# CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 2 - PowerPoint PPT Presentation

1 / 53

CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 2 School of Innovation, Design and Engineering Mälardalen University 2011. Content Languages, Alphabets and Strings Strings & String Operations Languages & Language Operations Regular Expressions

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 2

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

#### Presentation Transcript

CDT314

FABER

Formal Languages, Automata and Models of Computation

Lecture 2

School of Innovation, Design and Engineering Mälardalen University

2011

• Content

• Languages, Alphabets and Strings

• Strings & String Operations

• Languages & Language Operations

• Regular Expressions

• Finite Automata, FA

• Deterministic Finite Automata, DFA

## Languages, Alphabets and Strings

### Languages

• defined over an alphabet:

A language is a set of strings

A String is a sequence of letters

• An alphabet is a set of symbols

### Alphabets and Strings

• We will use small alphabets:

Strings

## Operations on Strings

=

w

a

a

a

L

x abba

1

2

n

=

v

b

b

b

y  bbbaaa

L

1

2

m

### String Operations

Concatenation (sammanfogning)

xy abbabbbaaa

Reverse (reversering)

Example:

Longest odd length palindrome in a natural language:

saippuakauppias

(Finnish: soap sailsman)

Length:

Examples:

### Recursive Definition of Length

• For any letter:

• For any string :

• Example:

=

=

u

aab

,

u

3

=

=

v

abaab

,

v

5

=

=

uv

aababaab

8

=

+

=

+

=

uv

u

v

3

5

8

Example:

### Proof of Concatenation Length

• Claim:

• Proof: By induction on the length

• Induction basis:

• From definition of length:

• Inductive hypothesis:

for

• Inductive step: we will prove

for

### Inductive Step

• Write , where

• From definition of length:

• From inductive hypothesis:

• Thus:

END OF PROOF

### Empty String

• A string with no letters:

• (Also denoted as )

• Observations:

### Substring (delsträng)

• Substring of a string:

• a subsequence of consecutive characters

• String Substring

prefix

suffix

### Prefix and Suffix

• Suffixes

Prefixes

(String repeated n times)

n

=

• Example:

• Definition:

w

ww...

w

}

n

### The (Kleene* star) Operation

• the set of all possible strings from alphabet

[* Kleene is pronounced "clay-knee“]

http://en.wikipedia.org/wiki/Kleene_star

{

}

S

=

l

*

,

a

,

b

,

aa

,

ab

,

ba

,

bb

,

aaa

,

aab

,

K

The + (Kleene plus) Operation

:the set of all possible strings from the

alphabet except

{

}

S

=

a

,

b

}

{

S

=

oj

,

fy

,

usch

{

S

=

l, oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch

*

}

+

+

S

S

K

Example

=

S

-

l

*

{

=

oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch

}

K

## Operations on Languages

### Language

• A language is any subset of

• Example:

• Languages:

{

}

S

=

a

,

b

{

}

S

=

l

*

,

a

,

b

,

aa

,

ab

,

ba

,

bb

,

aaa

,

K

{

}

l

{

}

a

,

aa

,

aab

l

{

,

abba

,

baba

,

aa

,

ab

,

aaaaaa

}

### Example

• An infinite language

{

}

S

=

l

*

,

a

,

b

,

aa

,

ab

,

ba

,

bb

,

aaa

,

aab

,

K

Complement:

### Operations on Languages

• The usual set operations

Definition:

Examples:

• Definition:

Example

### Repeat

• Definition:

• Special case:

• Definition:

• Example:

• Definition

+

1

2

=

L

L

U

L

U

L

{

}

=

-

l

L

*

## Regular Expressions

Primitive regular expressions:

Given regular expressions and

are Regular Expressions

### Regular Expressions: Recursive Definition

A regular expression:

Not a regular expression:

### Examples

Building Regular Expressions

• Zero or more.

• a* means "zero or more a's."

• To say "zero or more ab's," that is,

• {, ab, abab, ababab, ...}, you need to say (ab)*.

• ab*denotes {a, ab, abb, abbb, abbbb, ...}.

Building Regular Expressions

• One or more.

• Since a* means "zero or more a's", you can use aa* (or equivalently, a*a) to mean "one or more a's.“

• Similarly, to describe "one or more ab's," that is,

• {ab, abab, ababab, ...}, you can use ab(ab)*.

Building Regular Expressions

• Any string at all.

• To describe any string at all (with = {a, b, c}), you can use (a+b+c)*.

• Any nonempty string.

• This can be written as any character from followed by any string at all: (a+b+c)(a+b+c)*.

Building Regular Expressions

• Any string not containing....

• To describe any string at all that doesn't contain an a (with = {a, b, c}), you can use (b+c)*.

• Any string containing exactly one...

• To describe any string that contains exactly one a, put "any string not containing an a," on either side of the a, like this: (b+c)*a(b+c)*.

### Languages of Regular Expressions

language of regular expression

Example

### Definition

• For primitive regular expressions:

### Definition (continued)

• For regular expressions and

### Example

Regular expression:

### Example

• Regular expression

### Example

• Regular expression

 { all strings with at least

two consecutive 0 }

### Example

• Regular expression

= { all strings without

two consecutive 0 }

### Example

• Regular expression

• (consists of repeating 1’s and 01’s).

### Example

= { all strings without

two consecutive 0 }

Equivalent solution:

(In order not to get 00 in a string, after each 0 there must be an 1,

which means that strings of the form 1....101....1

are repeated. That is the first parenthesis. To take into account strings that end with 0, and those consisting of 1’s solely,

the rest of the expression is added.)

### Equivalent Regular Expressions

• Regular expressions and

Definition:

are equivalent if

= { all strings without

two consecutive 0 }

### Example

and

are equivalent

regular expressions.

### Lennart Salling’s Video Resources

• http://www.math.uu.se/~salling/Lennart Salling

• http://www.math.uu.se/~salling/AUTOMATA_DV/index.html

• Introduktion: http://www2.math.uu.se/~salling/Movies/Intro%20to%20Automata.mov

• Program, strings, integers and integerfunctions

• http://www2.math.uu.se/~salling/Movies/StringsNumbersAndFunctions.mov

• http://www.youtube.com/embed/VM5SUcyY4sI?hl=en&fs=1 Kan alla problem lösas av program?

• http://www.youtube.com/embed/acsUNSkzUgg?hl=en&fs=1 Vad har stora och små oändligheter med saken att göra?

• http://www.youtube.com/embed/2abHjjS8Tqc?hl=sv&fs=1 Hur ser problem ut som inte kan lösas av program?

• http://www.youtube.com/embed/y-zMnV3G9pg?hl=sv&fs=1 Hur kan man visa att ett problem inte kan l&ouml;sas av program?

### Lennart Salling’s Video Video Resources

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Closure_properties/Closure_properties.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/FA_to_RegExpression/FA_to_RegExpression.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Famous_undecidable_and_decidable_problems/Famous_undecidable_and_decidable_problems.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Universal_Turing_machines/Universal_Turing_machines.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Turing_machines/Turing_machines.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Unrestricted_grammar/Unrestricted_grammar.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Pumping%20CFL/Pumping%20CFL.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/PDA/PDA.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/CFG/CFG.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Nonregularity/Nonregularity.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Reduction_of_number_of_states/Reduction_of_number_of_states.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/Closure_properties/Closure_properties.mov

• http://www2.math.uu.se/~salling/AUTOMATA_DV/media/2011/FA_to_RegExpression/FA_to_RegExpression.mov

### More Video’s 1

• http://www.youtube.com/watch?v=WMN_wz-b3K0&feature=related Accept and decide (TM)