CSCI2400

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CSCI-2400 Models of Computation
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Computation
memory
CPU
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temporary memory
input memory
CPU
output memory
Program memory
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Example
temporary memory
input memory
CPU
output memory
Program memory
compute
compute
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temporary memory
input memory
CPU
output memory
Program memory
compute
compute
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temporary memory
input memory
CPU
output memory
Program memory
compute
compute
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temporary memory
input memory
CPU
Program memory
output memory
compute
compute
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Automaton
temporary memory
Automaton
input memory
CPU
output memory
Program memory
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Different Kinds of Automata

• Automata are distinguished by the temporary
  memory
• Finite Automata no temporary memory
• Pushdown Automata stack
• Turing Machines random access memory

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Finite Automaton
temporary memory
input memory
Finite Automaton
output memory
Vending Machines (small computing power)
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Pushdown Automaton
Stack
Push, Pop
input memory
Pushdown Automaton
output memory
Programming Languages (medium computing power)
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Turing Machine
Random Access Memory
input memory
Turing Machine
output memory
Algorithms (highest computing power)
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Power of Automata
Finite Automata
Pushdown Automata
Turing Machine
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We will show later in class

• How to build compilers for programming languages
• Some computational problems cannot be solved
• Some problems are hard to solve

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Mathematical Preliminaries
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• Mathematical Preliminaries
• Sets
• Functions
• Relations
• Graphs
• Proof Techniques

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SETS

A set is a collection of elements
We write
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Set Representations C a, b, c, d, e, f, g,
h, i, j, k C a, b, , k S 2, 4, 6,
S j j gt 0, and j 2k for some kgt0
S j j is nonnegative and even
finite set
infinite set
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A 1, 2, 3, 4, 5
Universal Set All possible elements
U
1 , , 10
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• Set Operations
• A 1, 2, 3 B 2, 3, 4, 5
• Union
• A U B 1, 2, 3, 4, 5
• Intersection
• A B 2, 3
• Difference
• A - B 1
• B - A 4, 5

U
A-B
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• Complement
• Universal set 1, , 7
• A 1, 2, 3 A 4, 5, 6, 7

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A
A
6
3
1
2
5
7
A A
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even integers odd integers
Integers
1
odd
0
5
even
6
2
4
3
7
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DeMorgans Laws
A U B A B
U
A B A U B
U
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Empty, Null Set

S U S S S - S
- S
U
Universal Set
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Subset
A 1, 2, 3 B 1, 2, 3, 4,
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Proper Subset
B
A
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Disjoint Sets
A 1, 2, 3 B 5, 6
A
B
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Set Cardinality

• For finite sets

A 2, 5, 7 A 3
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Powersets
A powerset is a set of sets
S a, b, c
Powerset of S the set of all the subsets of S
2S , a, b, c, a, b, a, c, b,
c, a, b, c
Observation 2S 2S ( 8 23 )
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Cartesian Product
A 2, 4 B 2, 3, 5 A
X B (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3),
(4, 4) A X B A B Generalizes to more
than two sets A X B X X Z
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FUNCTIONS
domain
range
B
A
f(1) a
a
1
2
b
c
3
f A -gt B
If A domain then f is a total function
otherwise f is a partial function
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RELATIONS
R (x1, y1), (x2, y2), (x3, y3),
xi R yi e. g. if R gt 2 gt 1,
3 gt 2, 3 gt 1 In relations xi can
be repeated
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Equivalence Relations

• Reflexive x R x
• Symmetric x R y y R x
• Transitive x R Y and y R z
  x R z
• Example R
• x x
• x y y x
• x y and y z x z

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Equivalence Classes
For equivalence relation R equivalence
class of x y x R y Example
R (1, 1), (2, 2), (1, 2), (2, 1),
(3, 3), (4, 4), (3, 4), (4, 3)
Equivalence class of 1 1, 2 Equivalence
class of 3 3, 4
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GRAPHS
A directed graph

• Nodes (Vertices)
• V a, b, c, d, e
• Edges
• E (a, b), (b, c), (c, a), (b, d), (d,
  c), (e, d)

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Labeled Graph
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Walk
Walk is a sequence of adjacent edges
(e, d), (d, c), (c, a)
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Path
Path is a walk where no edge is repeated Simple
path no node is repeated
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Cycle
e
base
b
3
1
d
a
2
c
Cycle a walk from a node (base) to
itself Simple cycle only the base node is
repeated
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Euler Tour
A cycle that contains each edge once
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Hamiltonian Cycle
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base
e
1
b
4
d
a
2
3
c
A simple cycle that contains all nodes
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Finding All Simple Paths
f
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Step 1
e
b
f
d
a
c
(c, a) (c, e)
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Step 2
e
b
f
d
a
c
(c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c,
e), (e, d)
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Step 3
e
b
f
d
a
c
(c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c,
e), (e, d) (c, e), (e, d), (d, f)
Repeat the same for each starting node
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Trees
root
parent
leaf
child
Trees have no cycles
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root
Level 0
Level 1
Height 3
leaf
Level 2
Level 3
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Binary Trees
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PROOF TECHNIQUES

• Proof by induction
• Proof by contradiction

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Induction
We have statements P1, P2, P3,

• If we know
• for some k that P1, P2, , Pk are true
• for any n gt k that
• P1, P2, , Pn imply Pn1
• Then
• Every Pi is true

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Proof by Induction

• Inductive basis
• Find P1, P2, , Pk which are true
• Inductive hypothesis
• Lets assume P1, P2, , Pn are true,
• for any n gt k
• Inductive step
• Show that Pn1 is true

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Example
Theorem A binary tree of height n
has at most 2n leaves.
Proof let l(i) be the number of
leaves at level i
l(0) 1 l(3) 8
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• We want to show l(i) lt 2i
• Inductive basis
• l(0) 1 (the root node)
• Inductive hypothesis
• Lets assume l(i) lt 2i for all i 0, 1, , n
• Induction step
• we need to show that l(n 1) lt 2n1

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Induction Step
Level
hypothesis l(n) lt 2n
n
n1
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Induction Step
Level
hypothesis l(n) lt 2n
n
n1
l(n1) lt 2 l(n) lt 2 2n 2n1
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Remark
Recursion is another thing Example of recursive
function f(n) f(n-1) f(n-2) f(0) 1,
f(1) 1
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Proof by Contradiction

• We want to prove that a statement P is true
• we assume that P is false
• then we arrive at an incorrect conclusion
• therefore, statement P must be true

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Example
Theorem is not rational Proof Ass
ume by contradiction that it is rational
n/m n and m have no common
factors We will show that this is impossible
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n/m 2 m2 n2
n is even n 2 k
Therefore, n2 is even
m is even m 2 p
2 m2 4k2
m2 2k2
Thus, m and n have common factor 2
Contradiction!