Unambiguous automata inference by means of states-merging methods

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Unambiguous automata inference by means of
states-merging methods

• François Coste, Daniel Fredouille
• fcostedfredoui_at_irisa.fr
• http//www.irisa.fr/symbiose
• IRISA-INRIA,
• Campus de Beaulieu
• 35042 Rennes Cedex
• France

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Definitions
I- Automata inference

• Alphabet ? a,b
• Word abbabbabbaaa ?
• Language L
• Automaton

?
Labaab
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Classes of automata (1/3)
I- Automata inference

• Nondeterministic Automata (NFA)
• Deterministic Automata (DFA)
• one outgoing transition per input symbol

Labaab
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Classes of automata (2/3)
I- Automata inference

• Unambiguous Automata (UFA) SH85
• one acceptance per word

Labaab

• NFA ? UFA ? DFA

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Automata inference
I- Automata inference
Examples
Counter-examples
Labaab
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Why this study ?
I- Automata inference

• State of the art DFA inference
• Our goal introducing some amount of
  non-determinism
• Why ?
• NFA ltlt DFA
• inferring with less data
• inferring explicit representations
• Method
• extending classical DFA inference algorithm

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II - Study of the DFA inference framework
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Search space for NFAs DMV94
II - The DFA search space
UA
MCA
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Counter-examples compatibility
II - The DFA search space
UA
(incompatible)
(compatible)
MCA
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The search space for DFA
II - The DFA search space
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Merging for determinisation procedure
II - The DFA search space
? q1,q2 ? Q, ? w ? ? w ? pref(q1) ? w ?
pref(q2) ? state-merging(q1,q2)
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Merging for determinization procedure
II - The DFA search space
? q1,q2 ? Q, ? w ? ? w ? pref(q1) ? w ?
pref(q2) ? state-merging(q1,q2)
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Merging for determinization procedure
II - The DFA search space
? q1,q2 ? Q, ? w ? ? w ? pref(q1) ? w ?
pref(q2) ? state-merging(q1,q2)
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Deterministic merging operator
state-merging merging for determinization
II - The DFA search space

• Very commonly used OG92, LPP98,...
• Demonstration of formal properties
• Merging for determinization
• Enables to reach the closest DFA from the
  original NFA
• Deterministic merging
• Enables to reach all derived DFA from a given DFA
• ... (see tech. rep.)

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IV - From DFA to UFA inference or how to
introduce some amount of non- determinism in
inference
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Inferring non-deterministic representations the
choice of UFA
III - DFA to UFA inference

• Why UFA ?
• unity in the search space (like DFA)

UA
NFA
UFA
DFA
MCA(aaaaa)
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Merging for disambiguisation procedure
III - DFA to UFA inference
? q1,q2 ? Q, ? w1,w2 ? ? w1 ? pref(q1) ? w1
? pref(q2) ? w2 ? suff(q1) ? w2 ? suff(q2) ?
state-merging(q1,q2)
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Merging for disambiguisation procedure
III - DFA to UFA inference
? q1,q2 ? Q, ? w1,w2 ? ? w1 ? pref(q1) ? w1
? pref(q2) ? w2 ? suff(q1) ? w2 ? suff(q2) ?
state-merging(q1,q2)
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Unambiguous merging state-merging merging
for disambiguisation
III - DFA to UFA inference

• Finer operator than merging for determinization
• Demonstration of formal properties
• Merging for disambiguisation
• Enables to reach the closest UFA from the
  original NFA
• unambiguous merging
• Enables to reach all derived UFA from a given UFA
• ... (see tech. rep.)

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IV - Comparative experiments - Inference
algorithms - Benchmarks - Experimental results
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Algorithms
IV - Comparative experiments

• UFA
• Hill-climbing heuristic
• DFA
• EDSM heuristic LPP98
• RFSA
• DeLeTe II DLT01

• Hill-climbing heuristic

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Counter-example use for DFA and UFA inference
IV - Comparative experiments

• Compatibility DMV94
• generalization of , stopped by
• Functionality AS95
• generalization of and , stopped by
  empty intersection

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Benchmarks
IV - Comparative experiments

• DLT01
• Generation DFA, NFA, Regular Expression
• 4 sizes of training sample
• 30 languages generated for each generation mode
  and sample size
• UFA generator
• Evaluation based on
• average recognition level on test sets
• matches between recognition level

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Results
IV - Comparative experiments

• Best algorithms w.r.t. benchmarks
• DFA bench UFA inference with hill-climbing
• UFA bench DFA inference with hill-climbing ?
  UFA inference with hill-climbing
• NFA bench RFSA inference
• Reg. Expr. RFSA inference ? DFA inference
  with hill-climbing

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Results
IV - Comparative experiments

• Heuristic
• Hill-climbing gtgt EDSM when inferring DFAs for
  NFA/Regular Expression/UFA bench.
• Counter-examples
• Compatibility ? Functionality

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IV - Comparative experiments
Results (matches)
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Conclusion

• UFA inference
• Merging for disambiguisation
• Heuristic
• Comparison with EDSM DeLeTe II

Perspectives

• Speeding up the algorithm
• Application
• Using properties of the DFA/UFA space

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References

• AS95 Alquézar, Sanfeliu, Incremental
  grammatical inference from positive and negative
  data using unbiased finite state automata,
  SSPR94
• DMV94 Dupont and al. What is the search space
  of the regular inference ?, ICGI 94
• DLT00 Denis and al., Learning regular
  languages using nondeterministic automata,
  ICGI 00
• SH85 Stearns, Hunt, On the equivalence and
  containment problems for unambiguous regular
  expressions, regular grammars and finite
  automata, SIAM vol 14
• tech. rep. Coste, Fredouille What is the
  search space for NFA, UFA and DFA inference ?,
  IRISA