Introduction
Chapter 0
Three Central Areas
1.
Automata
2.
Computability
3.
Complexity
Complexity Theory
•
Central Question:
What makes some problems
computationally hard and other easy?
–
Sorting (
easy
)
–
Computers can easily sort a billion
items in seconds.
–
Scheduling (
hard
)
–
With real constraints scheduling a
thousand courses to avoid room and instructor
conflicts can take years of computation.
•
Cryptography (secret encryption) depends on the
existence of sufficiently hard problems.
Computability
•
Based on the work of Kurt Gödel, Alan Turing,
and Alonzo Church.
•
Some very basic problems cannot be solved by
computers.
–
Example: Determining if a mathematical
statement is true or false.
•
Instead of hard vs. easy, Computability centers
on
solvable
vs.
unsolvable
problems.
Automata Theory
•
We construct a simple, formal and precise
model of a
computer
.
–
The precision and formality allow us to prove the
computability of problems
•
Three types of
computer machines
1.
Finite Automata
2.
Pushdown Automata
3.
Turing Machine
Sets
{7, 57, 7, 7, 21, 7} = {21, 7, 57}
Order does not matter
Repetition is not a factor
Infinite Sets
Subsets
Set Descriptions
Set Operations
Venn Diagrams
Venn Diagrams
Venn Diagrams
Venn Diagrams
•
What would A

B look like?
A
B
Sequences (Tuples)
Order and repetition matter
(7, 21, 57) different than (57, 21, 7)
(4,3) ordered pair,
whereas {4,3} set of size 2
(11, 21, 3, 24, 57) is a 5

tuple
Functions (Relations)
Domain & Range
Example
Binary Function (two inputs)
g(
x,y
) = z
where
x
,
y
and
z
are in {0,1,2,3}
Predicate
•
function whose range is {True, False}
•
Example:
f(
a,b
) = a
Beats
b
f(
scissors,paper
) = scissors
Beats
paper = TRUE
Binary Relation
•
Predicate whose domain is a set of ordered pairs
P: A x A
答牵攠F慬獥a
Example:
A = {paper, stone, scissors}
(scissors, stone)
F慬獥
⡳t潮攬捩獳潲猩s
T牵r
Predicates can be described as sets
S= { (
scissors,stone
), (paper, stone), (stone, scissors) }
Equivalence Relation
Binary Relation that is
1.
Reflexive
:
f(
x,x
)
is always true
2.
Symmetric
: if
f(
x,y
)
is true then
f(
y,x
)
is true
3.
Transitive
: if
f(
x,y
)
and
f(
y,z
)
are both true
then
f(
x,z
)
is true.
Example
f(
a,b
) = a Beat b;
•
Not reflexive:
f(
paper,paper
)
= false
•
Not symmetric:
f(paper, stone)
= true,
but
f(stone, paper)
= false
•
Not transitive
:
f(paper, stone)
and
f(stone, scissors)
is true,
but
f(paper, scissors)
is false
Example
f(a, b)
瑲略映愠㰽Ⱐ潴桥t睩w攠f慬ae
f(a, b)
愠㰽
R = {(
a,b
)  a <= b}
•
Always reflexive:
(
a,a
)
is always true for all a’s
•
Not always symmetric:
If
(
a,b
)
is true then
(
b,a
)
might be false.
•
Always transitive:
(
a,b
)
and
(
b,c
)
imply
(
a,c
)
Question
•
Is this an equivalence relation (i.e., reflexive,
symmetric, and transitive)?
f(
a,b
)
a != 1 AND b != 1
R = {(
a,b
)  a != 1 AND b != 1}
Equivalence Relations
•
f: S x S
{True, False}
•
S = {s
1
,
s
2
, s
3
, s
4
, s
5
}
f
s
1
s
2
s
3
s
4
s
5
s
1
True
True
s
2
True
True
s
3
True
True
True
s
4
True
True
True
s
5
True
True
True
Equivalence Relations & Automata
•
We are interested in functions that can separate a set
of items into disjoint classes.
•
{{
s
1
,
s
2
}, {
s
3
,
s
4
,
s
5
}}
•
In automata theory, the items will be languages.
•
The functions will be
automata and Turing machines
•
The classes will be
–
Regular languages
–
Context

free languages
–
Context

sensitive languages
Equivalence & Automata Theory
The theoretical “heart” of computation
•
Inputs, outputs and algorithms can all be
encoded as strings of a language
•
Solving problems can be reduced to defining
languages (set of strings)
that correspond to
correct solutions to a problem.
•
Then, solving a problem reduces to creating an
automata
that can recognize if a string is in a
particular language
–
i.e., is a string a solution to the encoded problem?
Graph Theory
Terminology
:
•
Nodes/vertices
•
Edges
•
Degree
•
Self

loop
•
Directed vs.
Undirected Graphs
•
Labeled Graph
•
Subgraph
Graph Theory
•
Paths, cycles, trees
•
Outdegree
,
Indegree
•
Directed path
•
Strongly connected
Meaningful graphs
•
Directed graphs can show relations
Equivalence Relations as Graphs
•
Example to be drawn
Strings & Languages
•
Defining alphabets (symbol domain)
String terminology
•
Length
w =
abc
,
w = 3
•
Empty string
ɛ
•
Reverse
w =
abc
,
w
R
=
cba
•
Substrings of
w = {ɛ, a, b, c,
ab
,
bc
,
abc
}
–
ba
,
cb
,
ca
,
cba
,
and
ac
are not substrings of
w
,
but they are subsets if
w
were considered a set
–
a
c
is not a substring, but it is subsequence (confusing)
•
Lexicographic order
{
ɛ
, 0, 1, 00, 01, 10, 11, 000, … }
Language
•
A language is a set of strings
•
Languages can be explicitly described
Animals = {cat, dog, cow, …}
{w  a
n
, n > 1} = {a,
aa
,
aaa
,
aaaa
, … }
•
Languages can also be described using
–
grammar rules
and
–
automata
that can recognize strings in the
language.
Boolean Logic
•
Review on your own, but
Boolean Logic
Proof by Construction
•
State your claim. Be very specific and describe
an object you claim exists
•
Construct the object. Show details and
describe parameters
•
Conclusion. Restate your goal in constructing
the object.
Example
Proof By Contradiction
•
Assume that the theorem is false
–
Even though you know it to be true
•
Show that this assumption leads to a
contradiction with something that has already
been proven true
Example
•
Prove that
•
First, assume that it is rational, i.e.,
•
Where m and n are integers
•
Also, m/n is reduced, so one of them has to be
even.
Example
•
Prove that
•
Do some algebra
•
We know that m must be even. Why?
Example
•
Prove that
•
We know that n must be even. Why?
•
Thus, m and n are even. Why is this a
contradiction?
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