Regular Expression Question Solution

8/10/2019 Regular Expression Question Solution

1/68

Dr. Shakir Al Faraji

Theory of Computation

Thank you, Shakir

Shakir Al Faraji

Computer Science Dept.,

Petra University

Amman - Jordan.email: [email protected]


2/68


IMPORTANT NOTES

Students

This presentation is designed to be used inclass as part of a guided discovery

sequence. It is not self-explanatory! Pleaseuse it only for revision purposes after havingtaken the class. Simply going through theslides will teach you nothing. You must beactively thinking, doing and questioning to

learn!

Thank you, Shakir.


3/68


Course Strategy

Be Warned: This is not a coursethat spoon-feeds students.

Students are expected to beinvestigative and resourceful.

Reading books and other research

of topics are expected.

Thank you, Shakir.


4/68


Material

There is a book:

Hopcroft, Rajeev,& Ullman 3edEdition

(2007), Addison Wesley

These were the lecture notes.Well, apart from the slides.

Thank you, Shakir.


5/68


Regular Expression


6/68


Definition

A regu lar exp ress ion, or RE,describes strings of characters

(words or phrases or anyarbitrary text). It's a pattern thatmatches certain strings and

doesn't match others.A regularexpressionis a set ofcharacters thatspecify a pattern.ORLanguage defining symbols.


7/68Dr. Shakir Al Faraji

Definition Cont

Regular expressions are used togeneratepatterns of strings. A

regular expression is an algebraicformula whose value is a pattern

consisting of a set of strings,called the language of the

expression.


8/68Dr. Shakir Al Faraji

Operands in a regular

expression

Operands in a regular expression can be:charactersfrom the alphabet over which

the regular expression is defined.variableswhose values are any pattern

defined by a regular expression.

epsi lonwhich denotes the empty stringcontaining no characters.

nul lwhich denotes the empty set ofstrings.


9/68


Operators used in

regular expressions

Union: If R1 and R2 are regularexpressions, then R1 | R2 (also written asR1 U R2 or R1 + R2) is also a regularexpression.

L(R1|R2) = L(R1) U L(R2).Concatenation: If R1 and R2 are regular

expressions, then R1R2 (also written asR1.R2) is also a regular expression.L(R1R2) = L(R1) concatenated with L(R2).


10/68


Operators used in

regular expressions

Kleene closure: If R1 is a regularexpression, then R1* (the Kleene closure

of R1) is also a regular expression.L(R1*) = epsilon U L(R1) U L(R1R1) U L(R1R1R1) U ...

Closure has the highest precedence,

followed by concatenation, followed byunion.


11/68


Examples

The set of strings over {0,1} that end in

3 consecutive 1's.

(0 | 1)*111OR(0 + 1)*111

The set of strings over {0,1} that have

at least one 1.0*1 (0 + 1) *


12/68


Examples Cont.

The set of strings over {0,1} that have atmost one 1.

0* | 0* 1 0*

The set of strings over {A..Z,a..z} thatcontain the word "main".

Let = A | B | ... | Z | a | b | ... | z

* main *


13/68


Examples Cont

.

The set of strings over {A..Z,a..z} that

contain 3 x's.

* x* x* x*


14/68


Examples Cont

.

The set of identifiers in Pascal.

Let = A | B | ... | Z | a | b | ... | zLet = 0 | 1 | 2 | 3 ... | 9

( | )*


15/68


Examples Cont

.

The set of real numbers in Pascal.

Let = 0 | 1 | 2 | 3 ... | 9

Let = 'E' * | epsilonLet = '+' | '-' | epsilonLet = '.' * | epsilon

*


16/68


Examples-Cont.

Consider = { a }

L is a language that each word isof odd length

a (aa)*


17/68


Examples-Cont.

Consider = { a, b }

L is a language that each wordmust start with the letter b

b (a+b)*


18/68


19/68


Examples-Cont.

Consider = { a, b, c }

L = { a, c, ab, cb, abb, cbb,abbb,cbbb, abbbb, cbbbb . . . }

L - language ((a+c) b*)


20/68


Examples-Cont.

Consider (a+b)*a(a+b)*

L = language of all words over the= { a, b }that have an ain them

L = { a, aa, ba, aab, aba, baa,

bba, aaaa, aaba, abaa . . . }


21/68


Examples-Cont.

Consider the following RE

(a+b)* a(a+b)* a(a+b)*

L = language of all words over the= { a, b }that have at least two

asin them


22/68


Examples-Cont.

(a+b)* a(a+b)* a(a+b)* = b*ab*a(a+b)*

?(a+b)* a(a+b)* a(a+b)* = (a+b)*ab*ab*

?

(a+b)* a(a+b)* a(a+b)* = b*a(a+b)*ab*


23/68


Examples-Cont.

(a+b)* a(a+b)* b(a+b)*

?


24/68


Examples-Cont.

(a+b)* a(a+b)* b(a+b)*

?

Language of all words that have at

least one aand at least one b!!!!


25/68


Examples-Cont.

(a+b)* a(a+b)* b(a+b)*

What about the word ba!!!!


26/68


Examples-Cont.

(a+b)* a(a+b)* b(a+b)*

What about the word ba!!!!

MUST BE

(a+b)* a(a+b)* b(a+b)* + (a+b)* b(a+b)* a(a+b)*


27/68


Examples-Cont.

IS

(a+b)* a(a+b)* b(a+b)* + (a+b)* b(a+b)* a(a+b)*

SAME AS

(a+b)* a(a+b)* b(a+b)* + bb*aa*


28/68


Examples-Cont.

b* + ab*

( + a )b*

b* + ab* = ( + a )b*


29/68


More on RE.

Defini t ion

If Sand Tare sets of strings of letters, we

define the product set of strings of letters tobeST= {all combinations of a string from S

concatenated with a string fromT}


30/68


More on RE - Cont.

If S= { a, aa, aaa} , T= { bb, bbb}then

ST= {abb,abbb,aabb,aabbb,aaabb,aaabbb }


31/68


Languages Associatedwith RE.

Defini t ionlanguage associated with the RE just is a single

letter is that one-letter word alone and the languageassociated with is just {

}, a one-word language.if r1is a RE associated with the languageL1and r2

is a RE associated with the language L2then:the RE (r1)(r2) is associated with the product L1L2RE (r1+r2) is associated with the language formed by

the union of sets L1and L2RE (r1)* is associated with the language L1*


32/68


Finite Languages areRegular.

Theorem

If L is a finite language, then L can be definedby a regular expression. In other words, allfinite languages are regular.ProofLet L = { aa, ab, ba, bb }

RE isaa+ab+ba+bb(a+b)(a+b)


33/68


Examples.

Can you describe the following RE

(a+b)* (aa+bb) (a+b)*


34/68


Examples.

Can you describe the following RE

(a+b)* (aa+bb) (a+b)*

All strings of as and bs that at some point

contain a double letter.


35/68


Examples.

= { a, b }What strings do not contain a double

letter?


36/68


Examples.


letter? (ab)*


37/68


Examples.


letter? (ab)*

Is it correct ?


38/68


Examples.


letter? (ab)*

Is it correct ? NO


39/68


Examples.


letter? (ab)*

Is it correct ? NO

(+b)(ab)*(+a)


40/68


Examples.

( a + b* )* = (a + b )* ?

( aa + ab*)* = (aa + ab)* ?

( a* b* )* = (a + b )* ?


41/68


Examples.

( a + b* )* = (a + b )* YES

( aa + ab*)* = (aa + ab)* NO

( a* b* )* = (a + b )* YES


42/68


Examples.

[aa + bb + ( ab + ba) (aa+bb)*(ab + ba) ]*


43/68


Examples.

[aa + bb + ( ab + ba) (aa+bb)*(ab + ba) ]*

EVEN-EVENtype1= aatype2= bbtype3= (ab+ba)(aa+bb)*(ab+ba)

E = [ type1+ type2+ type3] *


44/68


What Regular Expressions AreExactly - Terminology

Basically, a regular expression is apattern describing a certain amount

of text. Their name comes from themathematical theoryon which theyare based. But we will not dig intothat. Since most people including

myself are lazy to type, you willusually find the name abbreviated toregex or regexp.


45/68


What Regular Expressions AreExactlyCont.

This first example is actually a perfectlyvalid regex. It is the most basic pattern,

simply matching the literal text regex. A"match" is the piece of text, or sequence ofbytes or characters that pattern was foundto correspond to by the regexprocessingsoftware. Matches are highlighted in blue onthis site.


46/68



b[A-Z0-9._ -] @[A-Z0-9._ -] \.[A-Z]{2,4}\b


47/68



b[A-Z0-9._ -] @[A-Z0-9._ -] \.[A-Z]{2,4}\b

is a more complex pattern. It describes a

series of letters, digits, dots, percentagesigns and underscores, followed by an atsign, followed by another series of letters,digits, dots, percentage signs andunderscores, finally followed by a single dotand between two and four letters. In otherwords: this pattern describes an emailaddress.


48/68



With the above regular expression pattern,you can search through a text file to findemail addresses, or verify if a given string

looks like an email address. In this tutorial, Iwill use the term "string" to indicate the textthat I am applying the regular expression to.


49/68



The term "string" or "character string" isused by programmers to indicate asequence of characters. In practice, you

can use regular expressions withwhatever data you can access using theapplication or programming language youare working with.

Wh t R l E i A


50/68



A regular expression uses metacharacters(characters that assume special meaning formatching other characters) such as *, [ ], $

and .. For example, the RE [Hh]ello!* wouldmatch Helloand hel loand Hello!(andhello!!! ! !). The RE [Hh](ello|i)!* would matchHelloand Hiand Hi!(and so on). A backslash

(\) disables the special meaning of thefollowing character, so you could match thestring [Hel lo]with the RE \[Hello\].


51/68


How can I use regularexpressions?

Many text editors allow regular-expressionsearch-and-replace. EditPlusfor Windowshas this capability, as does BBEditfor the

Macintosh.EditPlus

The EditPlus search-replace window has acheckbox called Regular expression. To

use regular expressions in your search,simply check this box.


52/68


How can I use regularexpressions?Cont.

BBEdit

BBEdits search-replace window also hassuch a checkbox; its label, however, is

Use grep.Grep. What an odd term. The word grep is fromthe creators of the UNIX operating system, someof the first implementers of regular expressions.

UNIX programmers delighted in reducing longcommands to meaningless acronyms; grep issaid to have meant general regular expression

print.


53/68


Defining regular expressionpatterns

The way regular-expression patterns workis by creating a special little language in

which ordinary symbols take on special

meanings. This guide will go through thespecial meanings little by little, withexamples. You will get the most from thisguide if you read all the way through it. To

be sure you do, Ive left a very importantpiece of informationhow to replace what

you findfor the end.


54/68


Defining regular expressionpatterns-Cont

Dot, question mark, star, plus, andbackslash

Imagine that you have a book of letters, andyou need to tag all the salutations.Salutations fall into a pattern: the wordDear, a name, and a colon (or possibly a

comma, but well stick with a colon fornow). Obviously, the problem with findingthis via ordinary search is that the namecould be anything.


55/68



Regular expressions have a way of sayingany character: the dot, or period. To

find a three-letter word beginning andending with b, for example, you couldsearch onb.b . This would find bib

or bob or bub, but not bud or

dub or bulb.


56/68



Note that whitespace characters such asspace or tab can also be located by the dot.Sob.bwould find words with a space or

tab between them. (Quick exercise: wherewouldb.bmatch in the precedingsentence? There are two possibilities!)The hard return, however, is not matched

by a dot; more on hard returns later.This still wont solve our salutation problem,though: names are made up out of a variablenumber of letters, not just one.
http://www.textartisan.com/articles/regex.htmlhttp://www.textartisan.com/articles/regex.html


57/68



Regular expressions have several ways tosay not just one: the question mark (?),the star or asterisk (*), and the plus (+).

The question mark means zero or one,the star means zero or more, and theplus means one or more. These marksare like adjectives; they modify other

characters. Whats more, theyre likeadjectives in some foreign languages, inthat they come immediately afterthecharacter they modify.

fi i l i


58/68



Regular expressions have several ways tosay not just one: the question mark (?),the star or asterisk (*), and the plus (+).

The question mark means zero or one,the star means zero or more, and theplus means one or more. These marksare like adjectives; they modify other

characters. Whats more, theyre likeadjectives in some foreign languages, inthat they come immediately afterthecharacter they modify.

fi i l i


59/68



So the regular expression Ba?will match BorBabut not Baaor a. The regular expressionBa*will match Bor Baor Baa, up to any

number of as. The regular expressionBa+will match Baor Baaand on up, but itwill not match Bby itself, since the plussign demands at least one a.

Combining the dot with the plus or starsolves our salutation problem. The regularexpression Dear .+:will find anyimaginable business-letter salutation.

D fi i l i


60/68



But what if you actually want to look for adot, a star, a question mark, or a plus?How can you find them, if theyve got

special meanings?Any special regular-expression character loses its

special meaning if there is a backslash (\) beforeit. So \.will find a real period, like the one at the

end of this sentence. The backslash works onitself, too; to find a real backslash, put \\in your

search.

Wh l d f


61/68


What weve learned so far

(Metacharacters)

Character Regular-expression meaning

. Any character, including space or tab

? Zero or one of the preceding character

* Zero or more of the preceding

character

+

One or more of the preceding

character\ Negates the special meaning of the

following character


62/68


Metacharacters

As youve learned, the backslash negates anyspecial meaning that the character

following it has to a regular expression. It

has another function, too: it can turnordinary characters into special ones.

Consider the tab. You dont see it on the

screen the way you see ordinary letters;you see what it does.


63/68


Metacharacters-Cont.

If you turn on the show-invisibles function,however, you generally see an indicationthat there is a character there.

Regular expressions let you access theseinvisible characters (usually calledmetacharacters):


64/68



Metacharacter Meaning

\n Newline (or paragraph

mark, or however you

think of it)

\t Tab character

\s Any whitespace

character (tab, space, ornewline)


65/68



For purposes of modifiers like star andplus, these metacharacters act like singlecharacters. So \n+finds one or more

newlines.A special caution with BBEdit: Because of

ancient OS wars, Macs and non-Macstreat newlines differently. If a regularexpression containing \nisnt finding

what you think it should, try replacing \nin your search pattern with \r.


66/68



Depending on your regular-expressionengine or editing program, there may beother metacharacters available to you.

Read the manual or help pages for details.In addition, a few more special regular-

expression characters provide usefulfunctions. Remember that to look for theactual character, you must precede it witha backslash.


67/68



Depending on your regular-expressionengine or editing program, there may beother metacharacters available to you.

Read the manual or help pages for details.In addition, a few more special regular-

expression characters provide usefulfunctions. Remember that to look for theactual character, you must precede it witha backslash.


68/68

END

Documents

Regular Expression Question Solution