~Victor Hugo

Questions?
Other questions:
MaryJo.Nott@Bloorgroup.com

THANK
YOU!
Some images courtesy of Wikipedia and "All Gizah Pyramids" by Ricardo Liberato - All Gizah Pyramids. Licensed under CC BY-SA 2.0
via Commons - https://commons.wikimedia.org/wiki/File:All_Gizah_Pyramids.jpg#/media/File:All_Gizah_Pyramids.jpg

Introducing: A Complete Algebra of Data

More Related Content

What's hot

Similar to Introducing: A Complete Algebra of Data

More from Inside Analysis

Recently uploaded

Introducing: A Complete Algebra of Data