Finite state subautomata Application to Electronic Dictionaries - PowerPoint PPT Presentation

Title: Finite state subautomata Application to Electronic Dictionaries


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Finite state subautomataApplication to
Electronic Dictionaries
Lamia Tounsi Polytech'Tours, Computer Science
laboratory François Rabelais University of
Tours, France Lamia.tounsi_at_univ-tours.fr
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Motivation

• 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

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Finite state subautomataApplication to
Electronic Dictionnaries

• Mathematical preliminaries
• Automaton
• Subautomaton
• Research of subautomata
• Smallest closed subautomaton
• Smallest subautomaton
• Application to automata representing dictionaries
• Indexation and Compression
• Conclusion

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Automaton

• A deterministic acyclic automaton A lt?, Q, ?,
  qi, qf gt
• ? 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) q?Q ????, ?(p,?) q
• Succ(p) q?Q ?w??, ?(p,w) q
• Pred(p) q?Q ????, ?(q,?) p
• Pred(p) q?Q ?w??, ?(q,w) p
• Height
• H(qf)0
• H(p)Maxq ?Succ(p) H(q)1

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Automaton
An automaton that recognizes the flexion of nine
verbs
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Source (E) Initial State (p)

• Let E ??

E
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Source (E) Initial State (p)

• Let E ??
• AP(E) w path from qi to p, p ? E

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Source (E) Initial State (p)

• Let E ??
• AP(E) w path from qi to p, p ? E
• AN(E)p ?Q/ ?w ?AP(E), p ?w

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Source (E) Initial State (p)

• Let E ??
• AP(E) w path from qi to p, p ? E
• AN(E)p ?Q/ ?w ?AP(E), p ?w
• source(E) ? AN (E)
• Source(E)
• H(source(E)) Minq?AN (E)(H(q))

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Source (E) Initial State (p)

• Let E ??
• AP(E) w path from qi to p, p ? E
• AN(E)p ?Q/ ?w ?AP(E), p ?w
• source(E) ? AN (E)
• source(E)
• H(source(E)) Minq?AN (E)(H(q))
• Let p ?Q, p ?qi
• IS(p) Source(Pred(p))

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Source (E) Initial State (p)

• 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)) Maxq?PN (E)(H(q))
• Let p ?Q, p ?qi
• FS(p) Sink(Succ(p))

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Subautomaton (SA)

• Alt?, Q, ?, si, sf gt 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
An automaton that recognizes the flexion of nine
verbs
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Subautomaton (SA)
SA
An automaton that recognizes the flexion of nine
verbs
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Subautomaton (SA)
SA
An automaton that recognizes the flexion of nine
verbs
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Subautomaton (SA)
An automaton that recognizes the flexion of nine
verbs
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Closed subautomaton (CSA)

• Let Q ? Q and si, sf two distinct states
• A subautomaton Alt?, Q, ?, si, sf gt is a
  closed subautomaton iff
• ?q ? Q \ si Pred(q) ? Q
• ?q ? Q \ sf Succ(q) ? Q

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Closed subautomaton (CSA)
?CSA
An automaton that recognizes the flexion of nine
verbs
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Closed subautomaton (CSA)
CSA
An automaton that recognizes the flexion of nine
verbs
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Closed subautomaton (CSA)
CSA
An automaton that recognizes the flexion of nine
verbs
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Smallest Closed subautomaton (SCSA)

• Let Q ? Q and si, sf two distinct states
• A closed subautomaton Alt?, Q, ?, si, sf gt
• 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)
?SCSA
SCSA
SCSA
SCSA
An automaton that recognizes the flexion of nine
verbs
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Smallest subautomaton (SSA)

• Let p ? Q \si, sf
• The subautomaton Alt?, Q, ?, si, sf gt
• is SSA(p) iff
• A strictly contains p
• ? Alt?, Q, ?, si, sf gt wich strictly
  contains p Q ? Q

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Smallest subautomaton (SSA)
SSA(6)
SSA(18)
An automaton that recognizes the flexion of nine
verbs
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Finite state subautomataApplication to
Electronic Dictionaries

• Mathematical preliminaries
• Automaton
• Subautomaton
• Research of subautomata
• Smallest closed subautomaton (SCSA)
• Smallest subautomaton (SSA)
• Application to automata representing dictionaries
• Indexation and Compression
• Conclusion

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Research SCSA

• Property 1.
• (si, sf ) is a SCSA iff IS(sf) si FS(si) sf
• Property 2. (Associativity)
• If EE1?E2 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 ? Q
• P.IS initial state associated to p.
• P.FSmin minimal final state associated to p,
  assuming that p is the initial state of a SCSA.
• P.FSmax maximal final state associated to p,
  assuming that p is the initial state of a SCSA.
• Property 4.
• ?pgtqi, (p.IS,p) is a SCSA iff p.IS.FSmin ?p?
  p.IS.FSmax
• Complexity Algorithm O (n2)

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Research SCSA
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Research SCSA
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Research SSA

• Let Alt?, Q, ?, si, sf gt 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
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SSA associated to grey states
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SSA associated to grey states
Source
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SSA associated to grey states
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SSA associated to grey states
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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 repeat
• 2 repeat
• 3 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 sequences
• 6 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. 11-2298-313, 1982.
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All Subautomata of an automaton
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All Subautomata of an automaton
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All Subautomata of an automaton
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All Subautomata of an automaton
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All Subautomata of an automaton
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All Subautomata of an automaton
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Finite state subautomataApplication to
Electronic Dictionaries

• Mathematical preliminaries
• Automaton
• Subautomaton
• Research of subautomata
• Smallest closed subautomaton (SCSA)
• Smallest subautomaton (SSA)
• Application to automata representing dictionaries
• Indexation and Compression
• Conclusion

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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
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Internal structure of automata
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Experimental Results
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Finite state subautomataApplication to
Electronic Dictionnaries

• Mathematical preliminaries
• Automaton
• Subautomaton
• Research of subautomata
• Smallest closed subautomaton
• Smallest subautomaton
• Application to automata representing dictionaries
• Factorisation, 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
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Factorization
a
b
c
?
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Factorisation
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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
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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 s
• 4 Delete s from A and from DAWG
• 5 Update the DAWG

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Compression FCM
FCM
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Compression FCNM
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Compression FCDic
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Best Compressions
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Best Compressions
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Finite state subautomataApplication to
Electronic Dictionaries

• Mathematical preliminaries
• Automaton
• Subautomaton
• Research of subautomata
• Smallest closed subautomaton
• Smallest subautomaton
• Application to automata representing dictionaries
• Factorisation, 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