CS514 Introductory Lecture

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CS514 -- Introductory Lecture

• Instructor Bob Amar, raa4_at_cs.wustl.edu
• READING FOR NEXT TIME
• Rosen, Chapters 1.1 - 1.3

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The Big Question

• What Is
• Computer Science?

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The Big Question, cont.

• Not really about computers
• Thats computer/electrical engineering
• Not really about science
• Programs are often written for reasons that are
  far from experimental
• Not really about programs
• Programs are implementations of algorithms
• algorithm stepwise procedure for doing something

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The Big Question, cont.

• Computer Science is really about computability
• Can a certain class of problems be decided
  (solved) with a computer program?
• If so, how fast?
• If not, why not?
• In CS514, we will lay down a theoretical
  groundwork for the design and analysis of
  algorithms for a wide variety of problems

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Course Overview

• First, we will explore the realm of formal logic
  and boolean algebra
• How can we make a computer?
• Second, we will explore modern computer hardware
  and its design
• How does a computer compute?
• Finally, we will discuss modern computer science
  algorithms
• What can a computer compute?

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CS514 Mission Statement

• By taking this course, you will jump-start your
  graduate career by acquiring the fundamentals to
  attack any area or application of computer
  science.

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So...

• lets get started!

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Why Logic?

• We start with logic because it cleanly bridges
  the gap between mathematics and the processes
  used to solve problems
• Let us first see how to think logically about
  certain entertaining mathematical problems
• Always two questions for understanding the
  implications of a problem
• Can we find an answer?
• If so, why is it the answer? Why does it work?

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Logic As A System

• As pattern-recognizers, we tend to solve problems
  by working outside the system
• The MIU Puzzle (thanks to Douglas Hofstadter)
  Start with the string MI
• 1. If string ends in I, you may add U to the
  end
• 2. The string Mx may be replaced with Mxx
  where x is any sequence of characters
• 3. Within a string, III may be replaced by U
  (but not the other way around!)
• 4. Within a string, UU may be deleted
• Can you make the string MU? Try it!

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Logic As A System, cont.

• After some fiddling, you should begin to feel
  that MU cannot be made -- but why?
• Answer on next slide!
• The core of solving a problem in computing really
  boils down to capturing the essence of its
  solution in a program
• No hand-waving will work -- a computer cant
  synthesize truths outside the system of its
  programming -- we have to capture those first

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The Nature of Logical Proof

• Statement MI cannot make MU
• The key element is that you can never get rid of
  all the Is -- in other words, the number of Is
  is never a multiple of 3 (in particular, never 0)
• For any string, you can double the number of Is
  using Rule 2 -- this never produces a multiple of
  3 unless the number of Is is initially a
  multiple of 3!
• You can reduce the number of Is by 3 using Rule
  3
• Neither of these can change one I into zero Is,
  and no other rules change the number of Is
• This is a proof -- sound logical argument

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Seemingly Simple Questions...

• Heres an old mathematical adage
• If a and b are odd integers, then a b is odd
• Try it and see if you believe it
• Heres the question WHY is this true?
• Just because it works for every example you can
  think of doesnt mean it works for every example
• You have to think outside the box -- outside of
  what you can observe -- and find the essence

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Do You Believe This?

• Statement If a is odd and b is odd, then the
  quantity a b must be odd.
• a is odd exactly when there is an integer m such
  that a 2m1
• Also, there is an integer n where b2n1
• a b (2m1)(2n1) 4mn 2m 2n 1 2(2mn
  m n) 1
• Thus, there is an integer p 2mn m n such
  that a b 2p 1, thus a b is odd

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Next Time

• We will introduce a symbolism for logic and a
  formal system that will help take some of the
  thinking out of applying logic
• During the course, we will see formal proof
  techniques and ways of structuring logical
  arguments for things we want to prove