1

Theory of Computation

計算理論

2

Instructor: 顏嗣鈞

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

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

Time: 9:10-12:10 PM, Monday

Place: BL 103

Office hours: by appointment

Class web page: http://www.ee.ntu.edu.tw/~yen/courses/TOC-2008.htm

 

3

Automata, Computability and Complexity: Theory and Applications

by Elaine A. Rich

Publisher: Pearson Pub. Date: October 2007 ISBN-13: 9780132288064

Textbook

4

• :

Introduction to Automata Theory, Languages, and Computation

John E. Hopcroft, Rajeev Motwani,

Jeffrey D. Ullman,

(2nd Ed. Addison-Wesley, 2001)

Reference

5

1st Edition

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

Jeffrey D. Ullman,

(Addison-Wesley, 1979)

6

3rd Edition• :

Introduction to Automata Theory, Languages, and Computation

John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman,

(Addison-Wesley, 2006)

7

Grading

HW/Project : 20%

Midterm exam.: 40%

Final exam.: 40%

8

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.

9

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.

10

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)

11

Applications

Theoretical Computer ScienceAutomata Theory, Formal Languages,

Computability, Complexity …

Com

pile

r

Pro

g.

lan

gu

ag

es

Com

m.

pro

toco

ls

circuits

Patte

rn

reco

gn

ition

Sup

erv

isory

co

ntro

l

Qu

an

tum

co

mpu

ting

Com

pu

ter-

Aid

ed

V

erifi

catio

n ...

12

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

13

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?

14

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?

15

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;}

}

16

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?

17

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.

18

Our focus

AutomataComputability

Complexity

19

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.

20

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.

 

21

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.

22

Who should take this course?

YOU

23

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.

24

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.

25

Computation

CPU memory

26

CPU

input memory

output memory

Program memory

temporary memory

27

CPU

input memory

output memoryProgram memory

temporary memory

3)( xxf

compute xx

compute xx 2

Example:

28

CPU

input memory

output memoryProgram memory

temporary memory

3)( xxf

compute xx

compute xx 2

2x

29

CPU

input memory

output memoryProgram memory

temporary memory3)( xxf

compute xx

compute xx 2

2x

42*2 z82*)( zxf

30

CPU

input memory

output memoryProgram memory

temporary memory3)( xxf

compute xx

compute xx 2

2x

42*2 z82*)( zxf

8)( xf

31

Automaton

CPU

input memory

output memory

Program memory

temporary memory

Automaton

32

Different Kinds of Automata

Automata are distinguished by the temporary memory

• Finite Automata: no temporary memory

• Pushdown Automata: stack

• Turing Machines: random access memory

33

input memory

output memory

temporary memory

Finite

Automaton

Finite Automaton

Example: Vending Machines

(small computing power)

34

input memory

output memory

Stack

Pushdown

Automaton

Pushdown Automaton

Example: Compilers for Programming Languages

(medium computing power)

Push, Pop

35

input memory

output memory

Random Access Memory

Turing

Machine

Turing Machine

Examples: Any Algorithm

(highest computing power)

36

Finite

Automata

Pushdown

Automata

Turing

Machine

Power of Automata

Less power More power

Solve more

computational problems