Finite state subautomata Application to Electronic Dictionaries Lamia Tounsi Polytech'Tours, Computer Science laboratory François Rabelais University.

Finite state subautomataApplication to Electronic Dictionaries

Lamia TounsiPolytech'Tours, Computer Science laboratory

François Rabelais University of Tours, France

[email protected]

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Motivation

o DFSA are widely used in Natural Language processing

Find all sub structures in a given FSA.

Search of subautomata in a DFSA• Decompose a very large FSA into smaller ones• Discover frequently occurring data • Reduce memory consumption

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Plan

Mathematical preliminaries • Automaton• Subautomaton

Research of subautomata• Smallest closed subautomaton• Smallest subautomaton

Application to automata representing dictionaries Indexation and Compression Conclusion

Finite state subautomataApplication to Electronic Dictionnaries

Mathematical preliminaries •Automaton•Subautomaton

Research of subautomata•Smallest closed subautomaton•Smallest subautomaton

Application to automata representing dictionariesIndexation and Compression Conclusion

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Automaton

A deterministic acyclic automaton A =<, Q, , qi, qf > is the alphabet• Q is the finite set of states is the transition function: : Q Q• qi is the initial state (qi Q)• qf is the final state (qf Q)

Let a and w * : (p, )=p (p, wa)= ( (p,w),a)

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Successors & predecessors

Succ(p) = {qQ : , (p,)= q}Succ*(p) = {qQ : w*, (p,w)= q}

Pred(p) = { qQ : , (q,)= p}Pred*(p) = { qQ : w*, (q,w)= p}

Height : • H(qf)=0• H(p)=Max{q Succ(p)} H(q)+1

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Automaton

An automaton that recognizes the flexion of nine verbs

H(14)=4

H(13)=5

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Source (E) & Initial State (p)

Let E E

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Let E • AP(E)={ w path from qi to p, p E}

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• AN(E)={p Q/ w AP(E), p w}

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source(E) AN (E)

Source(E) :

H(source(E)) =MinqAN (E)(H(q))

Source(E)

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source(E) AN (E)

source(E) :

H(source(E)) =MinqAN (E)(H(q))

Let p Q, p qi

IS(p) = Source(Pred(p))

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Source(q2, q3, q5) = Source(q3, q4) = q2

Source(q3, q4, q5) = Source(q3, q4, q5 , q6) = q1

IS(q3)= q2

IS(q5)= q1

IS(q6)= q1

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Sink (E) & Final State (p)

Let E • PP(E) = { w path from p to qf, p E}

• PN(E) = {p Q/ w PP(E), p w}

Sink(E) PN (E)

Sink(E) :

H(Sink(E)) =MaxqPN (E)(H(q))

Let p Q, p qi

FS(p) = Sink(Succ(p))

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Subautomaton (SA)

A’=<, Q’, ’, si, sf > is a sub automaton of A iff:• Q’ Q

• {si, sf } Q’

Q’ Q’ ’:

(q, ) Q’ : ’ (q, ) = (q, )

q Q’ : q Succ*(si) and q Pred*(sf)

q Q’ \ {si, sf } : Succ(q) Q’ and Pred(q) Q’

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Subautomaton (SA)


SA

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Subautomaton (SA)


SA

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Subautomaton (SA)


SA

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Subautomaton (SA)


SA

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Closed subautomaton (CSA)

Let Q Q’ and si, sf two distinct states:

A subautomaton A’=<, Q’, ’, si, sf > is a closed subautomaton iff :

q Q’ \ {si}: Pred(q) Q’

q Q’ \ {sf}: Succ(q) Q’

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CSA

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CSA

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CSA

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Smallest Closed subautomaton (SCSA)

Let Q Q’ and si, sf two distinct states:

A closed subautomaton A’=<, Q’, ’, si, sf >is a smallest closed subautomaton iff :

(si, q) is CSA q= sf

q Q’ :

(q, sf) is CSA q= si

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Smallest Closed subautomaton (SCSA)


SCSASCSASCSA SCSA

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Smallest subautomaton (SSA)

Let p Q \{si, sf}

The subautomaton A’=<, Q’, ’, si, sf >

is SSA(p) iff :- A’ strictly contains p A’’=<, Q’’, ’’, s’’i, s’’f > wich strictly

contains p : Q’ Q’’

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Smallest subautomaton (SSA)


SSA(6) SSA(18)



Research of subautomata•Smallest closed subautomaton (SCSA)•Smallest subautomaton (SSA)


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Research SCSA

Property 1.

(si, sf ) is a SCSA iff IS(sf)= si & FS(si)= sf

Property 2. (Associativity)

If E=E1E2 and E1 , E2 then

Source(E)= Source(Source(E1),Source(E2))

Property 3. (Hierarchy between two SCSA )• Either, they have no common transitions,• Either, one is strictly included in the other.

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Research SCSA

Let p Q1. P.IS : initial state associated to p.2. P.FSmin : minimal final state associated to p, assuming

that p is the initial state of a SCSA.3. P.FSmax : maximal final state associated to p, assuming

that p is the initial state of a SCSA.

Property 4.

p>qi, (p.IS,p) is a SCSA iff p.IS.FSmin p p.IS.FSmax

Complexity Algorithm : O (n2)

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Research SCSA

FSminis

FSmax

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Research SCSA

FSminis

FSmax

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Research SSA

Let A’=<, Q’, ’, si, sf > be a subautomaton

Property 5.E Q’ \ {sf}: Succ*(si)Pred*(E) Q’

E Q’ \ {si}: Pred*(sf)Succ*(E) Q’

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SSA associated to grey states

E

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Source

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SinkSource

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SinkSource

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Source Sink

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Research SSA

Property 6.

Let p, p’, q, q’ Q• {p, p’} Pred(q) and {q, q’} Succ(p)• H(p’) ≥ H(p) and H(q’) ≤ H(q)

p and q belong to the same SSA

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All Subautomata of an automaton

Algorithm input: A - output: subautomata

1: repeat2: repeat3: Detect, store and replace each parallels by one transition;4: Detect, store and replace each sequences by one transition;5: until the automaton is freed from all its parallels and sequences6: Detect, store and replace each smallest subautomata by one transition;7: until The automaton A is reduced to one single transition

Valdez J., Tarjan R. E., Lawler E. L., The recognition of series-parallel digraphs, SIAM J. Comput. 11-2:298-313, 1982.

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45


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Research of subautomata•Smallest closed subautomaton (SCSA)•Smallest subautomaton (SSA)


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Dictionaries and automata

10 dictionaries : Lexicographic order of words

• 6 Delaf : French, English, Serbian, German, Polylexicaux English, French cities.

• 4 Web : Frech, Hungarian, Bulgarian and Portuguese.

Properties of automata:Finit set of states, Acyclic, deterministic, unique initial

state, unique final state, minimal.

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Internal structure of automata

d

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Internal structure of automata

d

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Experimental Results

Finite state subautomataApplication to Electronic Dictionnaries



Application to automata representing dictionariesFactorisation, indexation and compression Conclusion

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Factorisation, indexation and compression

The reseach of subautomata detects sequences and parallels

Sequence subautomaton

Parallel subautomaton

Proposal: - The application of the direct acyclic word graph, initially dedicated for

indexing text, to index the subautomata,- heuristic to select the most interesting substructure to factorize.

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Storage of an automaton

c

c

d

d

1 1 a 8

2 0 c 3

3 1 a 5

4 0 b 6

5 1 b 7

6 1 c 10

7 1 c 9

8 1 b 11

9 1 d 0

10 1 d 11

11 1 b 0

Boolean Character

log2(|Σ|) Address arrival state

log2(Max address+1)

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Factorization

c

c

d

d

b

1 1 a 5

2 0 c 3

3 1 a 7

4 0 6

5 1 b 6

6 1 b 0

7 1 0

a cb

1 1 a 8

2 0 c 3

3 1 a 5

4 0 b 6

5 1 b 7

6 1 c 10

7 1 c 9

8 1 b 11

9 1 d 0

10 1 d 11

11 1 b 0

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Factorisation

b

Factorization

bb

Factorization

c

c

d

d

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How can we choose the subautomata to factorize ?

- The best candidates to be factorized are those which increase memory storage efficiency and reduce the size of the initial automaton

Profit = saved memory – Consumed memory

- The memory space is saved by elimination of all occurrences of the substructure

- The memory space is consumed by the extention of the alphabet and the index.

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Directed Acyclic word graph (DAWG)

Computations of frequency and profit associated to each sequence with a DAWG

DAWG (aabba)

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Greedy Algorithm of Compression

Algorithm input: A - Output: A, Alphabet

1: Iterative process 2: Select the best sequence s from the DAWG 3: Extend the alphabet to represent s4: Delete s from A and from DAWG5: Update the DAWG

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Compression FCM

FCM

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Compression FCNM

FCNM

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Compression FCDic

FCDic

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Best Compressions

1024

FCNMFCNM

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Best Compressions

1024

FCNMFCNM




Application to automata representing dictionariesFactorisation, indexation and compression Conclusion

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Conclusion

Research of two kinds of smallest subautomata

Statistical analysis of the internal structure of some automata associated to dictionnaries

Method of compression based on factorizations of sequences or parallel subautomata

A minimised automaton does not always lead to the better compression.

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Future works

Factorization of more kinds of subautomata,

Find a way to deminimised an automaton in order to get a better compression,

Work on alternative encoding of automata, for example a depth first codage

Finite state subautomata Application to Electronic Dictionaries Lamia Tounsi Polytech'Tours, Computer Science laboratory François Rabelais University.

Documents

q q q i

predq q q q

s f q q

q pred

q succp

alphabet q

initial state q i q

p q w ape