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Theory of Computation

計算理論

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Instructor: 顏嗣鈞

E-mail: yen@cc.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-2009.html

 

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TEXTBOOK

Introduction to Automata Theory, Languages, and Computation 3rd Edition

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

(Addison-Wesley, 2006)

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Introduction to Automata Theory, Languages, and Computation

John E. Hopcroft, Rajeev Motwani,

Jeffrey D. Ullman,

(2nd Ed. Addison-Wesley, 2001)

2nd Edition

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

Jeffrey D. Ullman,

(Addison-Wesley, 1979)

1st Edition

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Grading

HW/Project : 20%

Midterm exam.: 40%

Final exam.: 40%

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A simple computer

BATTERY

SWITCH

input: switch

output: light bulb

actions: flip switch

states: on, off

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A simple “computer”

BATTERY

SWITCH

off onstart

f

f

input: switch

output: light bulb

actions: f for “flip switch”

states: on, off

bulb is on if and only if there was an odd number of flips

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Another “computer”

BATTERY

off offstart

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inputs: switches 1 and 2

actions: 1 for “flip switch 1”actions: 2 for “flip switch 2”

states: on, off

bulb is on if and only if both switches were flipped an odd number of times

1

2

1

off on

1

1

2 2 2 2

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A design problem

Can you design a circuit where the light is on if and only if all the switches were flipped exactly the same number of times?

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BATTERY

1

2

3

5

?

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A design problem

• Such devices are difficult to reason about, because they can be designed in an infinite number of ways

• By representing them as abstract computational devices, or automata, we will learn how to answer such questions

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These devices can model many things

• They can describe the operation of any “small computer”, like the control component of an alarm clock or a microwave

• They are also used in lexical analyzers to recognize well formed expressions in programming languages:

ab1 is a legal name of a variable in C5u= is not

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Different kinds of automata

• This was only one example of a computational device, and there are others

• We will look at different devices, and look at the following questions:– What can a given type of device compute, and

what are its limitations?– Is one type of device more powerful than

another?

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Some devices we will seefinite automata Devices with a finite amount of memory.

Used to model “small” computers.

push-down automata

Devices with infinite memory that can be accessed in a restricted way.

Used to model parsers, etc.

Turing Machines Devices with infinite memory.

Used to model any computer.

time-bounded Turing Machines

Infinite memory, but bounded running time.

Used to model any computer program that runs in a “reasonable” amount of time.

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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