Post on 18-Jan-2016
Natural Language Processing
Lecture 4 : Regular Expressions and Automata
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• A language is a set of strings• String: A sequence of letters
Examples: “cat”, “dog”, “house”, …
Defined over an alphabet:
Definitions
zcba ,,,,
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Alphabets and Strings
• We will use small alphabets:• Strings
abbaw
bbbaaav
abu
ba,
baaabbbaaba
baba
abba
ab
a
Regular Expressions
• In computer science, RE is a language used for specifying text search string.
• A regular expression is a formula in a special language that is used for specifying a simple class of string.
• Formally, a regular expression is an algebraic notation for characterizing a set of strings.
• RE search requires a pattern that we want to search for, and a corpus of texts to search through.
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Basic Regular Expression Patterns
• The use of the brackets [] to specify a disjunction of characters.
• The use of the brackets [] plus the dash - to specify a range.
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Basic Regular Expression Patterns
• Uses of the caret ^ for negation or just to mean ^
• The question-mark ? marks optionality of the previous expression.
• The use of period . to specify any character
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Disjunction, Grouping, and Precedence
• Disjunction• /cat|dog• Precedence• /gupp(y|ies)• To find the English article the• /the/• /[tT]he/• /[^a-zA-Z][tT]he[^a-zA-Z]/
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Aliases for common sets of characters
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Regular expression operators for counting
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Some characters that need to be backslashed
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Finite State Automata
• FSAs recognize the regular languages represented by regular expressions SheepTalk: /baa+!/
• Directed graph with labeled nodes and arc transitions
•Five states: q0 the start state, q4 the final state, 5 transitions
q0q0 q4q4q1q1 q2q2 q3q3
b aa
a !
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Formally
• FSA is a 5-tuple consisting of Q: set of states {q0,q1,q2,q3,q4}: an alphabet of symbols {a,b,!} q0: A start state F: a set of final states in Q {q4} (q,i): a transition function mapping Q
x to Q
q0q0 q4q4q1q1 q2q2 q3q3
b a
a
a !
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• FSA recognizes (accepts) strings of a regular language baa! baaa! baaaa! …
• Tape Input: a rejected input
a b a ! b
q0q0 q4q1q1 q2q2 q3b a
aa !
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State Transition Table for SheepTalk
StateInput
b a !
0 1 Ø Ø
1 Ø 2 Ø
2 Ø 3 Ø
3 Ø 3 4
4 Ø Ø Ø
baa! baaa! baaaa! baaaaa! ...
q0q0 q4q4q1q1 q2q2 q3q3
b aa
a !
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Non-Deterministic FSAs for SheepTalk
q0q0 q4q4q1q1 q2q2 q3q3
b a a a !
q0q0 q4q4q1q1 q2q2 q3q3
b a a !
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Finite Accepter
• Input
“Accept” or“Reject”
String
FiniteAutomata
Output
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Transition Graph
•
initialstate
final state“accept”state
transition
abba -Finite Accepter
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
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Initial Configuration
•
1q 2q 3q 4qa b b a
5q
a a bb
ba,
Input Stringa b b a
ba,
0q
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Reading the Input
•
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
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•
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
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•
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
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•
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b b a
ba,
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0q 1q 2q 3q 4qa b b a
Output: “accept”
5q
a a bb
ba,
a b b a
ba,
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Rejection
•
1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
0q
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•
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
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•
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
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•
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b a
ba,
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0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
Output:“reject”
a b a
ba,
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Another Example
a
b ba,
ba,
0q 1q 2q
a ba
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a
b ba,
ba,
0q 1q 2q
a ba
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a
b ba,
ba,
0q 1q 2q
a ba
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a
b ba,
ba,
0q 1q 2q
a ba
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a
b ba,
ba,
0q 1q 2q
a ba
Output: “accept”
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Rejection
a
b ba,
ba,
0q 1q 2q
ab b
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a
b ba,
ba,
0q 1q 2q
ab b
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a
b ba,
ba,
0q 1q 2q
ab b
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a
b ba,
ba,
0q 1q 2q
ab b
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a
b ba,
ba,
0q 1q 2q
ab b
Output: “reject”
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Formalities
• Deterministic Finite Accepter (DFA)
FqQM ,,,, 0Q
0q
F
: set of states
: input alphabet
: transition function
: initial state
: set of final states
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About Alphabets
• Alphabets means we need a finite set of symbols in the input.
• These symbols can and will stand for bigger objects that can have internal structure.
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Input Aplhabet
•
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
ba,
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Set of States
Q
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
543210 ,,,,, qqqqqqQ
ba,
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Initial State
0q
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
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Set of Final States
F
0q 1q 2q 3qa b b a
5q
a a bb
ba,
4qF
ba,
4q
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Transition Function
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
QQ :
ba,
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10 , qaq
2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q 1q
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50 , qbq
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
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0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
32 , qbq
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Transition Function
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
a b
0q
1q
2q
3q
4q
5q
1q 5q
5q 2q
2q 3q
4q 5q
ba,5q5q5q5q
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Extended Transition Function(Reads the entire string) *
QQ *:*
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
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20 ,* qabq
3q 4qa b b a
5q
a a bb
ba,
ba,
0q 1q 2q
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40 ,* qabbaq
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
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50 ,* qabbbaaq
1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
0q
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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
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Example
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
abbaML M
accept
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Another Example
0q 1q 2q 3q 4qa b b a
5q
a a bb
ba,
ba,
abbaabML , M
acceptacceptaccept
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More Examples
a
b ba,
ba,
0q 1q 2q
}0:{ nbaML n
accept trap state
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ML = { all substrings with prefix }ab
a b
ba,
0q 1q 2q
accept
ba,3q
ab
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ML = { all strings without substring }001
0 00 001
1
0
1
10
0 1,0
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Regular Languages
• A language is regular if there is a DFA such that
• All regular languages form a language family
LM MLL
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Example
• The language• is regular:
*,: bawawaL
a
b
ba,
a
b
ba
0q 2q 3q
4q
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Dollars and Cents