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CDT314

FABER

Formal Languages, Automata and Models of Computation

Lecture 2

School of Innovation, Design and Engineering Mälardalen University

2011

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Content

Languages, Alphabets and Strings

Strings & String Operations

Languages & Language Operations

Regular Expressions

Finite Automata, FA

Deterministic Finite Automata, DFA

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Languages, Alphabets and Strings

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defined over an alphabet:

Languages

zcba ,,,,

A language is a set of strings

A String is a sequence of letters

An alphabet is a set of symbols

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Alphabets and Strings

We will use small alphabets:

abbaw

bbbaaav

abu

baaabbbaaba

baba

abba

ab

aStrings

ba,

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Operations on Strings

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String Operations

m

n

bbbv

aaaw

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21

y bbbaaax abba

mn bbbaaawv 2121

Concatenation (sammanfogning)

xy abbabbbaaa

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12aaaw nR

naaaw 21 ababaaabbb

Reverse (reversering)

bbbaaababa

Example:

Longest odd length palindrome in a natural language:

saippuakauppias

(Finnish: soap sailsman)

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String Length

naaaw 21

1

2

4

a

aa

abba

nw Length:

Examples:

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Recursive Definition of Length

For any letter:

For any string :

Example:

1a

1wwawa

41111

11111

1

aab

abbabba

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Length of Concatenation

vuuv

853

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vuuv

aababaabuv

5,

3,

vabaabv

uaabuExample:

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Proof of Concatenation Length

Claim:

Proof: By induction on the length

Induction basis:

From definition of length:

vuuv

v

1v

vuuuv 1

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Inductive hypothesis:

vuuv

nv

1nv

vuuv

Inductive step: we will prove

for

for

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Inductive Step

Write , where

From definition of length:

From inductive hypothesis:

Thus:

wav 1, anw

1

1

wwa

uwuwauv

wuuw

vuwauwuuv 1

END OF PROOF

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Empty String

A string with no letters: (Also denoted as )

Observations:

}{{}

0

abbaabbaabba

www

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Substring (delsträng)

Substring of a string:

a subsequence of consecutive characters

String Substring

bbab

b

abba

ab

abbab

abbab

abbab

abbab

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Prefix and Suffix

Suffixesabbab

abbab

abba

abb

ab

a

b

ab

bab

bbab

abbab uvw

prefix

suffix

Prefixes

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Repetition

Example:

Definition:

n

n www... w

abbaabbaabba 2

0w

0abba

}

(String repeated n times)w

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The (Kleene* star) Operation

the set of all possible strings from alphabet

*

,,,,,,,,,*

,

aabaaabbbaabaaba

ba

[* Kleene is pronounced "clay-knee“]

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

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The + (Kleene plus) Operation

: the set of all possible strings from the

alphabet except

,ba

,,,,,,,,,* aabaaabbbaabaaba

*

,,,,,,,, aabaaabbbaabaaba

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Example

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

*

, fyoj , usch

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

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Operations on Languages

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Language

A language is any subset of

Example:

Languages:

*

,,,,,,,,*

,

aaabbbaabaaba

ba

},,,,,{

,,

aaaaaaabaababaabba

aabaaa

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Example

An infinite language }0:{ nbaL nn

Labb

aaaaabbbbb

aabb

ab

L

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Operations on Languages

The usual set operations

aaaaaabbbaaaaaba

ababbbaaaaaba

aaaabbabaabbbaaaaaba

,,,,

}{,,,

},,,{,,,

,,,,,,, aaabbabaabbaa ,,,,,,,,,* aabaaabbbaabaaba

LL *Complement:

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Reverse

}:{ LwwL RR

ababbaabababaaabab R ,,,,

}0:{

}0:{

nabL

nbaLnnR

nn

Examples:

Definition:

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Concatenation

Definition: 2121 ,: LyLxxyLL

baaabababaaabbaaaab

aabbaaba

,,,,,

,,,

Example

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Repeat

Definition:

Special case:

n

n LLLL

bbbbbababbaaabbabaaabaaa

babababa

,,,,,,,

,,,, 3

0

0

,, aaabbaa

L

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Example

}0:{ nbaL nn

}0,:{2 mnbabaL mmnn

2Laabbaaabbb

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Star-Closure (Kleene *)

Definition:

Example:

210* LLLL

,,,,

,,,,

,,

,

*,

abbbbabbaaabbaaa

bbbbbbaabbaa

bbabba

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Positive Closure

Definition

*L 21

LLL

,,,,

,,,,

,,

,

abbbbabbaaabbaaa

bbbbbbaabbaa

bba

bba

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Regular Expressions

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Regular Expressions: Recursive Definition

1

1

21

21

*

r

r

rr

rr

are Regular Expressions

,,Primitive regular expressions:

2rGiven regular expressions and 1r

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Examples

)(* ccbaA regular expression:

baNot a regular expression:

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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, ...}.

cba ,,Building Regular Expressions

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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)*.

cba ,,Building Regular Expressions

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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)*.

cba ,,

Building Regular Expressions

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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)*.

cba ,,Building Regular Expressions

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Languages of Regular Expressions

,...,,,,,*)( bcaabcaabcacbaL

Example

rL rlanguage of regular expression

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Definition

For primitive regular expressions:

aaL

L

L

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Definition (continued)

For regular expressions and

1r 2r

2121 rLrLrrL

2121 rLrLrrL

** 11 rLrL

11 rLrL

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Example *aba

*abaL *aLbaL

*aLbaL

*aLbLaL

*aba

,...,,,, aaaaaaba

,...,,,...,,, baababaaaaaa

Regular expression:

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Example

Regular expression

bbabar *

,...,,,,, bbbbaabbaabbarL

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Example

Regular expression

bbbaar **

}0,:{ 22 mnbbarL mn

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Example

Regular expression

*)10(00*)10( r

)(rL { all strings with at least

two consecutive 0 }

1,0

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Example

Regular expression

(consists of repeating 1’s and 01’s).

)0(*)011(1 r

)(rL = { all strings without

two consecutive 0 }

1,0

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Example

L = { all strings without

two consecutive 0 }

)0(*1)0(**)011*1(2 r

(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....1are 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 solution:

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Equivalent Regular Expressions

Regular expressions and

1r 2r

)()( 21 rLrL are equivalent if

Definition:

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Example L = { all strings without

two consecutive 0 }

)0(*1)0(**)011*1(2 r

LrLrL )()( 21 1r 2randare equivalent

regular expressions.

)0(*)011(1 r

Lennart Salling’s Video Resourceshttp://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 integer functions

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

http://www.youtube.com/embed/EuhbL0MPryU Vad handlar kursen om?

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/eQoP_kuNgXU?hl=sv&fs=1 Vad har språk och beräkningar med varandra att göra?

http://www.youtube.com/embed/-4K72wsQmPI?hl=sv&fs=1 Reguljära språk, vad är det?

http://www.youtube.com/embed/R2bHYnBXWFs?hl=sv&fs=1 Vilka automater är specialiserade på reguljära språk?

http://www.youtube.com/embed/2PNyEWl1AI0?hl=sv&fs=1 Varför icke-determinism?

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/WMN_wz-b3K0?hl=sv&fs=1

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

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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

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More Video’s 1

www.youtube.com/watch?v=MzDG34w0LLA&feature=player_embedded SubsetConstruction

http://www.youtube.com/watch?v=acsUNSkzUgg&feature=related Infinities

http://www.youtube.com/watch?v=y-zMnV3G9pg&feature=related Rice'sTheorem

http://www.youtube.com/watch?v=R2bHYnBXWFs&feature=related Finite Automata

http://www.youtube.com/watch?v=eQoP_kuNgXU&feature=related Strings and Languages

http://www.youtube.com/watch?v=-4K72wsQmPI&feature=related Regular Languages

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

 

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More Video’s 2

http://www.youtube.com/watch?v=RYNN-tb9WxI&feature=related

Regular Expression to NFA

http://www.youtube.com/watch?v=shN_kHBFOUE&feature=related 2 - Convert Regular Expression to Finite-State Automaton

http://www.youtube.com/watch?v=dlH2pIndNrU&feature=related Convert Regular Expression to DFA

http://www.youtube.com/watch?v=5a_pO3NYJl0 Great Principles of Computing - Peter J. Denning

http://www.youtube.com/watch?v=60P7717-XOQ&feature=related Stephen Wolfram: Computing a theory of everything

http://www.youtube.com/watch?v=cCdbZqI1r7I&feature=related Computing Beyond Turing - Jeff Hawkins

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