Sandro Spina, John Abela

1
Mutually compatible and incompatible merges for
the search of the smallest consistent DFA

• Sandro Spina, John Abela
• Department of CS AI,
• University of Malta.

Francois Coste INRIA/IRISA, Campus de Beaulieu,
35042 Rennes Cedex, France
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Evidence Driven State Merging

• The motivation behind our work was that of
  improving the (greedy) heuristic used by EDSM.
  Work was also carried out on diversification of
  the search strategy.
• EDSM Price98 is very effective at inferring
  regular languages, except when training data is
  sparse.
• According to Price98, Abbadingo style problems
  can be solved with high confidence (0.93) when
  the number of matched state labels is greater
  than 10.
• EDSM determines its merge sequence (greedily) by
  using a heuristic which compares language
  suffixes between two states in a DFA.
• Three Complementary Tracks
• Improve on Heuristic Score.
• Improve on Search Strategy.
• Combine these two.

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Sharing Evidence

• Q. Whenever EDSM does not correctly infer the
  target language, can we (using a greedy
  depth-first search) improve the learners merge
  path by gathering and combining information
  (state label matches) from multiple valid merges?
  Does the combination of their evidence scores
  result in valuable information? Can this
  information be used to guide the search?
• We think so !!! Some of the initial results are
  encouraging.
• Target Size Convergence Improves Drastically
• Classification Rate Does Not Improve Consistently
• EDSM score ? focuses on single merge analysis
• S-EDSM score ? score is a combination of single
  merge analysis

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Pairwise Compatible Merges

• Let M be the set of all possible merges.
• A merge M ltq1,q2gt, is said to be valid if all the
  states in the subtree of q1 are state compatible
  with the corresponding states in the subtree of
  q2.
• Let M1, M2 2 M be two valid merges
• We define the relation ? µ M X M as follows
• M1?M2 if M2 remains a valid merge in the
  hypothesis obtained by applying M1
• If M1?M2 , we say that M1 is pairwise compatible
  to M2

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Pairwise Compatible Merges (Simple) Example
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Mutual Compatible Merges

• Suppose that M1, M2 and M3 2 M where M1 ? M2
  and M2 ? M3
• This does not necessarily imply that M1?M3. This
  is because some states in M2 can be labelled
  differently by M1 and M3.
• Therefore ? is not transitive.
• In order to make ? a transitive relation (
  denoted as ? ), M1 ? M3 needs to be checked as
  well to create the set M1, M2, M3
• Set cardinality of mutual compatible merges can
  direct S-EDSM s heuristic score. This is
  currently not implemented.

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S-EDSM Algorithm
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Initial Results

• Shared Evidence Driven State Merging (S-EDSM)
  implements only pairwise compatibility by
  creating classes of M1 ? M2 Mn for the top
  30 valid merges. Scores are recalculated and the
  best merge is determined and executed. Various
  strategies can be implemented.
• In terms of classification rate we are still not
  consistently performing better than classic EDSM.
• S-EDSM approximates better the target size of the
  target automaton. However this improvement does
  NOT help on its own. Its only (possibly) an
  indication of a direction to follow.

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Results II ( 400 State Target Size Convergence )

• This graph documents 10 consecutive problems
  downloaded from
• Gowachin. Training set consisted of 20,000
  strings.

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Results III (256 State Target Size Classification
)
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Pairwise Incompatible Merges for Search
Classical Search Tree

?
m1?M

?

?
m2?M
m3?M


?
m5?M
m4?M





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Pairwise Incompatible Merges for Search
Candidates limitation after backtrack

?
m1?M

?

m2?M
m3?M?I(m1)
?



m5?M?I(m2)

?
?
m4?M



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Pairwise Incompatible Merges for Search

• Rationale
• A merge m ? I(m) may be tried after m.
• Introduces diversity in the search
• Edsm I(m) may be computed Coste
  Fredouille,ICGI00
• S-Edsm I(m) is available for free
• Significant improvement when applied to the 3
  first choices.
• Best application of scheme after the choice
  m3?M?I(m1) ?
• After merging m3 ()
• After not merging m3 (?)

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Future Directions

• Develop a calculus to describe merge
  interactions. Implement all the relations and
  functions ( mutual compatibility, dominance, etc.
  ) of the calculus. Analyse the results achieved
  from these different implementations.
• Combine heuristic with better search strategies
  and study the best combination of heuristic and
  search strategy. Introduction of diversity in the
  exploration of the search space by limiting
  choice of candidate merges after a backtrack.
• Noisy Data !! Can S-EDSM perform better by
  combining information between different merges.
  Maybe with information gathered from merge
  interactions S-EDSM can discover noise in the
  training set.
• Ultimately we want to see how far we can push, in
  terms of data sparseness, DFA learning.
• Thank you.