ICS 482 Natural Language Processing Regular Expression and Finite Automata

March 1, 2009March 1, 2009 Dr. Muhammed Al-mulhemDr. Muhammed Al-mulhem 11

ICS 482ICS 482Natural Language ProcessingNatural Language Processing

Regular Expression andRegular Expression andFinite AutomataFinite Automata

Muhammed Al-MulhemMuhammed Al-Mulhem

March 1, 2009March 1, 2009


Regular ExpressionsRegular Expressions

Regular expression (RE): A formula for Regular expression (RE): A formula for specifying a set of strings. specifying a set of strings.

String: A sequence of alphanumeric String: A sequence of alphanumeric characters (letters, numbers, spaces, tabs, characters (letters, numbers, spaces, tabs, and punctuation). and punctuation).


Regular Expression PatternsRegular Expression Patterns

RE RE String matchedString matched

woodchuckswoodchucks ““interesting links to interesting links to woodchuckswoodchucks and lemurs”and lemurs”

aa ““SSaarah Ali stopped by Mona’s”rah Ali stopped by Mona’s”

Ali says,Ali says, ““My gift please,” My gift please,” Ali says,Ali says,””

bookbook ““all our pretty all our pretty bookbooks”s”

!! ““Leave him behindLeave him behind!!” said Sami” said Sami


Specify Options and Range using Specify Options and Range using [ ] and -[ ] and -

RERE MatchMatch

[wW]ood[wW]ood Wood or woodWood or wood

[abc][abc] ““a”, “b”, or “c”a”, “b”, or “c”

[A-Z] an uppercase letter

[a-z] a lowercase letter

[0-9] a single digit


RE OperatorsRE OperatorsRERE DescriptionDescription

a*a* Zero or more a’sZero or more a’s

a+a+ One or more a’sOne or more a’s

a?a? Zero or one a’sZero or one a’s

[ab]*[ab]* Zero or more a’s or b’s. Matches aaa.., ababab.., Zero or more a’s or b’s. Matches aaa.., ababab.., bbbb..bbbb..

[0-9]+[0-9]+ Sequence of one or more digits.Sequence of one or more digits.

.. Wildcard expression-matches any single character.Wildcard expression-matches any single character.

\b\b Matches a word boundary. Matches the but not otherMatches a word boundary. Matches the but not other


Sidebar: ErrorsSidebar: Errors

Find all instances of the word “the” in a Find all instances of the word “the” in a text.text.– /the//the/

What About ‘The’What About ‘The’

– /[tT]he//[tT]he/What about ‘Theater”, ‘Another’What about ‘Theater”, ‘Another’



The process we just went through was The process we just went through was based on: based on: – Matching strings that we should not have Matching strings that we should not have

matched (there, then, other)matched (there, then, other)False positivesFalse positives

– Not matching things that we should have Not matching things that we should have matched (The)matched (The)

False negativesFalse negatives



Reducing the error rate for an application Reducing the error rate for an application often involves two efforts often involves two efforts – Increasing accuracyIncreasing accuracy (minimizing false (minimizing false

positives)positives)– Increasing coverageIncreasing coverage (minimizing false (minimizing false

negatives)negatives)


Regular expressionsRegular expressionsBasic regular expression patternsBasic regular expression patterns

Perl-based syntax (slightly different from other Perl-based syntax (slightly different from other notations for regular expressions)notations for regular expressions)

Disjunctions Disjunctions [abc][abc]

Ranges Ranges [A-Z][A-Z]

Negations Negations [^Ss][^Ss]

Optional characters Optional characters ?, +?, + and and **

Wild cards Wild cards ..

Anchors Anchors \b\b and and \B\B

Disjunction, grouping, and precedence Disjunction, grouping, and precedence ||


Preceding character or nothing Preceding character or nothing using ?using ?

March 1, 2009March 1, 2009 Dr. Muhammed Al-mulhemDr. Muhammed Al-mulhem 111104/21/23 11

WildcardWildcard

March 1, 2009March 1, 2009 Dr. Muhammed Al-mulhemDr. Muhammed Al-mulhem 121204/21/23 12

Negation using ^Negation using ^


Writing correct expressionsWriting correct expressions

Exercise: write a regular expression to match the Exercise: write a regular expression to match the English article “the”:English article “the”:

/the/ missed ‘The’

included ‘the’ in ‘others’/[tT]he/

/\b[tT]he\b/ Missed ‘the25’ ‘the_’

/[^a-zA-Z][tT]he[^a-zA-Z]/Missed ‘The’ at the beginning of a line


A more complex exampleA more complex example

Exercise: Write a Perl regular expression that Exercise: Write a Perl regular expression that will match “any PC with more than 500MHz and will match “any PC with more than 500MHz and 32 Gb of disk space for less than $1000”:32 Gb of disk space for less than $1000”:


Advanced Operators – Aliases for Advanced Operators – Aliases for common rangescommon ranges

Underscore: Correct figure 2.6


\ to Reference special characters\ to Reference special characters


Operators for countingOperators for counting


FFinite inite SState tate AAutomatautomata

FSAFSA recognizes the regular languages recognizes the regular languages represented by regular expressionsrepresented by regular expressions– SheepTalk: /baa+!/SheepTalk: /baa+!/

• Directed graph with labeled nodes and arc transitions

• Five states: q0 the start state, q4 the final state, 5 transitions

q0 q4q1 q2 q3

b a aa !


FormallyFormally

FSAFSA is a 5-tuple consisting of is a 5-tuple consisting of– QQ: set of states {q0,q1,q2,q3,q4}: set of states {q0,q1,q2,q3,q4} : an alphabet of symbols {a,b,!}: an alphabet of symbols {a,b,!}

– q0q0: A start state: A start state

– FF: a set of final states in Q {q4}: a set of final states in Q {q4} (q,i)(q,i): a transition function mapping : a transition function mapping Q x Q x

to Qto Q

q0 q4q1 q2 q3

b a

a

a !


FSA recognizes (FSA recognizes (acceptsaccepts) strings of a ) strings of a regular languageregular language– baa!baa!– baaa!baaa!– baaaa!baaaa!– ……

A rejected inputA rejected input

aa bb aa !! bb

q0 q4q1 q2 q3b a

aa !


State Transition TableState Transition Table

StateStateInputInput

bb aa !!

00 11 ØØ ØØ

11 ØØ 22 ØØ

22 ØØ 33 ØØ

33 ØØ 33 44

44 ØØ ØØ ØØq0 q4q1 q2 q3

b aa

a !

FSA can be represented FSA can be represented with State Transition Table with State Transition Table


Non-Deterministic FSAs for Non-Deterministic FSAs for SheepTalkSheepTalk

q0 q4q1 q2 q3

b a a a !

q0 q4q1 q2 q3

b a a !


A language is a set of A language is a set of stringsstrings

String:String: A sequence of lettersA sequence of letters

LanguagesLanguages


Tracing FSA - Initial ConfigurationTracing FSA - Initial Configuration

1q 2q 3q 4qa b b a

5q

a a bb

ba,

Input String

a b b a

ba,

0q


Reading the InputReading the Input

0q 1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b b a

ba,



5q

a a bb

ba,

a b b a

ba,



Output: “accept”

5q

a a bb

ba,

a b b a

ba,


RejectionRejection

1q 2q 3q 4qa b b a

5q

a a bb

ba,

a b a

ba,

0q



5q

a a bb

ba,

a b a

ba,



5q

a a bb

ba,

Output:“reject”

a b a

ba,


Another ExampleAnother Example

a

b ba,

ba,

0q 1q 2q

a ba


a

b ba,

ba,

0q 1q 2q

a ba


a

b ba,

ba,

0q 1q 2q

a ba

Output: “accept”


RejectionRejection

a

b ba,

ba,

0q 1q 2q

ab b


a

b ba,

ba,

0q 1q 2q

ab b


a

b ba,

ba,

0q 1q 2q

ab b

Output: “reject”


FormalitiesFormalities

Deterministic Finite Accepter (DFA)Deterministic Finite Accepter (DFA) FqQM ,,,, 0

Q

0q

F

: set of states

: input alphabet

: transition function

: initial state

: set of final states


About AlphabetsAbout Alphabets

Alphabets means we need a finite set of Alphabets means we need a finite set of symbols in the input.symbols in the input.

These symbols can and will stand for These symbols can and will stand for bigger objects that can have internal bigger objects that can have internal structure.structure.


Input Aplhabet Input Aplhabet


5q

a a bb

ba,

ba,

ba,


Set of States Set of States

Q


5q

a a bb

ba,

543210 ,,,,, qqqqqqQ

ba,


Initial State Initial State

0q

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q


Set of Final StatesSet of Final States

F

0q 1q 2q 3qa b b a

5q

a a bb

ba,

4qF

ba,

4q


Transition Function Transition Function


5q

a a bb

ba,

QQ :

ba,


10 , qaq

2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q 1q


50 , qbq

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q



5q

a a bb

ba,

ba,

32 , qbq


Transition FunctionTransition Function


5q

a a bb

ba,

a b

0q

1q

2q

3q

4q

5q

1q 5q

5q 2q

2q 3q

4q 5q

ba,5q5q5q5q


Extended Transition FunctionExtended Transition Function(Reads the entire string)(Reads the entire string)

*

QQ *:*


5q

a a bb

ba,

ba,


20 ,* qabq

3q 4qa b b a

5q

a a bb

ba,

ba,

0q 1q 2q


40 ,* qabbaq


5q

a a bb

ba,

ba,


50 ,* qabbbaaq

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q


50 ,* qabbbaaq

1q 2q 3q 4qa b b a

5q

a a bb

ba,

ba,

0q

Observation: There is a walk from to with label

0q 5qabbbaa


ExampleExample


5q

a a bb

ba,

ba,

abbaML M

accept


Another ExampleAnother Example


5q

a a bb

ba,

ba,

abbaabML ,, M

acceptacceptaccept


More ExamplesMore Examples

a

b ba,

ba,

0q 1q 2q

}0:{ nbaML n

accept trap state


ML = { all substrings with prefix }ab

a b

ba,

0q 1q 2q

accept

ba,3q

ab


ML = { all strings without substring }001

0 00 001

1

0

1

10

0 1,0


Regular LanguagesRegular Languages

A language is regular if there is a DFA A language is regular if there is a DFA such thatsuch that

All regular languages form a language All regular languages form a language familyfamily

LM MLL


ExampleExampleThe languageThe language

is regular:is regular:

*,: bawawaL

a

b

ba,

a

b

ba

0q 2q 3q

4q


Finite State AutomataFinite State Automata

Regular expressions can be viewed as a Regular expressions can be viewed as a textual way of specifying the structure of textual way of specifying the structure of finite-state automata.finite-state automata.


More FormallyMore Formally

You can specify an FSA by enumerating You can specify an FSA by enumerating the following things.the following things.– The set of states: QThe set of states: Q– A finite alphabet: A finite alphabet: ΣΣ– A start stateA start state– A set of accept/final statesA set of accept/final states– A transition function that maps QxA transition function that maps QxΣΣ to Q to Q

ICS 482 Natural Language Processing Regular Expression and Finite Automata

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