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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

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

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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:

String 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

Length of Concatenation

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)

Repetition

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


Reverse

Definition:

Examples:


Concatenation

  • Definition:

Example


Repeat

  • Definition:

  • Special case:


Example


Star-Closure (Kleene *)

  • Definition:

  • Example:


Positive Closure

  • 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/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?


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

  • www.youtube.com/watch?v=MzDG34w0LLA&feature=player_embeddedSubsetConstruction

  • http://www.youtube.com/watch?v=acsUNSkzUgg&feature=relatedInfinities

  • http://www.youtube.com/watch?v=y-zMnV3G9pg&feature=relatedRice'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=relatedRegular Languages

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


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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