CSCI%202670%20Introduction%20to%20Theory%20of%20Computing - PowerPoint PPT Presentation

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CSCI%202670%20Introduction%20to%20Theory%20of%20Computing

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Title: PowerPoint Presentation Author: Wim van Dam Last modified by: Shelby Funk Created Date: 8/27/2001 7:35:01 AM Document presentation format: Letter Paper (8.5x11 in) – PowerPoint PPT presentation

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Title: CSCI%202670%20Introduction%20to%20Theory%20of%20Computing


1
CSCI 2670Introduction to Theory of Computing
  • Instructor Shelby Funk

2
Today
  • Syllabus
  • Chapter 0
  • Homework due Tuesday, August 23
  • Read pages Chapter 0
  • You are responsible for all this material even if
    I dont cover it in class
  • Send me an e-mail telling me your favorite thing
    about Athens
  • If youre new to Athens, tell me your favorite
    thing about your most recent hometown

3
Course goals
  • Theoretically explore the capabilities and
    limitations of computers
  • Complexity theory
  • What makes some problems computationally hard and
    others easy?
  • Computability theory
  • What problems can be solved by a computer?
  • Automata theory
  • How can we mathematically model computation?

4
Sets, multisets and sequences
  • Set
  • Order and repetition dont matter
  • 7,4,7,3 3,4,7
  • Multiset
  • Order doesnt matter, repetition does
  • 7,4,7,3 3,4,7,7 ? 3,4,7
  • Sequence
  • Order and repetition matter
  • (7,4,7,3) ? (3,4,7,7)
  • Finite sequence of k elements may be called a
    k-tuple

5
Set notation
  • Union A?B
  • Intersection A?B
  • Complement A
  • Cartesian Product A?B
  • Also called cross product
  • Power set P (A)

6
Example
  • A 1,2, B2,3, U x?Nx lt 6
  • A?B
  • A?B
  • A
  • A?B
  • P (A)
  • A 1,2, B2,3, U x?Nx lt 6
  • A?B 1,2,3
  • A?B 2
  • A 3,4,5
  • A?B (1,2), (1,3), (2,2), (2,3)
  • P (A) Ø, 1, 2, 1,2

7
Function
  • Mechanism associating each input value with
    exactly one output value
  • Domain set of all possible input values
  • Range set containing all possible output values
  • f D ? R

n
f (n)
f 1, 2, 3, 4 ? 2, 4
1 2 3 4
2 4 2 4
f 1, 2, 3, 4 ? 1, 2, 3, 4
8
Relation
  • Predicate function whose output value is always
    either true or false
  • Relation predicate whose domain is the set
    AAA
  • If domain is all k-tuples of A, the relation is a
    k-ary relation on A

9
Graphs
10
Graphs
11
Graphs
12
Graphs
Subgraph
Binary tree
13
Directed graphs
1
2
(2,1),(3,1),(4,3),(5,2)
3
4
5
14
Alphabets and strings
  • Alphabet any finite set
  • ?1 1,2,3
  • ?2 ?,?,?
  • String finite sequence of symbols from the given
    alphabet
  • 1212123
  • ??????
  • Empty string, e, contains no symbols of the
    alphabet
  • Language a set of strings

15
Boolean logic
  • Conjunction (and) ?
  • Disjunction (or) ?
  • Negation (not) ?
  • Exclusive or (xor) ?
  • Equality ?
  • Implication ?

16
Proof techniques
  • Construction
  • Prove a there exists statement by finding the
    object that exists
  • Contradiction
  • Assume the opposite and find a contradiction
  • Induction
  • Show true for a base case and show that if the
    property holds for the value k, then it must also
    hold for the value k 1

17
Have a great weekend!
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