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

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

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3Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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

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5Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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)

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6Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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7Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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8Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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9Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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10Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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 recipients public key The private key decrypts the message

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11Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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12Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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

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14Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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15Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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

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17Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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

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19Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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

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

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22Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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23Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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

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

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26Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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27Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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28Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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29Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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)

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30Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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

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31Dr. Raymond Greenlaw – The Synergy of Computer Science and Mathematics 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 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 Rosen, Kenneth. Discrete Mathematics and its Applications. United States of America: McGraw-Hill, 1999. The Four Color Theorem. Thomas, Robin. December 2005 Multiple Articles, December 2005

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