Relevance of Maths for CS

1
Relevance of Maths for CS

• John Barnden
• School of Computer Science
• University of Birmingham
• Intro to Maths for CS
• 2014/15

2
Two Broad Types of Relevance

• Maths coming from the task performed by a program
• maths issues raised by CS concepts/techniques
  themselves

3
Task-Imposed Mathematics

• computation to help maths investigation itself,
  e.g.
• testing numbers for primehood
• proving mathematical theorems
• doing algebraic manipulations
• detecting and categorizing mathematical formulas
  in documents
• computational science, e.g.
• simulation of molecules, galaxies, the weather,
• massive statistical calculations in data mining
  and statistical approaches to natural language
  understanding, etc.

4
Task-Imposed Mathematics, contd

• maths involved in the mechanical, physical,
  visual and geographical aspects of robotics and
  computer games
• Maths involved in recognition and production of
  facial expressions, gesture, body language,
• calculation in general tools, e.g.
• statistical calculation in database systems
• calculations in word-processing and
  diagram-drawing software

5
Task-Imposed Mathematics, contd

• It's not just that you need mathematical
  sensitivity to understand the imposed task and
  program it more reliably and helpfully,
• but there can be major maths complications in
  exactly how the task is done
• e.g., clever ways of streamlining large
  scientific calculations.

6
Maths Issues raised by CS Techniques/Concepts
Themselves

• (there are tastes of all these issues in various
  modules in this school, some at undergraduate
  levels and some at master's level)
• Analysis of the "space complexity" and "time
  complexity" of algorithms (i.e. programs,
  basically)
• This concerns how much data an algorithm needs to
  store and how long the algorithm takes, as a
  function of the "size" of the problem.
• This matter is very important for making
  intelligent choices of algorithms for particular
  real-world problems.
• Even if you dont yourself work out the formulas,
  you need to understand them and their
  significance.
• issue introduced in the Foundations module

7
Maths Issues raised (contd)

• Maths involved in the calculations done by a
  data-structuring technique, e.g. in "hashing"
• Maths involved in the core concepts on which
  modern "relational database software is built
• the word "relational" alludes to an abstract
  notion of "relation" that brings in complicated
  mathematical reasoning.
• Maths involved in computer security, notably the
  cryptography involved in securely encrypting
  documents sent over the internet, incl. in
  activities like electronic voting.

8
Maths Issues raised (contd)

• Maths involved in the verification of programs -
  i.e. reliably establishing that a program does
  what it is meant to do, or uncovering often
  difficult-to-see cases where it might not.
• Especially difficult and important in modern,
  highly parallel and distributed computing,
  interfacing with awkward entities such as people!
• touched upon in Foundations

9
Maths Issues raised (contd)

• Maths (incl. pure, statistical and probabilistic,
  and calculus) involved in the nature of various
  advanced, non-traditional sorts of computational
  framework, such as neural networks, evolutionary
  computing and quantum computing
• Neural and evolutionary these are main types of
  natural or nature-inspired computing
• Relatedly maths involved in algorithms for
  automated learning-- "machine learning".

10
Maths Issues raised (contd)

• Maths involved in the general abstract notions of
  what programs, computers, computation, etc. are.
• this includes proofs of what can and cannot be
  done through computing (e.g. the "Halting
  Problem")
• and proofs about the equivalence or otherwise of
  different computational frameworks.
• Relatedly precise mathematical specifications of
  the meaning of individual programs.
• Statistical/probabilistic maths involved in the
  theory of information and orderedness (cf.
  "entropy").

11
Summary Two Broad Types of Relevance

• task-imposed maths performed by programs
• maths issues raised by CS concepts/techniques
  themselves