Automata Theory and Languages, Lecture notes of Theory of Automata

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Automata Theory and

Languages

SITE :

http://www.info.univ-tours.fr/˜mirian/

Automata Theory, Languages and Computation - M´

Introduction to Automata Theory

Automata theory :

the study of abstract computing devices, or ”machines”

Before computers (1930),

A. Turing

studied an abstract machine (

Turing

machine

) that had all the capabilities of today’ s computers (concerning what they

could compute).

His goal was to describe precisely the boundary between what a

computing machine could do and what it could not do. Simpler kinds of machines (

finite automata

) were studied by a number of

researchers and

useful for a variety of purposes

Theoretical developments bear directly on what computer scientists do today

Finite automata, formal grammars: design/ construction of software Turing machines: help us understand what we can expect from a software Theory of intractable problems: are we likely to be able to write a programto solve a given problem? Or we should try an approximation, a heuristic...

Automata Theory, Languages and Computation - M´

The Central Concepts of Automata Theory

Automata Theory, Languages and Computation - M´

Alphabet

A finite, nonempty set of symbols. Symbol:

Examples:

The binary alphabet:

0 ,^

The set of all lower-case letters:

{a, b,... , z

The set of all ASCII characters

Automata Theory, Languages and Computation - M´

Powers of an alphabet (1)

If^ Σ

is an alphabet, we can express the set of all strings of a certain length from that alphabet by using the exponential notation:

k Σ : the set of strings of length

k , each of whose is in

Examples:

:^

{ǫ

}, regardless of what alphabet

is. That is

ǫ^

is the only string of

length

If^

0 ,^

, then:

,^01

,^10

,^11

,^001

,^010

,^011

,^100

,^101

,^110

,^111

Note: confusion between

and

Σ^

is an alphabet; its members

and

are symbols

is a set of strings; its members are strings (each one of length 1)

Automata Theory, Languages and Computation - M´

Kleen star ∗ Σ : The set of all strings over an alphabet

{^0

,^1

ǫ,^

0 ,^

,^01

,^10

,^11

,^000

The symbol

∗^

is called

Kleene star

and is named after the mathematician and

logician Stephen Cole Kleene.+ Σ

Thus:

∗^

+^

ǫ}

Automata Theory, Languages and Computation - M´

Languages

If^

Σ^

is an alphabet, and

L

∗, then

L

is a (formal)

language

over

Language: A (possibly infinite) set of strings all of which are chosen from some∗ Σ A language over

need not include strings with all symbols of

Thus, a language over

is also a language over any alphabet that is a superset

of

Examples:

Programming language CLegal programs are a subset of the possible strings that can be formedfrom the alphabet of the language (a subset of ASCII characters) English or French

Automata Theory, Languages and Computation - M´

ırian Halfeld-Ferrari – p. 10/

Other language examples

1.^

The language of all strings consisting of

n

0 s followed by

n

1 s (

n^

{ǫ,

,^0011

,^000111

2.^

The set of strings of

0 s and

1 s with an equal number of each: {ǫ,

,^10

,^0011

,^0101

,^1001

3.^

is a language for any alphabet

4.^

∅, the empty language, is a language over any alphabet

5.^

{ǫ

}, the language consisting of only the empty string, is also a language over any alphabetNOTE:

∅ 6^

ǫ}

since

∅^

has no strings and

{ǫ

}^

has one

6.^

{w

|^

w^

consists of an equal number of

and

7.^

{^0

n^1

n^

|^ n

8.^

{^0

i^1

j^ |

i^

j}

Automata Theory, Languages and Computation - M´

ırian Halfeld-Ferrari – p. 11/

Important operators on languages:Concatenation

The

concatenation

of languages

L

and

M

, denoted

L.M

or just

LM

, is the set of

strings that can be formed by taking any string in

L

and concatenating it with any string

in^

M

Example

If^ L

,^10

,^111

}^

and

M

ǫ,^

}^ then

L.M

{^001

,^10

,^111

,^001001

,^10001

,^111001

Automata Theory, Languages and Computation - M´

Important operators on languages: Closure

The

closure

of a language

L

is denoted

L

∗^ and represents the set of those strings that

can be formed by taking any number of strings from

L

, possibly with repetitions (

i.e.

, the

same string may be selected more than once) and concatenating all of them.Examples:

If^

L^

=^

{^0

,^1

}^

then

L

∗^ is all strings of

and

If^

L^

=^

{^0

,^11

}^

then

L

∗^ consists of strings of

and

such that the

come in

pairs,

e.g.

,^011

,^11110

and

ǫ.

But not

or

Formally,

L

∗^ is the infinite union

S

i≥

L 0

i^ where

L

{ǫ

},^

1 L

L

, and for

i >

we

have

L

i^ =

LL... L

(the concatenation of

i^

copies of

L

Automata Theory, Languages and Computation - M´

ırian Halfeld-Ferrari – p. 14/

Regular Expressions and Languages

We define the regular expressions recursively. Basis

: The basis consists of three parts:

1.^

The constants

ǫ^

and

∅^

are regular expressions, denoting the language

{ǫ

}^ and

respectively. That is

L

(ǫ

ǫ}

and

L

2.^

If^ a^ is a symbol, then

a is a regular expression. This expression denotes the

language

{a

},^

i.e.

,^ L

( a

a}

NOTE: We use boldface font to denote an expression corresponding to a symbol

3.^

A variable, usually capitalised and italic such as

L

, is a variable, representing any

language.

Automata Theory, Languages and Computation - M´

ırian Halfeld-Ferrari – p. 16/

Regular Expressions and Languages

Induction

: There are four parts to the inductive step, one for each of the three operators

and one for the introduction of parentheses

1.^

If^

E^

and

F

are regular expressions, then

E

F

is a regular expression denoting

the union of

L

(E

)^ and

L

(F

). That is,

L

(E

F

L

(E

)^ ∪

L

(F

2.^

If^

E^

and

F

are regular expressions, then

EF

is a regular expression denoting the

concatenation of

L

(E

)^ and

L

(F

). That is,

L

(EF

L

(E

)L

(F

3.^

If^

E^

is a regular expression, then

E

∗^ is a regular expression denoting the closure

of

L

(E

). That is,

L

(E

L(

E

4.^

If^

E^

is a regular expression, then

(E

)^ is a regular expression denoting the same

as

E

. Formally,

L

E

L(

E

Automata Theory, Languages and Computation - M´

Automata Theory, Languages and Computation - M´