[PPT] - Regular Expressions (REs) Regular Expressions (REs) p.1/37 PowerPoint Presentation

SLIDE 1

Regular Expressions (REs)

Regular Expressions (REs) – p.1/37

SLIDE 2

Expressions

In arithmetic:

Regular Expressions (REs) – p.2/37

SLIDE 3

Expressions

In arithmetic:
1. expressions are constructed from numbers and variables

using arithmetic operations and parantheses

Regular Expressions (REs) – p.2/37

SLIDE 4

Expressions

In arithmetic:
1. expressions are constructed from numbers and variables

using arithmetic operations and parantheses

2. expressions represent computations with numbers

Regular Expressions (REs) – p.2/37

SLIDE 5

Expressions

In arithmetic:
1. expressions are constructed from numbers and variables

using arithmetic operations and parantheses

2. expressions represent computations with numbers
3. results of expressions evaluation are numbers

Regular Expressions (REs) – p.2/37

SLIDE 6

Expressions

In arithmetic:
1. expressions are constructed from numbers and variables

using arithmetic operations and parantheses

2. expressions represent computations with numbers
3. results of expressions evaluation are numbers
In automata theory:

Regular Expressions (REs) – p.2/37

SLIDE 7

Expressions

In arithmetic:
1. expressions are constructed from numbers and variables

using arithmetic operations and parantheses

2. expressions represent computations with numbers
3. results of expressions evaluation are numbers
In automata theory:
1. regular expressions are constructed from regular languages

using regular operations and parentheses

Regular Expressions (REs) – p.2/37

SLIDE 8

Expressions

In arithmetic:
1. expressions are constructed from numbers and variables

using arithmetic operations and parantheses

2. expressions represent computations with numbers
3. results of expressions evaluation are numbers
In automata theory:
1. regular expressions are constructed from regular languages

using regular operations and parentheses

2. regular expressions represent computations with regular

languages

Regular Expressions (REs) – p.2/37

SLIDE 9

Expressions

In arithmetic:
1. expressions are constructed from numbers and variables

using arithmetic operations and parantheses

2. expressions represent computations with numbers
3. results of expressions evaluation are numbers
In automata theory:
1. regular expressions are constructed from regular languages

using regular operations and parentheses

2. regular expressions represent computations with regular

languages

3. result of regular expressions evaluation are regular languages

Regular Expressions (REs) – p.2/37

SLIDE 10

Example

✁

✂ ✄ ☎ ✆ ✝

is an arithmetic expression constructed with

perations

✂

and

✆

; its value is the number 32

Regular Expressions (REs) – p.3/37

SLIDE 11

Example

✁

✂ ✄ ☎ ✆ ✝

is an arithmetic expression constructed with

perations

✂

and

✆

; its value is the number 32

✁

✂ ☎

✄

is a regular expression constructed from the languages

and

✂

using regular operations;

Regular Expressions (REs) – p.3/37

SLIDE 12

Example

✁

✂ ✄ ☎ ✆ ✝

is an arithmetic expression constructed with

perations

✂

and

✆

; its value is the number 32

✁

✂ ☎

✄

is a regular expression constructed from the languages

and

✂

using regular operations;

it evaluates to the language that contains all strings of

zero-s and 1 followed by zero-s

Regular Expressions (REs) – p.3/37

SLIDE 13

Regular expression evaluation

Similar to the evaluation of arithmetic expression

Regular Expressions (REs) – p.4/37

SLIDE 14

Regular expression evaluation

Similar to the evaluation of arithmetic expression

Identify first the constants (languages) involved.

Regular Expressions (REs) – p.4/37

SLIDE 15

Regular expression evaluation

Similar to the evaluation of arithmetic expression

Identify first the constants (languages) involved.
Example:
represents the language

✁

✂

and

✄

represents the language

✁ ✄ ✂

Regular Expressions (REs) – p.4/37

SLIDE 16

Regular expression evaluation

Similar to the evaluation of arithmetic expression

Identify first the constants (languages) involved.
Example:
represents the language

✁

✂

and

✄

represents the language

✁ ✄ ✂

Apply the regular operations on the languages thus identified

following their set-theory definitions.

Regular Expressions (REs) – p.4/37

SLIDE 17

Regular expression evaluation

Similar to the evaluation of arithmetic expression

Identify first the constants (languages) involved.
Example:
represents the language

✁

✂

and

✄

represents the language

✁ ✄ ✂

Apply the regular operations on the languages thus identified

following their set-theory definitions.

Example:
✁

✄ ✂

represents the language

✁ ☎✄ ✄ ✂

,

✆

represents the language

✁✞✝ ✄

✄
✄

✟ ✟ ✟ ✂

and

✁

✄ ✂

✆

represents the language

✁

✄

✄ ✄

✄

✄ ☎✄

☎✄

✄

☎✄

✟ ✟ ✟ ✂

Regular Expressions (REs) – p.4/37

SLIDE 18

Regular expression evaluation

Similar to the evaluation of arithmetic expression

Identify first the constants (languages) involved.
Example:
represents the language

✁

✂

and

✄

represents the language

✁ ✄ ✂

Apply the regular operations on the languages thus identified

following their set-theory definitions.

Example:
✁

✄ ✂

represents the language

✁ ☎✄ ✄ ✂

,

✆

represents the language

✁✞✝ ✄

✄
✄

✟ ✟ ✟ ✂

and

✁

✄ ✂

✆

represents the language

✁

✄

✄ ✄

✄

✄ ☎✄

☎✄

✄

☎✄

✟ ✟ ✟ ✂

Note: by convention, the symbol

f concatenation op.

is not written in regular expressions

Regular Expressions (REs) – p.4/37

SLIDE 19

Applications

1. Regular expressions provide a powerful method to

describe patterns searched in various texts

Regular Expressions (REs) – p.5/37

SLIDE 20

Applications

1. Regular expressions provide a powerful method to

describe patterns searched in various texts

2. Utilities such as AWK, GREP in Unix, modern

programming languages, such as PERL, text editors, all use regular expressions for their pattern description

Regular Expressions (REs) – p.5/37

SLIDE 21

Applications

1. Regular expressions provide a powerful method to

describe patterns searched in various texts

2. Utilities such as AWK, GREP in Unix, modern

programming languages, such as PERL, text editors, all use regular expressions for their pattern description

3. The lexicon of programming languages is specifi ed by

regular expressions. Hence, lexical analysis is done using regular expressions.

Regular Expressions (REs) – p.5/37

SLIDE 22

Applications

1. Regular expressions provide a powerful method to

describe patterns searched in various texts

2. Utilities such as AWK, GREP in Unix, modern

programming languages, such as PERL, text editors, all use regular expressions for their pattern description

3. The lexicon of programming languages is specifi ed by

regular expressions. Hence, lexical analysis is done using regular expressions.

4. Other examples?

Regular Expressions (REs) – p.5/37

SLIDE 23

For a given alphabet

The regular expression
describes the language

consisting of all strings of length 1 over

Regular Expressions (REs) – p.6/37

SLIDE 24

For a given alphabet

The regular expression
describes the language

consisting of all strings of length 1 over

✄

describes the language of all strings over the alphabet

Regular Expressions (REs) – p.6/37

SLIDE 25

For a given alphabet

The regular expression
describes the language

consisting of all strings of length 1 over

✄

describes the language of all strings over the alphabet

✄

✂

is the language of all strings over

that ends in

✂

Regular Expressions (REs) – p.6/37

SLIDE 26

For a given alphabet

The regular expression
describes the language

consisting of all strings of length 1 over

✄

describes the language of all strings over the alphabet

✄

✂

is the language of all strings over

that ends in

✂

Language
✄

☎ ✁

✄

✂ ☎

describes the language that consists of all strings that either start with 0 or end in 1

Regular Expressions (REs) – p.6/37

SLIDE 27

Precedence relation

Denoting

the rule describing the order of ops in the

evaluation of arithmetic and regular expressions we have:

Regular Expressions (REs) – p.7/37

SLIDE 28

Precedence relation

Denoting

the rule describing the order of ops in the

evaluation of arithmetic and regular expressions we have:

Arithmetic expressions:
✂
✁✂✁

✄ ✄ ✂

✁

☎ ✄ ✆ ✂

Regular Expressions (REs) – p.7/37

SLIDE 29

Precedence relation

Denoting

the rule describing the order of ops in the

evaluation of arithmetic and regular expressions we have:

Arithmetic expressions:
✂
✁✂✁

✄ ✄ ✂

✁

☎ ✄ ✆ ✂

Regular expressions:
✂
✁✁

✂

✁

✁ ✄ ✂ ✂

Regular Expressions (REs) – p.7/37

SLIDE 30

Precedence relation

Denoting

the rule describing the order of ops in the

evaluation of arithmetic and regular expressions we have:

Arithmetic expressions:
✂
✁✂✁

✄ ✄ ✂

✁

☎ ✄ ✆ ✂

Regular expressions:
✂
✁✁

✂

✁

✁ ✄ ✂ ✂

Note:

☎

denotes expressions enclosed in parentheses

Regular Expressions (REs) – p.7/37

SLIDE 31

Formal definitions

Consider

an alphabet. A regular expression (RE)
ver
is defi ned recursivelyby the rules:

Regular Expressions (REs) – p.8/37

SLIDE 32

Formal definitions

Consider

an alphabet. A regular expression (RE)
ver
is defi ned recursivelyby the rules:

Recursion basis:

Regular Expressions (REs) – p.8/37

SLIDE 33

Formal definitions

Consider

an alphabet. A regular expression (RE)
ver
is defi ned recursivelyby the rules:

Recursion basis:

1. For any
✁

✂

,

is a RE describing the language

✄

✂

☎ ✁

✂

Regular Expressions (REs) – p.8/37

SLIDE 34

Formal definitions

Consider

an alphabet. A regular expression (RE)
ver
is defi ned recursivelyby the rules:

Recursion basis:

1. For any
✁

✂

,

is a RE describing the language

✄

✂

☎ ✁

✂

2.

✝

is a RE describing the language

✄

✝

✂ ☎ ✁ ✝ ✂

Regular Expressions (REs) – p.8/37

SLIDE 35

Formal definitions

Consider

an alphabet. A regular expression (RE)
ver
is defi ned recursivelyby the rules:

Recursion basis:

1. For any
✁

✂

,

is a RE describing the language

✄

✂

☎ ✁

✂

2.

✝

is a RE describing the language

✄

✝

✂ ☎ ✁ ✝ ✂

3.

is a RE describing the empty language

✄

✂

☎

.

Regular Expressions (REs) – p.8/37

SLIDE 36

Formal definitions, continutation

Recursion body: If

✁

and

✂

are REs evaluating to the languages

✄

☎

and

✄

✂

☎

then:

Regular Expressions (REs) – p.9/37

SLIDE 37

Formal definitions, continutation

Recursion body: If

✁

and

✂

are REs evaluating to the languages

✄

☎

and

✄

✂

☎

then:

1.

✁

✁

✂

✂

is a RE evaluating to

✄

✁

✂ ✁ ✄

✂

✂

Regular Expressions (REs) – p.9/37

SLIDE 38

Formal definitions, continutation

Recursion body: If

✁

and

✂

are REs evaluating to the languages

✄

☎

and

✄

✂

☎

then:

1.

✁

✁

✂

✂

is a RE evaluating to

✄

✁

✂ ✁ ✄

✂

✂

2.

✁

✂

✂

✂

is a RE evaluating to

✄

✁

✂ ✂ ✄

✂

✂

Regular Expressions (REs) – p.9/37

SLIDE 39

Formal definitions, continutation

Recursion body: If

✁

and

✂

are REs evaluating to the languages

✄

☎

and

✄

✂

☎

then:

1.

✁

✁

✂

✂

is a RE evaluating to

✄

✁

✂ ✁ ✄

✂

✂

2.

✁

✂

✂

✂

is a RE evaluating to

✄

✁

✂ ✂ ✄

✂

✂

3.

✆

✁

is a RE evaluating to

✁✁ ✂ ✄ ✄

✁

✂

where

✄

✁

✂

☎

✄

✁

✂

☎

✁ ✂ ✄

✁

✂

and

✄

✁

✂ ✄ ☎ ✝

.

Regular Expressions (REs) – p.9/37

SLIDE 40

Note

A defi nition of the form expressed by rule (4) above is called an inductive defi nition

Regular Expressions (REs) – p.10/37

SLIDE 41

Potential confusions

Do not confuse the REs
and

✁

;

Regular Expressions (REs) – p.11/37

SLIDE 42

Potential confusions

Do not confuse the REs
and

✁

;

1.

✝

represents the language containing just the empty string

Regular Expressions (REs) – p.11/37

SLIDE 43

Potential confusions

Do not confuse the REs
and

✁

;

1.

✝

represents the language containing just the empty string 2.

is the RE that represents the the empty language

Regular Expressions (REs) – p.11/37

SLIDE 44

Potential confusions

Do not confuse the REs
and

✁

;

1.

✝

represents the language containing just the empty string 2.

is the RE that represents the the empty language
Note the distinction between
and

✄

☎

.

is an

expression and

✄

☎

is the set of strings specifi ed by

Regular Expressions (REs) – p.11/37

SLIDE 45

Example REs

1.

✆

✄

✆

,

Regular Expressions (REs) – p.12/37

SLIDE 46

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

Regular Expressions (REs) – p.12/37

SLIDE 47

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

Regular Expressions (REs) – p.12/37

SLIDE 48

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

Regular Expressions (REs) – p.12/37

SLIDE 49

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

Regular Expressions (REs) – p.12/37

SLIDE 50

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

Regular Expressions (REs) – p.12/37

SLIDE 51

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

Regular Expressions (REs) – p.12/37

SLIDE 52

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

Regular Expressions (REs) – p.12/37

SLIDE 53

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

Regular Expressions (REs) – p.12/37

SLIDE 54

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

Regular Expressions (REs) – p.12/37

SLIDE 55

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

6.

✄

✁

✄

,

Regular Expressions (REs) – p.12/37

SLIDE 56

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

6.

✄

✁

✄

,

✄

✄

✁

✄

✂ ☎ ✁

✄

✄

✄

✂

Regular Expressions (REs) – p.12/37

SLIDE 57

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

6.

✄

✁

✄

,

✄

✄

✁

✄

✂ ☎ ✁

✄

✄

✄

✂

7.

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄

,

Regular Expressions (REs) – p.12/37

SLIDE 58

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

6.

✄

✁

✄

,

✄

✄

✁

✄

✂ ☎ ✁

✄

✄

✄

✂

7.

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄

,

✄

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄ ✂ ☎ ✁

✁
starts and

ends with the same symbol

✂

Regular Expressions (REs) – p.12/37

SLIDE 59

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

6.

✄

✁

✄

,

✄

✄

✁

✄

✂ ☎ ✁

✄

✄

✄

✂

7.

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄

,

✄

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄ ✂ ☎ ✁

✁
starts and

ends with the same symbol

✂

8.

✁

✝ ✂ ✄ ✆

,

Regular Expressions (REs) – p.12/37

SLIDE 60

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

6.

✄

✁

✄

,

✄

✄

✁

✄

✂ ☎ ✁

✄

✄

✄

✂

7.

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄

,

✄

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄ ✂ ☎ ✁

✁
starts and

ends with the same symbol

✂

8.

✁

✝ ✂ ✄ ✆

,

✄

✁

✝ ✂ ✄ ✆ ✂ ☎ ✁

✄

✆ ✁ ✄ ✆ ✂

Regular Expressions (REs) – p.12/37

SLIDE 61

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

6.

✄

✁

✄

,

✄

✄

✁

✄

✂ ☎ ✁

✄

✄

✄

✂

7.

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄

,

✄

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄ ✂ ☎ ✁

✁
starts and

ends with the same symbol

✂

8.

✁

✝ ✂ ✄ ✆

,

✄

✁

✝ ✂ ✄ ✆ ✂ ☎ ✁

✄

✆ ✁ ✄ ✆ ✂

9.

✁

✝ ✂

✄

✁ ✝ ✂

,

Regular Expressions (REs) – p.12/37

SLIDE 62

Example REs

1.

✆

✄

✆

,

✄

✆

✄

✆

✂ ☎ ✁

✁
contains exactly a single 1

✂

2.

✂ ✆ ✄ ✂ ✆

,

✄

✂

✆ ✄ ✂ ✆ ✂ ☎ ✁

✁
contains at least one 1

✂

3.

✂ ✆

✄

✂ ✆

,

✄

✂

✆

✄

✂ ✆ ✂ ☎ ✁

✁
contains
✄

as a substring

✂

4.

✂

✂ ✂ ✆

,

✄

✂

✂ ✂ ✆ ☎ ✁

✁
is a string of even length

✂

5.

✂

✂ ✂ ✂ ✆

,

✄

✂

✂ ✂ ✂ ✆ ☎ ✁

✁

the length of

is a multiple of three

✂

6.

✄

✁

✄

,

✄

✄

✁

✄

✂ ☎ ✁

✄

✄

✄

✂

7.

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄

,

✄

✂

✆

✁

✄ ✂ ✆ ✄ ✁

✁

✄ ✂ ☎ ✁

✁
starts and

ends with the same symbol

✂

8.

✁

✝ ✂ ✄ ✆

,

✄

✁

✝ ✂ ✄ ✆ ✂ ☎ ✁

✄

✆ ✁ ✄ ✆ ✂

9.

✁

✝ ✂

✄

✁ ✝ ✂

,

✄

✁

✝ ✂

✄

✁ ✝ ✂ ✂ ☎ ✁ ✝ ✄ ☎✄ ✄ ✄

✄

✂

Regular Expressions (REs) – p.12/37

SLIDE 63

More example

10.

✂ ✄ ✁

,

✄

✂

✄ ✁ ☎

✁

; Note: concatenating

✁

to any set yields

✁

Regular Expressions (REs) – p.13/37

SLIDE 64

More example

10.

✂ ✄ ✁

,

✄

✂

✄ ✁ ☎

✁

; Note: concatenating

✁

to any set yields

✁

11.

✁ ✄

,

✄

✁

✄ ☎

✁

; by defi nition,

is in the star operation applied on any language;

if the language is empty

becomes the one element

Regular Expressions (REs) – p.13/37

SLIDE 65

Identities

If

is a RE then the following identities take place:

Regular Expressions (REs) – p.14/37

SLIDE 66

Identities

If

is a RE then the following identities take place:
✄
✁

✁ ☎

✄
☎

; adding the empty language to any

ther language will not change that language

Regular Expressions (REs) – p.14/37

SLIDE 67

Identities

If

is a RE then the following identities take place:
✄
✁

✁ ☎

✄
☎

; adding the empty language to any

ther language will not change that language
✄
☎
✄
☎

; concatenating any string with the empty string does not change that string

Regular Expressions (REs) – p.14/37

SLIDE 68

Note

✄
✁
☎
✄
☎

. Example: if

then

✄

☎
;

✄

✁
☎
✁
✁

Regular Expressions (REs) – p.15/37

SLIDE 69

Note

✄
✁
☎
✄
☎

. Example: if

then

✄

☎
;

✄

✁
☎
✁
✁
✄
✁

☎

✄
☎

. Example: if

then

✄

☎
✁

but

✄

✁

☎

✁

Regular Expressions (REs) – p.15/37

SLIDE 70

Application

REs are useful tools for the design of compilers

Regular Expressions (REs) – p.16/37

SLIDE 71

Application

REs are useful tools for the design of compilers
Language lexicon is described by REs.

Regular Expressions (REs) – p.16/37

SLIDE 72

Application

REs are useful tools for the design of compilers
Language lexicon is described by REs.

Example: numerical constants can be described by:

✁ ☎ ✄ ✆ ✄ ✝ ✂

✆

✁

✆

✟

✆

✁

✟
✆

✂

where

☎

✁

✄

✄ ✄ ✁☎✄ ✂☎✄ ✄ ✄ ☎☎✄ ✆ ✄ ✝ ✄ ✞☎✄ ✟ ✂

Regular Expressions (REs) – p.16/37

SLIDE 73

Application

REs are useful tools for the design of compilers
Language lexicon is described by REs.

Example: numerical constants can be described by:

✁ ☎ ✄ ✆ ✄ ✝ ✂

✆

✁

✆

✟

✆

✁

✟
✆

✂

where

☎

✁

✄

✄ ✄ ✁☎✄ ✂☎✄ ✄ ✄ ☎☎✄ ✆ ✄ ✝ ✄ ✞☎✄ ✟ ✂

From the lexicon description by REs one can generate

automatically lexical analyzers

Regular Expressions (REs) – p.16/37

SLIDE 74

New terminology

A RE

that evaluates to the language

✄

☎

is further said that specifi es the language

✄

☎

Regular Expressions (REs) – p.17/37

SLIDE 75

Equivalence with FA

REs and fi nite automata are equivalentin their

descriptive power

Regular Expressions (REs) – p.18/37

SLIDE 76

Equivalence with FA

REs and fi nite automata are equivalentin their

descriptive power

Any RE
can be converted into a fi nite automaton,

✁✄✂

, that recognizes the language specifi ed by

Regular Expressions (REs) – p.18/37

SLIDE 77

Equivalence with FA

REs and fi nite automata are equivalentin their

descriptive power

Any RE
can be converted into a fi nite automaton,

✁✄✂

, that recognizes the language specifi ed by

Vice-versa, any fi nite automaton recognizing a

language

✁

can be converted into a RE

✁

that specifi es the language

✁

Regular Expressions (REs) – p.18/37

SLIDE 78

Theorem 1.54

Language is regular iff some RE specifi es it

Regular Expressions (REs) – p.19/37

SLIDE 79

Theorem 1.54

Language is regular iff some RE specifi es it Proof idea: this proof has two parts:

Regular Expressions (REs) – p.19/37

SLIDE 80

Theorem 1.54

Language is regular iff some RE specifi es it Proof idea: this proof has two parts:

First part: we show that a language specified by a RE is regular,

i.e., there is a finite automaton that recognizes it.

Regular Expressions (REs) – p.19/37

SLIDE 81

Theorem 1.54

Language is regular iff some RE specifi es it Proof idea: this proof has two parts:

First part: we show that a language specified by a RE is regular,

i.e., there is a finite automaton that recognizes it.

Second part: we show that if a language is regular then there is a

RE that specifies it

Regular Expressions (REs) – p.19/37

SLIDE 82

Lemma 1.55

If a language is specifi ed by a RE, then it is regular.

Regular Expressions (REs) – p.20/37

SLIDE 83

Lemma 1.55

If a language is specifi ed by a RE, then it is regular. Proof idea: Assume that we have a RE

that evaluate to

the language

✁

.

Regular Expressions (REs) – p.20/37

SLIDE 84

Lemma 1.55

If a language is specifi ed by a RE, then it is regular. Proof idea: Assume that we have a RE

that evaluate to

the language

✁

.

1. We will show how to convert
into an NFA that recognizes
.

Regular Expressions (REs) – p.20/37

SLIDE 85

Lemma 1.55

If a language is specifi ed by a RE, then it is regular. Proof idea: Assume that we have a RE

that evaluate to

the language

✁

.

1. We will show how to convert
into an NFA that recognizes
.
2. Then by corollary 1.40, if an NFA recognizes
then
is regular

Regular Expressions (REs) – p.20/37

SLIDE 86

Proof

Convert

into an NFA
by the following six-steps proce-

dure:

Regular Expressions (REs) – p.21/37

SLIDE 87

Step 1:

If

✁
, then

✄

☎
✁

and the NFA

recognizing

✄

☎

is in Figure 1

✂ ✄ ✁ ✂

✄

✂

Figure 1: NFA

☎

recognizing

✁

✂

Regular Expressions (REs) – p.22/37

SLIDE 88

Step 1:

If

✁
, then

✄

☎
✁

and the NFA

recognizing

✄

☎

is in Figure 1

✂ ✄ ✁ ✂

✄

✂

Figure 1: NFA

☎

recognizing

✁

✂

Note: this is an NFA but not a DFA because it has states with no exiting arrow for each possible input symbol

Regular Expressions (REs) – p.22/37

SLIDE 89

Formal construction

Formally

✁
✁
✂

✁ ✁

✁

✂ ✁

✁
✂

✁ ☎

where:

✂

✁
☎
✂

✁

,

✂ ✁✄ ✁ ☎ ☎

✁

, for

✄

and

☎

Regular Expressions (REs) – p.23/37

SLIDE 90

Step 2:

If

then

✄

☎
✁

and the NFA

that recognizes

✄

☎

is in Figure 2

✂

✁

Figure 2: The NFA

☎

recognizing

✁ ✝ ✂

Regular Expressions (REs) – p.24/37

SLIDE 91

Formal construction

Formally

✁
✁

✁

✁

✂ ✁

✁

✁

✁

☎

where:

✂ ✁✄ ✁ ☎ ☎

✁

for any

✄

and

☎ ✁

Regular Expressions (REs) – p.25/37

SLIDE 92

Step 3:

If

✁

then

✄

☎
✁

, and the NFA

that recognizes

✄

☎

is in Figure 3

✂

Figure 3: he NFA

☎

recognizing

Regular Expressions (REs) – p.26/37

SLIDE 93

Formal construction

Formally

✁

✁ ✁

✁

✂ ✁

✁

✁ ☎

where:

✂ ✁✄ ✁ ☎ ☎

✁

for any

✄

and

☎

Regular Expressions (REs) – p.27/37

SLIDE 94

Step 4:

If

✁
✂

then

✄

☎
✄
✁

☎ ✁ ✄

✂

☎

.

Regular Expressions (REs) – p.28/37

SLIDE 95

Step 4:

If

✁
✂

then

✄

☎
✄
✁

☎ ✁ ✄

✂

☎

.

Note: in view with the inductive nature of

we may assume that:

Regular Expressions (REs) – p.28/37

SLIDE 96

Step 4:

If

✁
✂

then

✄

☎
✄
✁

☎ ✁ ✄

✂

☎

.

Note: in view with the inductive nature of

we may assume that:

1.

☎ ✁ ☎

✁

✄ ✂ ✄ ✁ ✁ ✄ ✄ ✁ ✄ ✂ ✁ ✂

is an NFA recognizing

✄

✁

✂

Regular Expressions (REs) – p.28/37

SLIDE 97

Step 4:

If

✁
✂

then

✄

☎
✄
✁

☎ ✁ ✄

✂

☎

.

Note: in view with the inductive nature of

we may assume that:

1.

☎ ✁ ☎

✁

✄ ✂ ✄ ✁ ✁ ✄ ✄ ✁ ✄ ✂ ✁ ✂

is an NFA recognizing

✄

✁

✂

2.

☎ ✂ ☎

✂

✄ ✂ ✄ ✁ ✂ ✄ ✄ ✂ ✄ ✂ ✂ ✂

is an NFA recognizing

✄

✂

✂

Regular Expressions (REs) – p.28/37

SLIDE 98

Step 4:

If

✁
✂

then

✄

☎
✄
✁

☎ ✁ ✄

✂

☎

.

Note: in view with the inductive nature of

we may assume that:

1.

☎ ✁ ☎

✁

✄ ✂ ✄ ✁ ✁ ✄ ✄ ✁ ✄ ✂ ✁ ✂

is an NFA recognizing

✄

✁

✂

2.

☎ ✂ ☎

✂

✄ ✂ ✄ ✁ ✂ ✄ ✄ ✂ ✄ ✂ ✂ ✂

is an NFA recognizing

✄

✂

✂

The NFA

recognizing

✄

✁
✂

☎

is given in Figure 4

Regular Expressions (REs) – p.28/37

SLIDE 99

NFA recognizing

✁

✂ ✁ ✂ ✄ ✄ ✄ ✂✆☎ ✂ ✄ ✄ ✄ ✂ ✂ ✂ ✂ ✝ ✝ ✄ ✄ ✄ ✄ ✄ ✄

Figure 4: Construction of

☎

to recognize

✄

✁

✁

✂

✂

Regular Expressions (REs) – p.29/37

SLIDE 100

Construction procedure

1.

☎

✁ ✄ ✄ ✂ ✁

✁

✁

✂

: That is, the states of

☎

are all states on

☎ ✁

and

☎ ✂

with the addition of a new state

✄ ✄

Regular Expressions (REs) – p.30/37

SLIDE 101

Construction procedure

1.

☎

✁ ✄ ✄ ✂ ✁

✁

✁

✂

: That is, the states of

☎

are all states on

☎ ✁

and

☎ ✂

with the addition of a new state

✄ ✄

2. The start state of

☎

is

✄ ✄

Regular Expressions (REs) – p.30/37

SLIDE 102

Construction procedure

1.

☎

✁ ✄ ✄ ✂ ✁

✁

✁

✂

: That is, the states of

☎

are all states on

☎ ✁

and

☎ ✂

with the addition of a new state

✄ ✄

2. The start state of

☎

is

✄ ✄

3. The accept states of

☎

are

✂ ☎ ✂ ✁ ✁ ✂ ✂

: That is, the accept states

f

☎

are all the accept states of

☎ ✁

and

☎ ✂

Regular Expressions (REs) – p.30/37

SLIDE 103

Construction procedure

1.

☎

✁ ✄ ✄ ✂ ✁

✁

✁

✂

: That is, the states of

☎

are all states on

☎ ✁

and

☎ ✂

with the addition of a new state

✄ ✄

2. The start state of

☎

is

✄ ✄

3. The accept states of

☎

are

✂ ☎ ✂ ✁ ✁ ✂ ✂

: That is, the accept states

f

☎

are all the accept states of

☎ ✁

and

☎ ✂

4. Define

✁

so that for any

✄ ✁

and any
✁

✂✁

:

✁

✄

✄

✂

☎ ✂ ✄☎✄✆✄✝✄☎✄✆✞ ✄☎✄✝✄✆✄☎✄✝✟ ✁ ✁

✄

✄

✂

✄

if

✄ ✁

✁

✁ ✂

✄

✄

✂

✄

if

✄ ✁

✂

✁ ✄ ✁ ✄ ✄ ✂ ✂ ✄

if

✄ ☎ ✄ ✄

and

☎

✝

✄

if

✄ ☎ ✄ ✄

and

✠

☎ ✝

.

Regular Expressions (REs) – p.30/37

SLIDE 104

Step 5:

If

✂

then

✄

☎
✄
✁

☎

✄
✂

☎

.

Regular Expressions (REs) – p.31/37

SLIDE 105

Step 5:

If

✂

then

✄

☎
✄
✁

☎

✄
✂

☎

.

Note: in view with the inductive nature of

we may assume that:

Regular Expressions (REs) – p.31/37

SLIDE 106

Step 5:

If

✂

then

✄

☎
✄
✁

☎

✄
✂

☎

.

Note: in view with the inductive nature of

we may assume that:

1.

☎ ✁ ☎

✁

✄ ✂ ✄ ✁ ✁ ✄ ✄ ✁ ✄ ✂ ✁ ✂

is an NFA recognizing

✄

✁

✂

Regular Expressions (REs) – p.31/37

SLIDE 107

Step 5:

If

✂

then

✄

☎
✄
✁

☎

✄
✂

☎

.

Note: in view with the inductive nature of

we may assume that:

1.

☎ ✁ ☎

✁

✄ ✂ ✄ ✁ ✁ ✄ ✄ ✁ ✄ ✂ ✁ ✂

is an NFA recognizing

✄

✁

✂

2.

☎ ✂ ☎

✂

✄ ✂ ✄ ✁ ✂ ✄ ✄ ✂ ✄ ✂ ✂ ✂

is an NFA recognizing

✄

✂

✂

.

Regular Expressions (REs) – p.31/37

SLIDE 108

NFA recognizing

✁

✂ ✁ ✂ ✄ ✄ ✄ ✂ ☎ ✂ ✄ ✄ ✄ ✄ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✁ ✁ ✁ ✁ ✂ ✝ ✂ ✝ ✄ ✄ ✄ ✄ ☎ ✝ ✂

Figure 5: Construction of

☎

to recognize

✄

✁

✂

✂

✂

Regular Expressions (REs) – p.32/37

SLIDE 109

Construction procedure

1.

☎
✁

✁

✂

. The states of

☎

are all states of

☎ ✁

and

☎ ✂

Regular Expressions (REs) – p.33/37

SLIDE 110

Construction procedure

1.

☎
✁

✁

✂

. The states of

☎

are all states of

☎ ✁

and

☎ ✂

2. The start state is the state

✄ ✁

f

☎ ✁

Regular Expressions (REs) – p.33/37

SLIDE 111

Construction procedure

1.

☎
✁

✁

✂

. The states of

☎

are all states of

☎ ✁

and

☎ ✂

2. The start state is the state

✄ ✁

f

☎ ✁

3. The accept states is the set

✂ ✂

f the accept states of

☎ ✂

Regular Expressions (REs) – p.33/37

SLIDE 112

Construction procedure

1.

☎
✁

✁

✂

. The states of

☎

are all states of

☎ ✁

and

☎ ✂

2. The start state is the state

✄ ✁

f

☎ ✁

3. The accept states is the set

✂ ✂

f the accept states of

☎ ✂

4. Define

✁

so that for any

✄ ✁

and any
✁

✂

:

✁

✄

✄

✂

☎ ✂ ✄✝✄✆✄☎✄✝✄✆✞ ✄☎✄✝✄✆✄☎✄✝✟ ✁ ✁

✄

✄

✂

✄

if

✄ ✁

✁

and

✄ ✠ ✁ ✂ ✁ ✁ ✁

✄

✄

✂

✄

if

✄ ✁ ✂ ✁

and

✠

☎ ✝ ✁ ✁

✄

✄

✂

✁ ✁ ✄ ✂ ✂ ✄

if

✄ ✁ ✂ ✁

and

☎

✝ ✁ ✂

✄

✄

✂

✄

if

✄ ✁

✂

.

Regular Expressions (REs) – p.33/37

SLIDE 113

Step 6:

If

✄
then

✄

☎
✁✁

✂✄ ✄

✁

☎

.

Regular Expressions (REs) – p.34/37

SLIDE 114

Step 6:

If

✄
then

✄

☎
✁✁

✂✄ ✄

✁

☎

.

Note: in view with the inductive nature of

we may assume that:

Regular Expressions (REs) – p.34/37

SLIDE 115

Step 6:

If

✄
then

✄

☎
✁✁

✂✄ ✄

✁

☎

.

Note: in view with the inductive nature of

we may assume that:

1.

☎ ✁ ☎

✁

✄ ✂ ✄ ✁ ✁ ✄ ✄ ✁ ✄ ✂ ✁ ✂

is an NFA recognizing

✄

✁

✂

Regular Expressions (REs) – p.34/37

SLIDE 116

Step 6:

If

✄
then

✄

☎
✁✁

✂✄ ✄

✁

☎

.

Note: in view with the inductive nature of

we may assume that:

1.

☎ ✁ ☎

✁

✄ ✂ ✄ ✁ ✁ ✄ ✄ ✁ ✄ ✂ ✁ ✂

is an NFA recognizing

✄

✁

✂

The NFA

recognizing

✄

✄
☎

is given in Figure 6

Regular Expressions (REs) – p.34/37

SLIDE 117

NFA recognizing

✂

✁ ✂ ✄ ✄ ✂ ✂ ✄ ✄ ✂ ✝ ✁ ✝ ✂ ✝

Figure 6: Construction of

to recognize

✄

✄
☎

Regular Expressions (REs) – p.35/37

SLIDE 118

Construction procedure

1.

☎

✁ ✄ ✄ ✂ ✁

✁

; that is, states of

☎

are the states of

☎ ✁

plus a new state

✄ ✄

Regular Expressions (REs) – p.36/37

SLIDE 119

Construction procedure

1.

☎

✁ ✄ ✄ ✂ ✁

✁

; that is, states of

☎

are the states of

☎ ✁

plus a new state

✄ ✄

2. Start state of

☎

is

✄ ✄

Regular Expressions (REs) – p.36/37

SLIDE 120

Construction procedure

1.

☎

✁ ✄ ✄ ✂ ✁

✁

; that is, states of

☎

are the states of

☎ ✁

plus a new state

✄ ✄

2. Start state of

☎

is

✄ ✄

3.

✂ ☎ ✁ ✄ ✄ ✂ ✁ ✂ ✁

; that is, the accept states of

☎

are the accept states of

☎ ✁

plus the new start state

Regular Expressions (REs) – p.36/37

SLIDE 121

Construction procedure

1.

☎

✁ ✄ ✄ ✂ ✁

✁

; that is, states of

☎

are the states of

☎ ✁

plus a new state

✄ ✄

2. Start state of

☎

is

✄ ✄

3.

✂ ☎ ✁ ✄ ✄ ✂ ✁ ✂ ✁

; that is, the accept states of

☎

are the accept states of

☎ ✁

plus the new start state

4. Define

✁

so that for any

✄ ✁

and
✁

✂

:

✁

✄

✄

✂

☎ ✂ ✄✆✄☎✄✝✄✆✄☎✄✝✄✝✄✝✞ ✄✝✄✝✄✝✄✝✄✝✄✝✄✝✄✝✟ ✁ ✁

✄

✄

✂

✄

if

✄ ✁

✁

and

✄ ✠ ✁ ✂ ✁ ✁ ✁

✄

✄

✂

✄

if

✄ ✁ ✂ ✁

and

✠

☎ ✝ ✁ ✁

✄

✄

✂

✁ ✁ ✄ ✁ ✂ ✄

if

✄ ✁ ✂ ✁

and

☎

✝ ✁ ✄ ✁ ✂ ✄

if

✄ ☎ ✄ ✄

and

☎

✝

✄

if

✄ ☎ ✄ ✄

and

✠

☎ ✝

.

Regular Expressions (REs) – p.36/37

SLIDE 122

Examples conversion

Convert the following REs into NFA following the procedure presented above 1.

☎

✁

☎

✄

to an NFA 2.

✁

☎ ☎ ✄

☎
Regular Expressions (REs) – p.37/37