Natural Language Processing Lecture 4 : Regular Expressions and Automata

2 A language is a set of strings String: A sequence of letters  Examples: “cat”, “dog”, “house”, …  Defined over an alphabet: Definitions

3 Alphabets and Strings We will use small alphabets: Strings

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.

5 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.

6 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

7 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]/

8 Aliases for common sets of characters

9 Regular expression operators for counting

10 Some characters that need to be backslashed

11 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 q0 q4 q1 q2 q3 ba a a!

12 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 q0 q4 q1 q2 q3 ba a a!

13 FSA recognizes (accepts) strings of a regular language  baa!  baaa!  baaaa!  … Tape Input: a rejected input aba!b q0 q4 q1 q2 q3 ba a a!

14 State Transition Table for SheepTalkSheepTalk State Input ba! 01ØØ 1Ø2Ø 2Ø3Ø 3Ø34 4ØØØ baa! baaa! baaaa! baaaaa !... q0 q4 q1 q2 q3 ba a a!

15 Non-Deterministic FSAs for SheepTalk q0 q4 q1 q2 q3 ba a a! q0 q4 q1 q2 q3 baa! 

16 Finite Accepter Input “Accept” or “Reject” String Finite Automata Output

17 Transition Graph initial state final state “accept” state transition abba -Finite Accepter

18 Initial Configuration Input String

12/21/ Reading the Input

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12/21/ Output: “accept”

12/21/ Rejection

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12/21/ Output: “reject”

12/21/ Another Example

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12/21/ Output: “accept”

12/21/ Rejection

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12/21/ Output: “reject”

12/21/ Formalities Deterministic Finite Accepter (DFA) : set of states : input alphabet : transition function : initial state : set of final states

12/21/ 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.

12/21/ Input Aplhabet

12/21/ Set of States

12/21/ Initial State

12/21/ Set of Final States

12/21/ Transition Function

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12/21/ Transition Function

12/21/ Extended Transition Function (Reads the entire string)

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12/21/ Observation: There is a walk from to with label

12/21/ Example accept

12/21/ Another Example accept

12/21/ More Examples accept trap state

12/21/ = { all substrings with prefix } accept

12/21/ = { all strings without substring }

12/21/ Regular Languages A language is regular if there is a DFA such that All regular languages form a language family

12/21/ Example The language is regular:

12/21/ Dollars and Cents