Transcript Object Oriented Programming & Mathematics

Object Oriented Programming
& Mathematics
The Beauty of Implementing
Abstract Structures
as
Abstract Structures.
Marc Conrad, University of Luton.
16.07.2015
Marc Conrad, University of Luton
1
Once upon a time...
Mathematics before
the 20th century
16.07.2015
Marc Conrad, University of Luton
2
there was two branches of
Mathematics
Mathematics before
the 20th century
16.07.2015
Marc Conrad, University of Luton
3
Axiomatic
Hilbert,
N. Bourbaki,
etc.
Mathematics before
the 20th century
16.07.2015
Marc Conrad, University of Luton
4
Algorithmic
Mathematics before
the 20th century
Turing,
Church,
etc.
16.07.2015
Marc Conrad, University of Luton
5
Pure Mathematics &
Computer Science
"pure" mathematics
Mathematics before
the 20th century
computer science
16.07.2015
Marc Conrad, University of Luton
6
With links in between.
"pure" mathematics
Mathematics before
the 20th century
computer
algebra etc.
computer science
16.07.2015
Marc Conrad, University of Luton
7
However, some topics of Computer
Science seemed to be unrelated to
mathematics...
"pure" mathematics
Mathematics before
the 20th century
computer science
software design
operating systems
16.07.2015
Marc Conrad, University of Luton
8
as e.g. object oriented programming.
"pure" mathematics
Mathematics before
the 20th century
object oriented programming is
a technique to solve the
"software crisis". It evolved in a
context completely unrelated to
mathematics.
16.07.2015
Marc Conrad, University of Luton
computer science
e.g. object oriented
programming
9
But object oriented programming is closer
to "axiomatic" mathematics than it
appeared in the first place.
object oriented programming...
... allows to implement abstract
structures in an "axiomatic" way.
"pure" mathematics
Mathematics before
the 20th century
computer science
16.07.2015
Marc Conrad, University of Luton
10
Example: A ring (abstract).

We cannot implement:







addition
negation
multiplication
inversion
"zero"
"one"
check if zero

We can implement:






16.07.2015
subtraction (because
of addition and
negation)
exponentiation a n
embedding of Z, Q.
check for equality
polynomials over this
ring
etc.
Marc Conrad, University of Luton
11
Example: A ring.

We cannot implement:







addition
negation
multiplication
inversion
"zero"
"one"
check if zero

We can implement:






16.07.2015
subtraction (because
of addition and
negation)
exponantiation a n
embedding of Z, Q.
check for equality
polynomials over this
ring
etc.
Marc Conrad, University of Luton
12
A "UML" approach to a ring.
Ring
Z

Q
Polynomial
Ring
The child classes implement (override) the missing
functionality of the parent class.
16.07.2015
Marc Conrad, University of Luton
13
A "UML" approach to a ring.
Ring
Z

Q
Polynomial
Ring
But things are more complicated, a polynomial is defined
over a ring. It both inherits and aggregates a ring.
16.07.2015
Marc Conrad, University of Luton
14
A "UML" approach to a ring.
Leads to multivariate
polynomials by
implementing univariate
polynomials!
Z

Ring
Q
Polynomial
Ring
But things are more complicated, a polynomial is defined
over a ring. It both inherits and aggregates a ring.
16.07.2015
Marc Conrad, University of Luton
15
A "UML" approach to a ring.
Ring
Element
Ring
Z

Q
Polynomial
Ring
And in order to perform computations we also
need a class for the elements of a ring.
16.07.2015
Marc Conrad, University of Luton
16
The practical side.




In order to get experience with the idea, an
experimental implementation in Java classes has
been developed.
Java is highly object oriented but not very
common in mathematical context.
Results can be viewed at
http://ring.perisic.com
The name of the Java package is consequently:
com.perisic.ring
16.07.2015
Marc Conrad, University of Luton
17
Results, Remarks, Conclusions




It is possible to work with abstract structures!
 E.g. Modular Ring R/p(x), where R is abstract.
 E.g. Quotient Field Quot(R), where R is abstract.
 Same amount of work as implementing Z/mZ or Q.
Multivariate polynomials can be used although only
univariate polynomials have been implemented (over R).
Complex structures can easily be derived as child classes:
 Cyclotomic fields, complex numbers, rational function
fields, ...
Concepts for automatic mapping from one ring to another.
16.07.2015
Marc Conrad, University of Luton
18
Results, Remarks, Conclusions



It is astonishing simple to implement complex
mathematical structures in an object oriented
environment "from scratch".
You are invited to experiment, contribute, or share
experiences. The package com.perisic.ring is available
and documented at http://ring.perisic.com.
Caveat: There are drawbacks: performance, decisions on
how to organise classes, implementing specialised
algorithms (primality testing, factoring, …)
16.07.2015
Marc Conrad, University of Luton
19
Results, Remarks, Conclusions

The experiments with the Java package com.ring.perisic
show that object oriented programming deserves a
closer look in the context of mathematics:



The mechanism of overriding and dynamic binding
allows protoyping of abstract mathematical
structures.
Object oriented programming should be a main
feature in CAS (as user defined functions a couple of
years ago).
Mathematical software should use object oriented
terminology instead of "reinventing the wheel".
 Disseminate object oriented concepts to the
mathematical community.
16.07.2015
Marc Conrad, University of Luton
20