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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
• If youre new to Athens, tell me your favorite

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