FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

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15-453
FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
2
YOU NEED
THE SYLLABUS
TODAYS CLASS NOTES
A BLUE SHEET OF PAPER
A MARKER
3
TODAYS CLASS WILL START AT 1202
YOU NEED TO PICK UP THE SYLLABUS, TODAYS CLASS
NOTES, A BLUE SHEET OF THICK PAPER, AND A MARKER
15-453
FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
4
DONT FORGET
THE SYLLABUS
TODAYS CLASS NOTES
A BLUE SHEET OF PAPER
A MARKER
5
TODAYS CLASS WILL START AT 1202
YOU NEED TO PICK UP THE SYLLABUS, TODAYS CLASS
NOTES, A BLUE SHEET OF THICK PAPER, AND A MARKER
15-453
FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
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15-453
FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
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Suppose A ? 1, 2, , 2n
with A n1
TRUE or FALSE There are always two numbers in A
such that one divides the other
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INSTRUCTORS
Manuel Blum
Lenore Blum
Luis von Ahn
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This class is about mathematical models of
computation
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WHY SHOULD I CARE?
WAYS OF THINKING
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WHY SHOULD I CARE?
THEORY CAN DRIVE PRACTICE
Mathematical models of computation predated
computers as we know them
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WHY SHOULD I CARE?
THIS STUFF IS USEFUL
PART 1
Automata and Languages
PART 2
Computability Theory
PART 3
Complexity and Applications
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HOMEWORK
Homework will be assigned every Thursday and will
be due one week after at the beginning of class
You must list your collaborators and references
in every homework assignment
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COURSE PROJECT
Choose a (unique) topic
Learn about your topic
Actually write progress reports
Meet with an instructor once a month
Prepare a 10-minute presentation
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Suppose A ? 1, 2, , 2n
with A n1
TRUE or FALSE There are always two numbers in A
such that one divides the other
TRUE
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HINT
THE PIGEONHOLE PRINCIPLE
If you put 6 pigeons in 5 holes then at least one
hole will have more than one pigeon
If there are more girls than guys and every girl
is dating a guy, there must be a guy thats
cheating
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HINT 2
Every integer a can be written as a 2km, where
m is an odd number
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Suppose A ? 1, 2, , 2n
with A n1
Write every number in A as a 2km, where m is an
odd number
between 1 and 2n-1
How many odd numbers in 1, , 2n-1?
n
Since A n1, there must be two numbers in A
with the same odd part
Say a1 and a2 have the same odd part m. Then a1
2im and a2 2jm, so one must divide the other
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This class will emphasize PROOFS
A good proof should be
Easy to understand
Correct
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We expect your proofs to have three levels
The first level should be a one-word or
one-phrase hint of the proof
(e.g. Proof by contradiction, Follows from
the pigeonhole principle)
The second level should be a short one-paragraph
description or longer hint
The third level should be the full proof
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DOUBLE STANDARDS?
During the lectures, my proofs will usually only
contain the first two levels and maybe part of
the third
Reasons
I have to keep you guys awake
I know my proofs are correct
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DETERMINISTIC FINITE AUTOMATA
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11
1
0
0,1
1
0111
111
1

0
0
1
The machine accepts a string if the process ends
in a double circle
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ANATOMY OF A DETERMINISTIC FINITE AUTOMATON
q1

q0
q2
The machine accepts a string if the process ends
in a double circle
start state (q0)
q3
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NOTATION
An alphabet S is a finite set (e.g., S 0,1)
A string over S is a finite-length sequence of
elements of S
For x a string, x is
the length of x
The unique string of length 0 will be denoted by
e and will be called the empty or null string
A language over S is a set of strings over S
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A finite automaton is a 5-tuple M (Q, S, ?, q0,
F)
Q is the set of states
S is the alphabet
? Q ? S ? Q is the transition function
q0 ? Q is the start state
F ? Q is the set of accept states
L(M) the language of machine M set of all
strings machine M accepts
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q0
0,1
L(M)
?
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0
0
1
q0
q1
1
w w has an even number of 1s
L(M)
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Build an automaton that accepts all and only
those strings that contain 001
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A language is regular if it is recognized by a
deterministic finite automaton
L w w contains 001 is regular
L w w has an even number of 1s is regular
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UNION THEOREM
Given two languages, L1 and L2, define the union
of L1 and L2 as L1 ? L2 w w ? L1 or w ? L2

Theorem The union of two regular languages is
also a regular language
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Theorem The union of two regular languages is
also a regular language
Proof Let M1 (Q1, S, ?1, q0, F1) be finite
automaton for L1 and M2 (Q2, S, ?2, q0, F2)
be finite automaton for L2
1
2
We want to construct a finite automaton M (Q,
S, ?, q0, F) that recognizes L L1 ? L2
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Idea Run both M1 and M2 at the same time!
F (q1, q2) q1 ? F1 or q2 ? F2
?( (q1,q2), ?) (?1(q1, ?), ?2(q2, ?))
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Theorem The union of two regular languages is
also a regular language
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1
q0,p0
q1,p0
1
0
0
0
0
1
q0,p1
q1,p1
1
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