1 Unit 1: Automata Theory and Formal Languages Readings 1, 2.2, 2.3

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Unit 1: Automata Theoryand Formal Languages

Readings 1, 2.2, 2.3

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What is automata theory

• Automata theory is the study of abstract computational devices

• Abstract devices are (simplified) models of real computations

• Computations happen everywhere: On your laptop, on your cell phone, in nature, …

• Why do we need abstract models?

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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

1

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 C

5u= 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 see

finite 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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Some highlights of the course

• Finite automata– We will understand what kinds of things a

device with finite memory can do, and what it cannot do

– Introduce simulation: the ability of one device to “imitate” another device

– Introduce nondeterminism: the ability of a device to make arbitrary choices

• Push-down automata– These devices are related to grammars, which

describe the structure of programming (and natural) languages

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• Turing Machines– This is a general model of a computer,

capturing anything we could ever hope to compute

– Surprisingly, there are many things that we cannot compute, for example:

– It seems that you should be able to tell just by looking at the program, but it is impossible to do!

Write a program that, given the code of another program in C, tells if this program ever outputs the word “hello”

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• Time-bounded Turing Machines– Many problems are possible to solve on a

computer in principle, but take too much time in practice

– Traveling salesman: Given a list of cities, find the shortest way to visit them and come back home

– Easy in principle: Try the cities in every possible order

– Hard in practice: For 100 cities, this would take 100+ years even on the fastest computer!

Hong Kong

Beijing

ShanghaiXian

Guangzhou

Chengdu

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Preliminaries of automata theory

• How do we formalize the question

• First, we need a formal way of describing the problems that we are interested in solving

Can device A solve problem B?

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Problems

• Examples of problems we will consider– Given a word s, does it contain the subword

“fool”?– Given a number n, is it divisible by 7?– Given a pair of words s and t, are they the

same?– Given an expression with brackets, e.g. (()()),

does every left bracket match with a subsequent right bracket?

• All of these have “yes/no” answers. • There are other types of problems, that ask

“Find this” or “How many of that” but we won’t look at those.

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Alphabets and strings

• A common way to talk about words, number, pairs of words, etc. is by representing them as strings

• To define strings, we start with an alphabet

• Examples

An alphabet is a finite set of symbols.

S1 = {a, b, c, d, …, z}: the set of letters in English

S2 = {0, 1, …, 9}: the set of (base 10) digits

S3 = {a, b, …, z, #}: the set of letters plus the special symbol #

S4 = {(, )}: the set of open and closed brackets

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Strings

• The empty string will be denoted by e• Examples

A string over alphabet S is a finite sequenceof symbols in S.

abfbz is a string over S1 = {a, b, c, d, …, z}

9021 is a string over S2 = {0, 1, …, 9}

ab#bc is a string over S3 = {a, b, …, z, #}

))()(() is a string over S4 = {(, )}

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Languages

• Languages can be used to describe problems with “yes/no” answers, for example:

A language is a set of strings over an alphabet.

L1 = The set of all strings over S1 that containthe substring “fool”

L2 = The set of all strings over S2 that are divisible by 7 = {7, 14, 21, …}L3 = The set of all strings of the form s#s where s is any

string over {a, b, …, z}L4 = The set of all strings over S4 where every ( can be

matched with a subsequent )

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Finite Automata

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Example of a finite automaton

• There are states off and on, the automaton starts in off and tries to reach the “good state” on

• What sequences of fs lead to the good state?

• Answer: {f, fff, fffff, …} = {f n: n is odd}• This is an example of a deterministic finite

automaton over alphabet {f}

off on

f

f

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Deterministic finite automata

• A deterministic finite automaton (DFA) is a 5-tuple (Q, S, d, q0, F) where– Q is a finite set of states– S is an alphabet– d: Q × S → Q is a transition function– q0 Î Q is the initial state– F Í Q is a set of accepting states (or final

states).

• In diagrams, the accepting states will be denoted by double loops

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Example

q0 q1 q21 0

0 0,11

alphabet S = {0, 1}set of states Q = {q0, q1, q2}initial state q0

accepting states F = {q0, q1}

state

s

inputs

0 1q0

q1

q2

q0 q1

q2

q2q2

q1

transition function :d

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Language of a DFA

The language of a DFA (Q, S, d, q0, F) is the set of all strings over S that, starting from q0 and following the transitions as the string is read leftto right, will reach some accepting state.

• Language of M is {f, fff, fffff, …} = {f n: n is odd}

off on

f

f

M:

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q0 q1

q0 q1

q0 q1 q2

0 0

1

1

0 1

1

0

1 0

0 0,11

What are the languages of these DFAs?

Examples

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Examples

• Construct a DFA that accepts the language

L = {010, 1}

( S = {0, 1} )

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Examples

• Construct a DFA that accepts the language

• Answer

L = {010, 1}

( S = {0, 1} )

qe

q0

q1

q01 q010

qdie 0, 1

0

1 0

0, 11

0 10, 1

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Examples

• Construct a DFA over alphabet {0, 1} that accepts all strings that end in 101

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Examples


• Hint: The DFA must “remember” the last 3 bits of the string it is reading

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Examples

0

1

…

…

……

qe

q0

q1

q00

q10

q01

q11

q000

q001

q101

q111

0

1

0

1

0

1

1

1

1

1

0


• Sketch of answer:

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Would be easier if…

• Suppose we could guess when the string we are reading has only 3 symbols left

• Then we could simply look for the sequence 101and accept if we see it

qdie

1 0 13 symbols left

This is not a DFA!

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Nondeterminism

• Nondeterminism is the ability to make guesses, which we can later verify

• Informal nondeterministic algorithm for language of strings that end in 101:

1. Guess if you are approaching end of input2. If guess is yes, look for 101 and accept if you see it3. If guess is no, read one more symbol and go to step 1

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Nondeterministic finite automaton

• This is a kind of automaton that allows you to make guesses

• Each state can have zero, one, or more transitions out labeled by the same symbol

1 0 1

0, 1

q0 q1 q2 q3

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Semantics of guessing

1 0 1

0, 1

q0 q1 q2 q3

• State q0 has two transitions labeled 1

• Upon reading 1, we have the choice of staying in q0 or moving to q1

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1 0 1

0, 1

q0 q1 q2 q3

• State q1 has no transition labeled 1

• Upon reading 1 in q1, we die; upon reading 0, we continue to q2

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1 0 1

0, 1

q0 q1 q2 q3

• State q3 has no transition going out

• Upon reading anything in q3, we die

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Meaning of automaton

1 0 1

0, 1

q0 q1 q2 q3

Guess if you are 3 symbols away from end of input

If so, guess you will see the pattern 101

Check that you are at the end of input

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Formal definition

• A nondeterministic finite automaton (NFA) is a 5-tuple (Q, S, d, q0, F) where– Q is a finite set of states– S is an alphabet– d: Q × S → subsets of Q is a transition function– q0 Î Q is the initial state– F Í Q is a set of accepting states (or final

states).

• Only difference from DFA is that output of d is a set of states

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Example

1 0 1

0, 1

q0 q1 q2 q3

alphabet S = {0, 1}states Q = {q0, q1, q2, q3}initial state q0

accepting states F = {q3}

stat

es

inputs0 1

q0

q1

q2

{q0, q1}

transition function :d

q3

{q0}

{q2}

Æ

Æ

Æ

{q3}

Æ

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Language of an NFA

The language of an NFA is the set of all strings forwhich there is some path that, starting from the initial state, leads to an accepting state as the string is read left to right.

1 0 1

0, 1

q0 q1 q2 q3

• Example

– 1101 is accepted, but 0110 is not

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NFAs are as powerful as DFAs

• Obviously, an NFA can do everything a DFA can do

• But can it do more?

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NFAs are as powerful as DFAs

• Obviously, an NFA can do everything a DFA can do

• But can it do more?

• TheoremA language L is accepted by some DFA if andonly if it is accepted by some NFA.

NO!

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Proof of theorem

• To prove the theorem, we have to show that for every NFA there is a DFA that accepts the same language

• We will give a general method for simulating any NFA by a DFA

• Let’s do an example first

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Simulation example

1 0

0, 1

q0 q1 q2NFA:

DFA: 1q0 q0 or q1

1

q0 or q2

1

0 0

0

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General method

NFA DFAstates q0, q1, …,

qn

, {Æ q0}, {q1}, {q0,q1}, …, {q0,…,qn}

one for each subset of states in the NFA

initial state

q0 {q0}

transitions

d d’({qi1,…,qik}, a) =

d(qi1, a) ∪…∪ d(qik, a)

accepting states

F Í Q F’ = {S: S contains some state in F}

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Proof of correctness

• Lemma

• Proof by induction on n• At the end, the DFA accepts iff it is in a

state that contains some accepting state of NFA

• By lemma, this is true iff the NFA can reach an accepting state

After reading n symbols, the DFA is in state {qi1,…,qik} if and only if the NFA is in one of the states qi1,…,qik

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Exercises

• Construct NFAs for the following languages over the alphabet {a, b, …, z}:– All strings that contain eat or sea or easy– All strings that contain both sea and tea– All strings that do not contain fool

Documents

1 Unit 1: Automata Theory and Formal Languages Readings 1, 2.2, 2.3