The Synergy of Computer Science and Mathematics

1. The Synergy of Computer Science and Mathematics Dr. Raymond Greenlaw Armstrong Atlantic State University School of Computing

2. Outline • Introduction • Influence of Math on Computer Science • Influence of Computer Science on Math • Computational Mathematics • Conclusions • References Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

3. Introduction • Computer science and mathematics are currently extensively used in many fields • Both have influenced the other and led to many advancements in each • This synergy between the two have even lead to the increase in interest in computational mathematics, a field that many consider to be the intersection between computer science and mathematics Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

4. Influence of Math on C.S. Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

5. Influence of Math on C.S. • Many computer science programs have spawned from mathematics departments • During their time under mathematics, they slowly built up reputation and students, with most eventually becoming their own department, then later, their own school • Many noted computer scientists have degrees in mathematics, including Knuth, Cook, Lamport, Backus (FORTRAN, BNF), Kay (GUI), and McCarthy (LISP) Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

6. Influence of Math on C.S. Why do C.S. students need Math? • Many people believe that mathematics is fundamental to computer science • Real world problems are often expressed mathematically and require problem solving skills • Studies have shown that mathematical ability has a strong correlation with success in introductory computer science courses Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

7. Influence of Math on C.S. • When the human brain is subjected to extended educational experience permanent, physical changes occur, creating neural pathways which aid in learning new things • The more repetitive the learning process, the strong and longer lasting the changes • The mind finds it extremely difficult to accept formal abstractions, yet at some point, they seem very real and the developer does not see them as abstract Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

8. Influence of Math on C.S. • Actual information learned in a class may go away, but the learning abilities are still there • True goal of education is to improve minds • Should enable the student to acquire abilities and skills to accomplish new things • Main benefit of learning and doing mathematics is to develop the ability to reason precisely and analytically about abstract structures, which is what computing deals with Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

9. Influence of Math on C.S. • ABET, the recognized accreditor for college and university programs in computing and other technological areas requires an accredited C.S. program to include a minimum of 15 hours of mathematics • These courses must include discrete mathematics, differential and integral calculus, and probability and statistics Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

10. Influence of Math on C.S. • An example of math playing a large role in computer science occurs when using public key cryptography • A published, publicly available key (the public key) is provided in an open directory • The person keeps a private key • To send an encrypted file or email message, the sender encrypts using the recipient’s public key • The private key decrypts the message Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

11. Influence of Math on C.S. • The keys are formed and secured based on the properties of prime numbers • Basically, the public key is a product of two random, large primes • The private key is the two primes themselves • This algorithm is secure because of the almost impossible task of factorizing the large number into the two primes • To break this encryption, all prime numbers less than the product must be checked Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

12. Influence of Math on C.S. Samples of Math in C.S. areas • Relational Databases – Rely on the ideas of set theory to understand usage • NP-Completeness – Heavy math basis • Network – Topology of networks, routing, and load analysis problems deal with graph theory • Cryptography – Encryption/decryption schemes rely on mathematical ideas and methods to provide security Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

13. Influence of C.S. on Math Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

14. Influence of C.S. on Math Computer algebra system (CAS) • Software program that allows symbolic mathematics, the manipulation of equations and expressions in symbolic format • Typical CASs store expressions as directed acyclic graphs, DAGs are also used in the parse trees for compilers • Popular examples include Maple, MATLAB, and Mathematica Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

15. Influence of C.S. on Math Maple • General purpose CAS • Developed in 1981 by a group at the University of Waterloo • Sold commercially since 1988 • Interpreted, dynamically typed language • Allows static scoping, where a variable always refers to its nearest binding Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

16. Influence of C.S. on Math Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

17. Influence of C.S. on Math MATLAB • Numerical computing environment and programming language • Has a toolbox to interface with Maple engine which turns it into a CAS • Invented in late 1970s by Cleve Moler, chairman of computer science at the University of New Mexico • Used mainly for linear algebra and numerical analysis Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

18. Influence of C.S. on Math Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

19. Influence of C.S. on Math Mathematica • CAS and powerful programming language • Written by Stephen Wolfram • First version released in 1988 • Uses a kernel which does all calculations and feeds results to multiple front ends • Based on term rewriting • Supports functional and procedural programming Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

20. Influence of C.S. on Math Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

21. The Four Color Theorem Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

22. The Four Color Theorem • Long standing conjecture proven with aid of the computer • Originally posed in the 1850s • Proven by Kenneth Appel and Wolfgang Haken in 1976 • States that the chromatic number (least number of colors required to color a graph) of a planar graph is no greater than four • Relies on case-by-case analysis carried out by computer Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

23. The Four Color Theorem • Showed that if the theorem was false, a counterexample would have to exist in one of 1,936 reducible configurations (later reduced to 1,476) • A later enhancement by Robertson, Sanders, Seymour, and Thomas reduced that number to 633 • Many disproofs use one region touching all others, not realizing that the surrounding regions can be colored with only three Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

24. The Four Color Theorem Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

25. Computational Mathematics Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

26. Computational Mathematics • Considered to be at the intersection of computer science and mathematics • Application of computer methods to simulate computer models and analyzing the results • Historically, the software and models were developed by people already working in the application area (engineers, scientists, etc.) • Developing and analyzing these models requires more than classic mathematics and elementary computer science Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

27. Computational Mathematics • Deals with issues such as • the implications of finite precision arithmetic • the efficiency, accuracy, and stability of numerical computations • the development and maintenance of mathematical software • the effects of modern developments in computer architectures and networks • Programs are available at institutions like Waterloo, Princeton, and CalTech Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

28. Computational Mathematics • Students in these programs study • Asymptotics, analysis, numerical analysis, and signal processing • Discrete mathematics, combinatorics, algorithms, computation geometry, and graphics • Mechanics and field theories • Optimization • Partial and ordinary differential equations • Stochastic modeling, probability, statistics, and information theory Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

29. Computational Mathematics • Sample applications from Waterloo include • Tornado tracking for meteorologists • Numerical simulation of flow in aeronautical engineering • Derivative pricing in computational finance • Modeling breaking waves • Medical imaging with functional magnetic resonance imaging (fMRI) Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

30. Conclusions • Computer science and mathematics have both influenced each other for many years • A number of advancements in each field can be directly linked to the other • Computational mathematics has successfully combined the two fields into one which is being used to produce quality software for many important fields Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics

31. References • Baldwin, Doug and Peter Henderson. “The Importance of Mathematics to the Software Practicioner.” IEEE Software. March/April 2002: 22-24. • Beaubouef, Theresa. “Why Computer Science Students Need Math.” Inroads: SIGCSE Bulletin. December 2002: 57-59. • Bruce, Kim, et al. “Why Math?” Communications of the ACM. September 2003: 41-44. • “Computational Mathematics.” University of Waterloo. December 2005 <http://www.math.uwaterloo.ca/navigation/CompMath/> • “Criteria for Accrediting Computer Science Programs (2005-2006).” ABET, Inc. • Devlin, Keith. “The Real Reason Why Software Engineers Need Math.” Communications of the ACM. October 2001: 21-22. • Devlin, Keith. “Why Universities Require Computer Science Students to Take Math.” Communications of the ACM. September 2003: 37-39. • “PACM Graduate Program.” Princeton University. December 2005 <http://www.pacm.princeton.edu/graduate.html> • Rosen, Kenneth. Discrete Mathematics and its Applications. United States of America: McGraw-Hill, 1999. • “The Four Color Theorem.” Thomas, Robin. December 2005 <http://www.math.gatech.edu/~thomas/FC/fourcolor.html> • Multiple Articles, December 2005 <http://wikipedia.org> Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics