Finite State Automata and Tries

Finite State Automata and Tries

Sambhav JainIIIT Hyderabad

Finite State Automata and Tries 2

Think !!!

• How to store a dictionary in computer?

• How to search for an entry in that dictionary?

– Say you have each word length exactly equal to 10 characters and can take any letter from ‘a-z’

Eg. aaaaaaaaaa, abcdefghij, …. etc Language = [a-z]{10} - RegEx


A Simple Way

• aaaaaaaaaa• aaaaaaaaab• aaaaaaaaac• ….• ….• ….• ….• zzzzzzzzzz

A Linear Sorted List of Entries


A Simple Way

• aaaaaaaaaa• aaaaaaaaab• aaaaaaaaac• ….• ….• ….• ….• zzzzzzzzzz

Character to be stored = 2610

= 1.41167096 × 1014

Each character take 1 Byte

~ 141 TB


Smart Way !

a b c d w x y z

a b c d w x y z

a b c d w x y z

……………………………………………..

……………………………………………..

……………………………………………..

………………………………..……………………………………………………………………………………….


Smart Way !

a b c d w x y z

a b c d w x y z

a b c d w x y z

……………………………………………..

……………………………………………..

……………………………………………..

………………………………..……………………………………………………………………………………….

•Total Storage = 26x10 = 260 bytes•Traverse 10 nodes


Does it work for Natural Language

• Oxford Advanced English Learner 20th Edition– A quarter of a million distinct English words,

excluding inflections, and words from technical and regional vocabulary not covered by the OED

• After inflections ? – eat,eats,eaten,eating …..

• What after multiple inflexion ???– beauty, beautiful, beautifully …


Example (Store & Search)

e

e

a

s

t

n

i n g


Example

e

e

a

s

t

n

i n g

b


Example

e

e

a

s

t

n

i n g

b

f

s

a


Example

e

e

a

s

t

n

i n g

b

f

s

a

tei

r

w


Inflectional morphology

• Deals with word forms of a root, when there is no change in lexical category.

• Each word form gives different values of features like gender, number, person, etc.


Paradigm

• For a given root, there are many word forms with different features.

• Ex. Forms of Hindi root laDakA (boy)

Direct Oblique

Singular laDakA laDake

Plural laDake laDakoM


Paradigm

- 'laDakoM' is plural with oblique case - given by feature structure {num=pl,

case=obl} - 'laDake' stands for two feature structures + Singular oblique (Ex. laDake ne kahA ...) - where oblique means 'laDake' is followed

by a postposition marker + plural direct case (Ex. laDake Aye)


Paradigmo Paradigms - What operation is done on root to obtain word forms - Model using pairs: (delete string, add string) | direct oblique ---|----------------------- sg | (O,O) (A,e) pl | (A,e) (A,oM) o List roots with paradigms they follow: - ghoDA follows paradigm laDakA - charkhA follows paradigm laDakA - laDakA follows paradigm laDakA•


l k | | a a | | D p | | -------- a | | | a A D | | | k ------- | | | | ------------ | I i | | | ------- | A e o | | | A | | | | | | A e o M M | M


Abstracting out suffixes

k l | | a a | | p D | | a --------- | | | D #1 a A | | k (#1) I

#1: Corresponds to paradigm for 'laDakA'


- Suffix trie (forward)

#1 | -------------- | | | e o A | M


• Can we further optimize our search ?- Use knowledge of paradigms

- Use suffix tree


• Store suffix tree in main memory• Store rest of the categorized by paradigm in

hard disk• Do backward search for suffix tree• Identify the paradigm• Search only in that paradigm set• Eg. if ‘–ing’ occur you first won’t be searching

word like home, cat, god …


Finite State Automata

• Trie is a data structure

• FSA is the computational approach

• Slight difference in representation – Putting characters on edges rather than nodes


+ / \ l / \ k + + a | | a | | + + D | | p | | + + a | | a | | + + k | | D | | + + \ / 0 \ / 0 +______ e/ \o \ A / \ \ (+) + (+) | |M (+)


FSAo A deterministic finite-state machine formally is - Q: A finite set of states (Ex.:{q0,q1,q2}) - SIGMA: A finite set of input alphabet (Ex.: {a,b,c}) - Start state: A state in Q, from which machine starts (Ex.: q0) - F: A set of accepting states (Ex.: {q2}) - DELTA (q,i): A transition function or transition matrix where: - q MEMBER Q, i MEMBER SIGMA, - DELTA(q,i) MEMBER Q

Thus, DELTA(q,i): Q x SIGMA --> Q


RECOGNITION Problem

• Till now we were handling only RECOGNITION problem

• If FSA reach a final state at the end of input string then EXIST

• Else NOT


• But we seek analyzed output• We want the machine to tell– Root– Gender– Number– Person– Case– Etc ……


Finite State TransducerFST is like the finite state automation defined earlier, except each arc is labelled by a pair of symbols: i:o where i: symbol in input string o: symbol output by FST when are is taken

+ Ex. arc in finite state transducer corresponding to 'e' in 'ladake'

e : ((+pl, -direct), (+sg, +dir)) q1 +----------------->--------------------+ q2

Two pairs of symbols: i : o - i is: 'e' - o is: '((+pl, -direct), (+sg, +dir))'

+ Ex. Morph Analyzer: Match input with i, if successful go ahead & produce o in output


o Formally: Finite state transducer - Q: Finite set of states q0, ..., qN - SIGMA_IN: Finite set of input symbols - SIGMA_OUT: Finite set of pairs output symbols - q0: Start state (q0 IN Q) - F: Set of final accepting states (F SUBSET Q) - DELTA (q, i:o) : For every state q, gives a set of states that can be reached from q with i in SIGMA_IN, and o in SIGMA_OUT.


Example

• on board


Tools for FSA

• Lex• OpenFST– (www.openfst.org/)

• AT&T FSM Toolkit – (http://www2.research.att.com/~fsmtools/fsm/)

http://www.openfst.org/



http://www2.research.att.com/~fsmtools/fsm/

Finite State Automata and Tries

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