A Brief History of Computer Science

A Brief History of Computer Science

William Klostermeyer

Typical ConversationA: What do you do?B: I’m a computer scientist.A: How come when I’m on screen X in MS Y.Z I can’t print file Q?

“Computer science is no more about computers than astronomy is about telescopes.”

E. Dijkstra (1972 Turing Award winner)

What is computer science?Sometimes called computing or computing science.

Not so much about computers, but computing.

Areas of Computer ScienceHardware/Architecture Building chips, machines, devicesSoftware Building programs, systems, databasesTheory Determines what is possible: underlies all other areas of computing.

History of computer sciencePre-dates modern computers by

more than 2000 years!Digital computers make it more

practical to compute.“Computer” was a job title for

people around time of WWII

Computers Electronic computers created

because of need for them: Ballistics computations Codebreaking Census calculations

Algorithm Precise set of instructions to solve

a problem. How to play blackjack: draw two cards; compute total while (total < 17) draw a card

add cards to total end

Mathematics of Algorithms Hilbert (1928): “Can every problem

be solved by a mechanical procedure?”

Turing (1936): NO! There exist problems no computer can solve.

Alan Turing: developedmodel of computation

Programs Computer programs implement

algorithms and are written in a computer language (Java, C, etc.)

Interested in efficient (FAST) algorithms/programs

Hard Problems Hilbert’s 10th problem (1900): Does

there exist an algorithm to find integer solutions to Diophantine equations?

x + 2y2 = 0 (use quadratic formula)

x6 + y6 = z6

Hilbert’s 10th problem Matiyasevich proved in 1970 that

no algorithm exists to solve arbitrary Diophantine equations.

Some problems yield themselves to algorithms, some don’t. Some yield themselves to EFFICIENT algorithms!

“Computer science is the systematic study of algorithmic processes”

From an ACM study (1989)

Ancient Algorithms Babylonians knew how to

approximate square roots (500 B.C.)

Newton’s method (1600’s) generalizes this to find zeroes of polynomial

“Numerical” algorithms

Euclid’s Algorithm 300 B.C. GCD(x, y) greatest common divisor of x & y

Still in use today!

GCD Algorithm Euclid(12, 9) = Euclid(9, 3) = Euclid(3, 0) = 3

Euclid(a, b): if b=0 return b else return Euclid (b, a mod b)

Prime Numbers x is prime if nothing divides x

evenly except 1 and x

Some primes:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Big primes useful in cryptography

Sieve of Eratosthenes 250 B.C. Algorithm to determine if x is prime:x=17. List all numbers from 2 to 16.Start with 2. If 2 does not divide 17,cross 2 and all multiples of 2 off list.Go to next uncrossed number on list and

repeat. If all numbers crossed off, then x is prime.

Primality Testing Eratosthenes method notfast for LARGE numbers(hundreds of digits) Fast probabilistic methodsdeveloped in ’70’s, ’80’s(have a small chance of error)

Breakthrough! 2002: after centuries of searching,

a fast algorithm found (polynomial-time

algorithm, no chance of error)IIT professor and two undergrads:

Uses of Prime Numbers Large primes needed in

cryptography (to securely send information over internet)

RSA encryption algorithm based on assumption that factoring large integers into primes is difficult.

Sending Messages Securely

Alice BobAlice wants to send to Bob, not reveal key

Sending Messages Securely

Alice Bob

Sending Messages Securely

Alice Bob

Sending Messages Securely

Alice Bob

How can Alice do it? Alice puts lock on box Alice sends box to Bob Bob puts his lock on Bob sends to Alice Alice removes her lock Alice sends to Bob Bob removes his lock!

RSA Algorithm1. Find P and Q, two large (e.g.,

1024-bit) prime numbers. 2. Choose E such that E is greater

than 1, E is less than PQ, and E and (P-1)(Q-1) are relatively prime. E must be odd.

3. Compute D such that (DE - 1) is evenly divisible by (P-1)(Q-1).

RSA cont.1. The encryption function is

C = (T^E) mod PQ, where C is the ciphertext (a positive integer), T is the plaintext (a positive integer).

2. Your public key is the pair (PQ, E). Your private key is the number D. D used in decrypting C back into T.

No known easy methods of calculating D, P, or Q given only (PQ, E) (your public key).

Factoring RSA security based on assumption that

FACTORING large integers is hard. Note that we can determine if an

integer is prime or not quickly, but factoring seems to require more work

In other words: can prove an integer is composite w/o showing factors

Graph Algorithms Euler 1736: Bridges of Konigsberg

Konigsberg = Graph

Graphs

“Concepts from graph theory may hold

the key for everything…”

Business Week, January 22, 2002

Shortest Routes Dijkstra’s Algorithm (1959): used

by www.mapquest.com

Faster algorithms found in 1990’s

Traveling Salesman Salesman wants to visit N cities

and return home Minimize total distance traveled

TSP Algorithms? No fast algorithm known to compute

best route for salesman Computing optimal route for 1000

cities would take centuries on fast computer

Must settle for near-optimal solutions: Can get very close to optimal if cities are in Euclidean plane (Arora 1999)

Map Coloring Can map be colored with 4 colors

so neighboring regions have different colors (1850’s)

4-color theorem Yes! (Appel & Haken 1970’s) Can be done quickly (1990’s) But some maps require only 3

colors!

Easy & Hard Which is harder?

Taking a test Grading a test

Proving a theorem Checking a proof

Hilbert and von Neumann pondered this.

P vs. NP P: problems that can be solved

quickly (shortest route) NP: problems whose solutions can

be checked quickly (TSP, 3-coloring)

Is this a 3-coloring?

P = NP? Believed P is not equal to NP $1,000,000 reward for proof: www.claymath.org

Most important problem in Computer Science: are certain problems intrinsically hard?

NP-complete problems Class of hard problems (unless P=NP) Occur in real world:

Scheduling problems, routing problems, Biological problems, etc.

Optimal solutions take too long to compute (we believe, if P not equal to NP)

Must settle for sub-optimal solutions.

Example Find best match for criminal’s DNA

sequence in a database:AATCCGATAGGATATTCCAGATCGATTACCGATAGACATGTACAGGCAATCAGATACAAATCCGAAAACCC

Sequence Assembly Assemble small overlapping

sequences into single long sequence

AAAATCGC, CGCATAAA, GCATCTCATT

CGCATAAAATCGCATCTATT

Can be hard if you start with lots of fragments

Complexity Theory Interested in categorizing how hard

(complex) problems are. Key concept is a “proof” that an

object satisfies some conditions. How might we “prove” that 11 is

prime?

See that none of 2, 3, 5, 7 divide 11.

Checking Proofs How do we grade a test? How do we check a proof?

xn + yn = zn, no solutions for n > 2

By reading it! By reading every word of it!

PCP Theorem 1990’s was found that proofs can

be checked (with high probability) by reading only a few characters.

(Proof must be in a standard form) “Holographic” proofs: each symbol

contains essence of proof

Amazing connection PCP Theorem also tells us that for

certain hard problems, finding sub-optimal solutions quickly is hard!

Hard to quickly find a route that is close to the longest possible route between two cities!

Conclusion Computer science is about:

Algorithms to solve problems!