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Theory of Computation

計算理論

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Instructor: 顏嗣鈞

E-mail: yen@ee.ntu.edu.tw

Web: http://www.ee.ntu.edu.tw/~yen

Time: 2:20-5:10 PM, Tuesday

Place: BL 112

Office hours: by appointment

Class web page: http://www.ee.ntu.edu.tw/~yen/courses/TOC-2006.htm

 

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• :

Introduction to Automata Theory, Languages, and Computation

John E. Hopcroft, Rajeev Motwani,

Jeffrey D. Ullman,

(2nd Ed. Addison-Wesley, 2001)

textbook

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1st Edition

Introduction to Automata Theory, Languages, and Computation John E. Hopcroft,

Jeffrey D. Ullman,

(Addison-Wesley, 1979)

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Grading

HW : 0-20%

Midterm exam.: 35-45%

Final exam.: 45-55%

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Why Study Automata Theory?

Finite automata are a useful model for important kinds of hardware and software:

• Software for designing and checking digital circuits.

• Lexical analyzer of compilers. • Finding words and patterns in large bodies of

text, e.g. in web pages.• Verification of systems with finite number of

states, e.g. communication protocols.

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Why Study Automata Theory? (2)

The study of Finite Automata and Formal Languages are intimately connected. Methods for specifying formal languages are very important in many areas of CS, e.g.:

• Context Free Grammars are very useful when designing software that processes data with recursive structure, like the parser in a compiler.

• Regular Expressions are very useful for specifying lexical aspects of programming languages and search patterns.

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Why Study Automata Theory? (3)

Automata are essential for the study of the limits of computation. Two issues:

• What can a computer do at all? (Decidability)

• What can a computer do efficiently? (Intractability)

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Applications

Theoretical Computer ScienceAutomata Theory, Formal Languages,

Computability, Complexity …

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Aims of the Course• To familiarize you with key Computer Science

concepts in central areas like- Automata Theory- Formal Languages- Models of Computation- Complexity Theory

• To equip you with tools with wide applicability in the fields of CS and EE, e.g. for- Complier Construction- Text Processing- XML

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Fundamental Theme

• What are the capabilities and limitations of computers and computer programs?– What can we do with

computers/programs?

– Are there things we cannot do with computers/programs?

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Studying the Theme

• How do we prove something CAN be done by SOME program?

• How do we prove something CANNOT be done by ANY program?

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Example: The Halting Problem (1)

Consider the following program. Does it terminate for all values of n 1?

while (n > 1) {if even(n) {

n = n / 2;} else {

n = n * 3 + 1;}

}

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Example: The Halting Problem (2)

Not as easy to answer as it might first seem.

Say we start with n = 7, for example:

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5,

16, 8, 4, 2, 1

In fact, for all numbers that have been tried

(a lot!), it does terminate . . .

. . . but in general?

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Example: The Halting Problem (3)Then the following important undecidability

result should perhaps not come as a total surprise:

It is impossible to write a program that decides if another, arbitrary, program terminates (halts) or not.

What might be surprising is that it is possible to prove such a result. This was first done by the British mathematician Alan Turing.

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Our focus

AutomataComputability

Complexity

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Topics1.  Finite automata, Regular languages, Regular

grammars: deterministic vs. nondeterministic, one-way vs. two-way finite automata, minimization, pumping lemma for regular sets, closure properties.

2.  Pushdown automata, Context-free languages, Context-free grammars: deterministic vs. nondeterministic, one-way vs. two-way PDAs, reversal bounded PDAs, linear grammars, counter machines, pumping lemma for CFLs, Chomsky normal form, Greibach normal form, closure properties.

3.

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Topics (cont’d)3. Linear bounded automata, Context-

sensitive languages, Context-sensitive grammars.

4. Turing machines, Recursively enumerable sets, Type 0 grammars: variants of Turing machines, halting problem, undecidability, Post correspondence problem, valid and invalid computations of TMs.

 

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Topics (cont’d)5. Basic recursive function theory 6. Basic complexity theory: Various

resource bounded complexity classes, including NLOGSPACE, P, NP, PSPACE, EXPTIME, and many more. reducibility, completeness.

7. Advanced topics: Tree Automata, quantum automata, probabilistic automata, interactive proof systems, oracle computations, cryptography.

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Who should take this course?

YOU

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Languages

The terms language and word are used in a strict technical sense in this course:

A language is a set of words.

A word is a sequence (or string) of symbols.

(or ) denotes the empty word, the sequence of zero symbols.

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Symbols and Alphabets

• What is a symbol, then?

• Anything, but it has to come from an alphabet which is a finite set.

• A common (and important) instance is = {0, 1}.

, the empty word, is never an symbol of an alphabet.

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Computation

CPU memory

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CPU

input memory

output memory

Program memory

temporary memory

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CPU

input memory

output memoryProgram memory

temporary memory

3)( xxf

compute xx

compute xx 2

Example:

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CPU

input memory

output memoryProgram memory

temporary memory

3)( xxf

compute xx

compute xx 2

2x

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CPU

input memory

output memoryProgram memory

temporary memory3)( xxf

compute xx

compute xx 2

2x

42*2 z82*)( zxf

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CPU

input memory

output memoryProgram memory

temporary memory3)( xxf

compute xx

compute xx 2

2x

42*2 z82*)( zxf

8)( xf

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Automaton

CPU

input memory

output memory

Program memory

temporary memory

Automaton

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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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input memory

output memory

temporary memory

Finite

Automaton

Finite Automaton

Example: Vending Machines

(small computing power)

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input memory

output memory

Stack

Pushdown

Automaton

Pushdown Automaton

Example: Compilers for Programming Languages

(medium computing power)

Push, Pop

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input memory

output memory

Random Access Memory

Turing

Machine

Turing Machine

Examples: Any Algorithm

(highest computing power)

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Finite

Automata

Pushdown

Automata

Turing

Machine

Power of Automata

Less power More power

Solve more

computational problems

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Mathematical Preliminaries

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Mathematical Preliminaries

• Sets

• Functions

• Relations

• Graphs

• Proof Techniques

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}3,2,1{AA set is a collection of elements

SETS

},,,{ airplanebicyclebustrainB

We write

A1

Bship

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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 > 0, and j = 2k for some k>0 }

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 }

1 2 34 5

A

U

6

78

910

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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

A-B

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A

• Complement

Universal set = {1, …, 7}

A = { 1, 2, 3 } A = { 4, 5, 6, 7}

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3

4

5

6

7

A

A = A

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024

6

1

3

5

7

even

{ even integers } = { odd integers }

odd

Integers

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DeMorgan’s Laws

A U B = A B

U

A B = A U BU

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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, 5 }

A B

U

Proper Subset: A B

UA

B

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Disjoint Sets

A = { 1, 2, 3 } B = { 5, 6}

A B =

UA 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

Powerset of S = the set of all the subsets of S

S = { a, b, c }

2S = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }

Observation: | 2S | = 2|S| ( 8 = 23 )

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Cartesian ProductA = { 2, 4 } B = { 2, 3, 5 }

A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 5) }

|A X B| = |A| |B|

Generalizes to more than two sets

A X B X … X Z

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FUNCTIONSdomain

123

a

bc

range

f : A -> B

A B

If A = domain

then f is a total function

otherwise f is a partial function

f(1) = a

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RELATIONS R = {(x1, y1), (x2, y2), (x3, y3), …}

xi R yi

e. g. if R = ‘>’: 2 > 1, 3 > 2, 3 > 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 ClassesFor 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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GRAPHSA directed graph

• Nodes (Vertices)

V = { a, b, c, d, e }

• Edges

E = { (a,b), (b,c), (b,e),(c,a), (c,e), (d,c), (e,b), (e,d) }

node

edge

a

b

c

d

e

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Labeled Graph

a

b

c

d

e

1 35 6

26

2

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Walk

a

b

c

d

e

Walk is a sequence of adjacent edges

(e, d), (d, c), (c, a)

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Path

a

b

c

d

e

Path is a walk where no edge is repeated

Simple path: no node is repeated

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Cycle

a

b

c

d

e

12

3

Cycle: a walk from a node (base) to itself

Simple cycle: only the base node is repeated

base

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Euler Tour

a

b

c

d

e1

23

45 6

78 base

A cycle that contains each edge once

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Hamiltonian Cycle

a

b

c

d

e1

23

4

5 base

A simple cycle that contains all nodes

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Treesroot

leaf

parent

child

Trees have no cycles

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root

leaf

Level 0

Level 1

Level 2

Level 3

Height 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 b that P1, P2, …, Pb are true

• for any k >= b that

P1, P2, …, Pk imply Pk+1

Then

Every Pi is true

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Proof by Induction• Inductive basis

Find P1, P2, …, Pb which are true

• Inductive hypothesis

Let’s assume P1, P2, …, Pk are true,

for any k >= b

• Inductive step

Show that Pk+1 is true

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Example

Theorem: A binary tree of height n

has at most 2n leaves.

Proof by induction:

let L(i) be the number of leaves at level i

L(0) = 1

L(1) = 2

L(2) = 4

L(3) = 8

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We want to show: L(i) <= 2i

• Inductive basis

L(0) = 1 (the root node)

• Inductive hypothesis

Let’s assume L(i) <= 2i for all i = 0, 1, …, k

• Induction step

we need to show that L(k + 1) <= 2k+1

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Induction Step

From Inductive hypothesis: L(k) <= 2k

Level

k

k+1

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L(k) <= 2k

Level

k

k+1

L(k+1) <= 2 * L(k) <= 2 * 2k = 2k+1

Induction Step

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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:

Assume 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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2

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= n/m 2 m2 = n2

Therefore, n2 is evenn is even

n = 2 k

2 m2 = 4k2 m2 = 2k2m is even

m = 2 p

Thus, m and n have common factor 2

Contradiction!

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