1 erice 2005, the analysis of patterns. grammatical inference 1 colin de la higuera grammatical...

1Erice 2005, the Analysis of Patterns. Grammatical Inference

1

Colin de la Higuera

Grammatical inference: techniques and

algorithms


2

Acknowledgements

• Laurent Miclet, Tim Oates, Jose Oncina, Rafael Carrasco, Paco Casacuberta, Pedro Cruz, Rémi Eyraud, Philippe Ezequel, Henning Fernau, Jean-Christophe Janodet, Thierry Murgue, Frédéric Tantini, Franck Thollard, Enrique Vidal,...

• … and a lot of other people to whom I am grateful


3

Outline

1 An introductory example2 About grammatical inference3 Some specificities of the task4 Some techniques and algorithms5 Open issues and questions


4

1 How do we learn languages?

A very simple example


5

The problem:

• You are in an unknown city and have to eat.

• You therefore go to some selected restaurants.

• Your goal is therefore to build a model of the city (a map).


6

The data

• Up Down Right Left Left Restaurant

• Down Down Right Not a restaurant

• Left Down Restaurant


7

Hopefully something like this:

N

N

R

u

u

d

r d

d,l

u,r

R


8

R

R

R

R

R

N

N

N

N

d

d

r d

N

d

u

u

u u

d

u

u

d


9

Further arguments (1)

• How did we get hold of the data?– Random walks– Following someone

•someone knowledgeable•Someone trying to lose us•Someone on a diet

– Exploring


10


• Can we not have better information (for example the names of the restaurants)?

• But then we may only have the information about the routes to restaurants (not to the “non restaurants”)…


11


What if instead of getting the information “Elimo” or “restaurant”, I get the information “good meal” or “7/10”?

Reinforcement learning: POMDP


12


• Where is my algorithm to learn these things?

• Perhaps should I consider several algorithms for the different types of data?


13


• What can I say about the result?

• What can I say about the algorithm?


14


• What if I want something richer than an automaton?– A context-free grammar– A transducer– A tree automaton…


15


• Why do I want something as rich as an automaton?

• What about– A simple pattern?– Some SVM obtained from features over the strings?

– A neural network that would allow me to know if some path will bring me or not to a restaurant, with high probability?


16

Our goal/idea

• Old Greeks:A whole is more than the sum of all parts

• Gestalt theoryA whole is different than the sum of all parts


17

Better said

• There are cases where the data cannot be analyzed by considering it in bits

• There are cases where intelligibility of the pattern is important


18

Nothing Lots

What do people know about formal language theory?


19

A small reminder on formal language theory

• Chomsky hierarchy• + and – of grammars


20

A crash course in Formal language theory

• Symbols• Strings• Languages• Chomsky hierarchy• Stochastic languages


21

Symbols

are taken from some alphabet

Stringsare sequences of symbols from


22

Languages

are sets of strings over

Languagesare subsets of *


23

Special languages

• Are recognised by finite state automata

• Are generated by grammars


24

a

b

a

b

a

b

DFA: Deterministic Finite State Automaton


25

a

b

a

b

a

b

ababL


26

What is a context free grammar?

A 4-tuple (Σ, S, V, P) such that:– Σ is the alphabet;– V is a finite set of non terminals;

– S is the start symbol;– P V (VΣ)* is a finite set of rules.


27

Example of a grammar

The Dyck1 grammar– (Σ, S, V, P)– Σ = {a, b}– V = {S}– P = {S aSbS, S }


28

Derivations and derivation trees

S aSbS aaSbSbS aabSbS aabbS aabb

a

a

b

b

S

SS

S

S


29

Chomsky Hierarchy

• Level 0: no restriction• Level 1: context-sensitive• Level 2: context-free• Level 3: regular


30

Chomsky Hierarchy• Level 0: Whatever Turing machines can do

• Level 1: – {anbncn: n }– {anbmcndm : n,m }– {uu: u*}

• Level 2: context-free– {anbn: n }– brackets

• Level 3: regular– Regular expressions (GREP)


31

The membership problem

• Level 0: undecidable• Level 1: decidable• Level 2: polynomial • Level 3: linear


32

The equivalence problem

• Level 0: undecidable• Level 1: undecidable• Level 2: undecidable• Level 3: Polynomial only when the representation is DFA.


33

4

1

3

1

2

1

2

1

2

13

2

b

b

a

a

a

b

4

3

2

1

PFA: Probabilistic Finite (state) Automaton


34

0.1

0.3

a

b

a

b

a

b

0.65

0.350.9

0.7

0.3

0.7

DPFA: Deterministic Probabilistic Finite (state) Automaton


35

What is nice with grammars?

• Compact representation• Recursivity• Says how a string belongs, not just if it belongs

• Graphical representations (automata, parse trees)


36

What is not so nice with grammars?

• Even the easiest class (level 3) contains SAT, Boolean functions, parity functions…

• Noise is very harmful:– Think about putting edit noise to language {w: |w|a=0[2]|w|b=0[2]}


37

2 Specificities of grammatical

inferenceGrammatical inference consists (roughly) in finding the (a) grammar or automaton that has produced a given set of strings (sequences, trees, terms, graphs).


38

Inductive Inference Pattern Recognition

Grammatical Inference

The field

Machine Learning

Computational linguistics Computational biology Web technologies


39

The data

• Strings, trees, terms, graphs• Structural objects• Basically the same gap of information as in programming between tables/arrays and data structures


40

Alternatives to grammatical inference

• 2 steps:– Extract features from the strings

– Use a very good method over n.


41

Examples of strings

A string in Gaelic and its translation to English:

• Tha thu cho duaichnidh ri èarr àirde de a’ coisich deas damh

•You are as ugly as the north end of a southward traveling ox


42


43


44

>A BAC=41M14 LIBRARY=CITB_978_SKBAAGCTTATTCAATAGTTTATTAAACAGCTTCTTAAATAGGATATAAGGCAGTGCCATGTAGTGGATAAAAGTAATAATCATTATAATATTAAGAACTAATACATACTGAACACTTTCAATGGCACTTTACATGCACGGTCCCTTTAATCCTGAAAAAATGCTATTGCCATCTTTATTTCAGAGACCAGGGTGCTAAGGCTTGAGAGTGAAGCCACTTTCCCCAAGCTCACACAGCAAAGACACGGGGACACCAGGACTCCATCTACTGCAGGTTGTCTGACTGGGAACCCCCATGCACCTGGCAGGTGACAGAAATAGGAGGCATGTGCTGGGTTTGGAAGAGACACCTGGTGGGAGAGGGCCCTGTGGAGCCAGATGGGGCTGAAAACAAATGTTGAATGCAAGAAAAGTCGAGTTCCAGGGGCATTACATGCAGCAGGATATGCTTTTTAGAAAAAGTCCAAAAACACTAAACTTCAACAATATGTTCTTTTGGCTTGCATTTGTGTATAACCGTAATTAAAAAGCAAGGGGACAACACACAGTAGATTCAGGATAGGGGTCCCCTCTAGAAAGAAGGAGAAGGGGCAGGAGACAGGATGGGGAGGAGCACATAAGTAGATGTAAATTGCTGCTAATTTTTCTAGTCCTTGGTTTGAATGATAGGTTCATCAAGGGTCCATTACAAAAACATGTGTTAAGTTTTTTAAAAATATAATAAAGGAGCCAGGTGTAGTTTGTCTTGAACCACAGTTATGAAAAAAATTCCAACTTTGTGCATCCAAGGACCAGATTTTTTTTAAAATAAAGGATAAAAGGAATAAGAAATGAACAGCCAAGTATTCACTATCAAATTTGAGGAATAATAGCCTGGCCAACATGGTGAAACTCCATCTCTACTAAAAATACAAAAATTAGCCAGGTGTGGTGGCTCATGCCTGTAGTCCCAGCTACTTGCGAGGCTGAGGCAGGCTGAGAATCTCTTGAACCCAGGAAGTAGAGGTTGCAGTAGGCCAAGATGGCGCCACTGCACTCCAGCCTGGGTGACAGAGCAAGACCCTATGTCCAAAAAAAAAAAAAAAAAAAAGGAAAAGAAAAAGAAAGAAAACAGTGTATATATAGTATATAGCTGAAGCTCCCTGTGTACCCATCCCCAATTCCATTTCCCTTTTTTGTCCCAGAGAACACCCCATTCCTGACTAGTGTTTTATGTTCCTTTGCTTCTCTTTTTAAAAACTTCAATGCACACATATGCATCCATGAACAACAGATAGTGGTTTTTGCATGACCTGAAACATTAATGAAATTGTATGATTCTAT


45


46


47


48

<book> <part> <chapter> <sect1/> <sect1> <orderedlist numeration="arabic"> <listitem/> <f:fragbody/> </orderedlist> </sect1> </chapter> </part> </book>


49

<?xml version="1.0"?><?xml-stylesheet href="carmen.xsl" type="text/xsl"?><?cocoon-process type="xslt"?><!DOCTYPE pagina [<!ELEMENT pagina (titulus?, poema)><!ELEMENT titulus (#PCDATA)><!ELEMENT auctor (praenomen, cognomen, nomen)><!ELEMENT praenomen (#PCDATA)><!ELEMENT nomen (#PCDATA)><!ELEMENT cognomen (#PCDATA)><!ELEMENT poema (versus+)><!ELEMENT versus (#PCDATA)>]><pagina><titulus>Catullus II</titulus><auctor><praenomen>Gaius</praenomen><nomen>Valerius</nomen><cognomen>Catullus</cognomen></auctor>


50


51

A logic program learned by GIFT

color_blind(Arg1) :- start(Arg1,X),p11(Arg1,X).

start(X,X). p11(Arg1,P) :- mother(M,P),p4(Arg1, M). p4(Arg1,X) :- woman(X),father(F,X),p3(Arg1,F). p4(Arg1,X) :- woman(X),mother(M,X),p4(Arg1,M). p3(Arg1,X) :- man(X),color_blind(X).


52

3 Hardness of the task

– One thing is to build algorithms, another is to be able to state that it works.

– Some questions:– Does this algorithm work?– Do I have enough learning data?– Do I need some extra bias?– Is this algorithm better than the other?

– Is this problem easier than the other?


53

Alternatives to answer these questions:

– Use well admitted benchmarks– Build your own benchmarks– Solve a real problem– Prove things


54

Use well admitted benchmarks

• yes: allows to compare

• no: many parameters

• problem: difficult to better

(also, in GI, not that many

about!)


55

Build your own benchmarks

• yes: allows to progress

• no: against one-self

• problem: one invents the

benchmark where one is best!


56

Solve a real problem

• yes: it is the final goal

• no: we don’t always know why

things work

• problem: how much pre-

processing?


57

Theory

• Because you may want to be able to say something more than « seems to work in practice ».


58

Identification in the limit

L Pres XA class of languages

A class of grammars

G

L A learnerThe naming function

yields

f()=g() yields(f)=yields(g)L((f))=yields(f)


59

f1 f2

h1 h2

fn

hn

fi

hi hn

L(hi)= L

L is identifiable in the limit in terms of G from Pres iff

LL, f Pres(L)


60

No quería componer otro Quijote —lo cual es fácil— sino el Quijote. Inútil agregar que no encaró nunca una transcripción mecánica del original; no se proponía copiarlo. Su admirable ambición era producir unas páginas que coincidieran palabra por palabra y línea por línea con las de Miguel de Cervantes.

[…]

“Mi empresa no es difícil, esencialmente” leo en otro lugar de la carta. “Me bastaría ser inmortal para llevarla a cabo.”

Jorge Luis Borges(1899–1986)Pierre Menard, autor del Quijote (El jardín de senderos que se

bifurcan) Ficciones


61

4 Algorithmic ideas


62

The space of GI problems

• Type of input (strings)• Presentation of input (batch)• Hypothesis space (subset of the regular grammars)

• Success criteria (identification in the limit)


63

Types of input

the cat hates the dogStrings:

StructuralExamples:

cat dog the the hates

(+)

(-)

Graphs:


64

Types of input - oracles

• Membership queries– Is string S in the target language?

• Equivalence queries– Is my hypothesis correct?– If not, provide counter example

• Subset queries– Is the language of my hypothesis a subset of the target language?


65

Presentation of input

• Arbitrary order• Shortest to longest• All positive and negative examples up to some length

• Sampled according to some probability distribution


66

Presentation of input

• Text presentation– A presentation of all strings in the target language

• Complete presentation (informant)– A presentation of all strings over the alphabet of the target language labeled as + or -


67

Hypothesis space

• Regular grammars– A welter of subclasses

• Context free grammars– Fewer subclasses

• Hyper-edge replacement graph grammars


68

Success criteria

• Identification in the limit– Text or informant presentation– After each example, learner guesses language

– At some point, guess is correct and never changes

• PAC learning


69

Theorem’s due to Gold• The good news

– Any recursively enumerable class of languages can be learned in the limit from an informant (Gold, 1967)

• The bad news– A language class is superfinite if it includes all finite languages and at least one infinite language

– No superfinite class of languages can be learned in the limit from a text (Gold, 1967)

– That includes regular and context-free


70

A picture

Little information

A lot of information

Poor languages Rich Languages

Sub-classes of reg, from pos

Mildly context sensitive, from queries

DFA, from queries

Context-free, from pos

DFA, from pos+neg


71

Algorithms

RPNI

K-Reversible

GRIDS

SEQUITUR

L*


72

4.1 RPNI

• Regular Positive and Negative Grammatical Inference

Identifying regular languages in polynomial time

Jose Oncina & Pedro García 1992


73

• It is a state merging algorithm;

• It identifies any regular language in the limit;

• It works in polynomial time;• It admits polynomial charac-teristic sets.


74

The algorithm

function rmerge(A,p,q)A = merge(A,p,q)while a, p,qA(r,a), pq do

rmerge(A,p,q)


75

A=PTA(X); Fr ={(q0,a): a };

K ={q0};

While Fr dochoose q from Frif pK: L(rmerge(A,p,q))X-=

then A = rmerge(A,p,q) else K = K {q}Fr = {(q,a): qK} – {K}


76

X+={, aaa, aaba, ababa, bb, bbaaa}

a

a

aa

b

b

b

a

a

a

ba b

a

X-={aa, ab, aaaa, ba}

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15


77

Try to merge 2 and 1

a

a

aa

b

b

b

a

a

a

ba b

a


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15


78

Needs more merging for determinization

aa

aa

b

b

b

a

a

a

b a ba


1,2

3

4

5

6

7

8

9

10

11

12

13

14

15


79

But now string aaaa is accepted, so the merge must be rejected

a

b

b a

a

a

ab

a


1,2,4,7

3,5,86

9, 11

10

12

13

14

15


80

Try to merge 3 and 1

a

a

aa

b

b

b

a

a

a

ba b

a


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15


81

Requires to merge 6 with {1,3}

a

a

aa

b

b

a

a

a

ba b

a


1,3

2

4

5

6

7

8

9

10

11

12

13

14

15

b


82

And now to merge 2 with 10

a

a

aa

b

aa

a

ba b

a


1,3,6

2

4

5

7

8

9

10

11

12

13

14

15

b


83

And now to merge 4 with 13

a

a

aa

b

a

ba b

a


1,3,6

2,104

5

7

8

9

11

12

13

14

15

b a


84

And finally to merge 7 with 15

a

a

aa

b

a

ba b

a


1,3,6

2,10

4,13

5

7

8

9

11

12

14

15

b


85

No counter example is accepted so the merges are kept

a

a

aa

bb

a ba


1,3,6

2,10

4,13

5

7,15

8

9

11

12

14

b


86

Next possible merge to be checked is {4,13} with {1,3,6}

a

a

aa

bb

a ba


1,3,6

2,10

4,13

5

7,15

8

9

11

12

14

b


87

a

a

ab

ba b

a


1,3,4,6,13

2,10

5

7,15

8

9

11

12

14

b

a

More merging for determinization is needed


88

a ba b

a


1,3,4,6,8,13

2,7,10,11,15

5

9

12

14

b

a

But now aa is accepted


89

So we try {4,13} with {2,10}

a

a

aa

bb

a ba


1,3,6

2,10

4,13

5

7,15

8

9

11

12

14

b


90

After determinizing, negative string aa is again accepted

a ba b

a


1,3,62,4,7,10,13,15

5,8

9,11 12

14

b

a


91

So we try 5 with {1,3,6}

a

a

aa

bb

a ba


1,3,6

2,10

4,13

5

7,15

8

9

11

12

14

b


92

But again we accept ab

aa

aa

b

b


1,3,5,6,12

2,9,10,14

4,13

7,15

8

11

b


93

So we try 5 with {2,10}

a

a

aa

bb

a ba


1,3,6

2,10

4,13

5

7,15

8

9

11

12

14

b


94

Which is OK. So next possible merge is {7,15} with {1,3,6}

a

a

a

ab

b


1,3,6

2,5,10

4,9,13

7,15

8,12

11,14

b


95

Which is OK. Now try to merge {8,12} with {1,3,6,7,15}

a a

a

ab

a


1,3,6,7,15

2,5,10

4,9,13

8,12

11,14

b

b


96

And ab is accepted

a

a

b

a


1,3,6,7,8,12,15

2,5,10,11,14

4,9,13

b

b


97

Now try to merge {8,12} with {4,9,13}

a a

a

ab

a


1,3,6,7,15

2,5,10

4,9,13

8,12

11,14

b

b


98

This is OK and no more merge is possible so the algorithm halts.

a a

a

b

a


1,3,6,7,11,14,15

2,5,10

4,8,9,12,13

b

b


99

Definitions

• Let be the length-lex ordering over *

• Let Pref(L) be the set of all prefixes of strings in some language L.

100

Erice 2005, the Analysis of Patterns. Grammatical Inference

100

Short prefixes

Sp(L)={uPref(L): (q0,u)=(q0,v) uv}

• There is one short prefix per useful state

0

1 2a

b

a

b b

aSp(L)={, a}

101


101

Kernel-sets

• N(L)={uaPref(L): uSp(L)}{}• There is an element in the Kernel-set for each useful transition

0

1 2a

b

a

b b

aN(L)={, a, b, ab}

102


102

A characteristic sample

• A sample is characteristic (for RPNI) if xSp(L) xuX+

xSp(L), yN(L), (q0,x)(q0,y)

z*: xzX+yzX-

xzX-yzX+

103


103

About characteristic samples

• If you add more strings to a characteristic sample it still is characteristic;

• There can be many different characteristic samples;

• Change the ordering (or the exploring function in RPNI) and the characteristic sample will change.

104


104

Conclusion

• RPNI identifies any regular language in the limit;

• RPNI works in polynomial time.

Complexity is in O(║X+║3.║X-║);

• There are many significant variants of RPNI;

• RPNI can be extended to other classes of grammars.

105


105

Open problems

• RPNI’s complexity is not a tight upper bound. Find the correct complexity.

• The definition of the characteristic set is not tight either. Find a better definition.

106


106

Algorithms

RPNI

K-Reversible

GRIDS

SEQUITUR

L*

107


107

4.2 The k-reversible languages

• The class was proposed by Angluin (1982).

• The class is identifiable in the limit from text.

• The class is composed by regular languages that can be accepted by a DFA such that its reverse is deterministic with a look-ahead of k.

108


108

Let A=(, Q, , I, F) be a NFA, we denote by AT=(, Q, T, F, I) the reversal automaton with:

T(q,a)={q’Q: q(q’,a)}

109


109

0 1

3

b2

4

a

ba

a a a

0 1

3

b2

4

a

ba

a a a

A

AT

110


110

Some definitions

• u is a k-successor of q if │u│=k and (q,u).

• u is a k-predecessor of q if │u│=k and T(q,uT).

is 0-successor and 0-predecessor of any state.

111


111

0 1

3

b2

4b

a

a a a

A

• aa is a 2-successor of 0 and 1 but not of 3.

• a is a 1-successor of 3.• aa is a 2-predecessor of 3 but not of 1.

a

112


112

A NFA is deterministic with look-ahead k iff q,q’Q: qq’(q,q’I) (q,q’(q”,a))

(u is a k-successor of q) (v is a k-successor of q’) uv

113


113

Prohibited:

2

1

a

a

u

u

│u│=k

114


114

Example

This automaton is not deterministic with look-ahead 1 but is deterministic with look-ahead 2.

0 1

3

b2

4

a

ba

a a a

115


115

K-reversible automata• A is k-reversible if A is deterministic and AT is deterministic with look-ahead k.

• Example

0 1

b

2ba

a

b

0 1

b

2ba

a

bdeterministic deterministic with look-ahead 1

116


116

Violation of k-reversibility• Two states q, q’ violate the k-reversibility condition iff– they violate the deterministic condition: q,q’(q”,a);

or– they violate the look-ahead condition: •q,q’F, uk: u is k-predecessor of both;

uk, (q,a)=(q’,a) and u is k-predecessor of both q and q’.

117


117

Learning k-reversible automata

• Key idea: the order in which the merges are performed does not matter!

• Just merge states that do not comply with the conditions for k-reversibility.

118


118

K-RL Algorithm (k-RL)

Data: k, X sample of a k-RL LA=PTA(X)While q,q’ k-reversibility violators do

A=merge(A,q,q’)

119


119

Let X={a, aa, abba, abbbba}

a

ab abb

aa

abbbbabbb abbbba

abba

a

b b b b a

a

a

k=2

Violators, for u= ba

120


120


a

ab abb

aa

abbbbabbb

abba

a

b b b b

a

a

a

k=2

Violators, for u= bb

121


121


a

ab abb

aa

abbb

abbaa

b b b

b

a

a

k=2

122


122

Properties (1)k0, X, k-RL(X) is a k-reversible language.

• L(k-RL(X)) is the smallest k-reversible language that contains X.

• The class Lk-RL is identifiable in the limit from text.

123


123

Properties (2)

• Any regular language is k-reversible iff (u1v)-1L (u2v)-1L and │v│=k

(u1v)-1L=(u2v)-1L

(if two strings are prefixes of a string of length at least k, then the

strings are Nerode-equivalent)

124


124

Properties (3)

• Lk-RL(X) L(k+1)-RL(X)

• Lk-TSS(X) L(k-1)-RL(X)

125


125

Properties (4)

The time complexity is O(k║X║3).

The space complexity is O(║X║).

The algorithm is not incremental.

126


126

Properties (4) Polynomial aspects

• Polynomial characteristic sets• Polynomial update time• But not necessarily a polynomial number of mind changes

127


127

Extensions

• Sakakibara built an extension for context-free grammars whose tree language is k-reversible

• Marion & Besombes propose an extension to tree languages.

• Different authors propose to learn these automata and then estimate the probabilities as an alternative to learning stochastic automata.

128


128

Exercises

• Construct a language L that is not k-reversible, k0.

• Prove that the class of k-reversible languages is not in TxtEx.

• Run k-RL on X={aa, aba, abb, abaaba, baaba} for k=0,1,2,3

129


129

Solution (idea)

• Lk={ai: ik}

• Then for each k: Lk is k-reversible but not k-1-reversible.

• And ULk = a*

• So there is an accumulation point…

130


130

Algorithms

RPNI

K-Reversible

GRIDS

SEQUITUR

L*

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4.4 Active Learning:

learning DFA from membership and

equivalence queries: the L* algorithm

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The classes C and H

• sets of examples• representations of these sets• the computation of L(x) (and h(x)) must take place in time polynomial in x.

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Correct learning

A class C is identifiable with a polynomial number of queries of type T if there exists an algorithm that:

1) LC identifies L with a polynomial number of queries of type T;

2) does each update in time polynomial in f and in xi, {xi} counter-examples seen so far.

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Algorithm L*

• Angluin’s papers• Some talks by Rivest• Kearns and Vazirani• Balcazar, Diaz, Gavaldà & Watanabe

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Some references

• Learning regular sets from queries and counter-examples, D. Angluin, Information and computation, 75, 87-106, 1987.

• Queries and Concept learning, D. Angluin, Machine Learning, 2, 319-342, 1988.

• Negative results for Equivalence Queries, D. Angluin, Machine Learning, 5, 121-150, 1990.

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The Minimal Adequate Teacher

• You are allowed:– strong equivalence queries;– membership queries.

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General idea of L*

• find a consistent table (representing a DFA);

• submit it as an equivalence query;• use counterexample to update the table;

• submit membership queries to make the table complete;

• Iterate.

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An observation table

a

a

abaab

1 000

010001

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The states (S) or test set

The transitions (T)

The experiments (E)

a

a

abaab

1 000

010001

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Meaning

(q0, . )F

L

a

a

abaab

1 000

010001

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141

(q0, ab.a) F

aba L

a

a

abaab

1 000

010001

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Equivalent prefixes

These two rows are equal,

hence

(q0,)= (q0,ab)

a

a

abaab

1 000

010001

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Building a DFA from a table

a

a

abaab

1 000

010001

aa

144


144

a

a

abaab

1 000

010001

aa

b

a

b

145


145

a

a

abaab

1 000

010001

aa

b

a

b

Some rules

This set is prefix-closed

This set is suffix-closed

S\S=T

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An incomplete table

a

a

abaab

1 00

01001

aa

b

a

b

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147

Good idea

We can complete the table by making membership queries...

u

v

?uvL ?

Membership query:

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A table is

closed if any row of T corresponds to some row in S

a

a

abaab

1 000

011001

Not closed

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149

And a table that is not closed

a

a

abaab

1 000

011001

aa

b

a

b

?

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What do we do when we have a table that is not

closed?

• Let s be the row (of T) that does not appear in S.

• Add s to S, and a sa to T.

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An inconsistent table

a

abaa

1 0ab

00000101

bbba 01

00

Are a and b equivalent?

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A table is consistent if

Every equivalent pair of rows in H remains equivalent in S after appending any symbol

row(s1)=row(s2)

a, row(s1a)=row(s2a)

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What do we do when we have an inconsistent

table?Let a be such that row(s1)=row(s2) but row(s1a)row(s2a)

• If row(s1a)row(s2a), it is so for experiment e

• Then add experiment ae to the table

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What do we do when we have a closed and consistent table ?

• We build the corresponding DFA

• We make an equivalence query!!!

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What do we do if we get a counter-example?

• Let u be this counter-example

wPref(u) do– add w to S a, such that waPref(u) add wa to T

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Run of the algorithm

a

b

1

1

1 Table is now closed

and consistent

b

a

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An equivalence query is made!

b

a

Counter example baa is returned

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a

b1

1

0baaba

baaa

bbbab

baab

1

01

1

11

Not consistent

Because of

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a

a

b1 1

1

0 0

baaba

baaa

bbbab

baab

1 1

0 1

1 0

1 1

Table is now closed and consistent

ba

baa

a

b

a

b b

a

0

0 0

1 1

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Proof of the algorithm

Sketch only

Understanding the proof is important for further algorithms

Balcazar et al. is a good place for that.

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Termination / Correctness

• For every regular language there is a unique minimal DFA that recognizes it.

• Given a closed and consistent table, one can generate a consistent DFA.

• A DFA consistent with a table has at least as many states as different rows in S.

• If the algorithm has built a table with n different rows in S, then it is the target.

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Finiteness

• Each closure failure adds one different row to S.

• Each inconsistency failure adds one experiment, which also creates a new row in S.

• Each counterexample adds one different row to S.

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Polynomial

• |E| n• at most n-1 equivalence queries• |membership queries| n(n-1)m where m is the length of the longest counter-example returned by the oracle

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Conclusion• With an MAT you can learn DFA

– but also a variety of other classes of grammars;

– it is difficult to see how powerful is really an MAT;

– probably as much as PAC learning.– Easy to find a class, a set of queries and provide and algorithm that learns with them;

– more difficult for it to be meaningful.

• Discussion: why are these queries meaningful?

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Algorithms

RPNI

K-Reversible

GRIDS

SEQUITUR

L*

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4.5 SEQUITUR

(http://sequence.rutgers.edu/sequitur/)(Neville Manning & Witten, 97)Idea: construct a CF grammar from a very long string w, such that L(G)={w}– No generalization– Linear time (+/-)– Good compression rates

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Principle

The grammar with respect to the string:– Each rule has to be used at least twice;

– There can be no sub-string of length 2 that appears twice.

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Examples

Sabcdbc

SAbAabA aa

S aAdAA bc

Saabaaab

SAaAA aab

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169

abcabdabcabd

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In the beginning, God created the heavens and the earth.

And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.

And God said, Let there be light: and there was light.

And God saw the light, that it was good: and God divided the light from the darkness.

And God called the light Day, and the darkness he called Night. And the evening and the morning were the first day.

And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters.

And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament: and it was so.

And God called the firmament Heaven. And the evening and the morning were the second day.

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• appending a symbol to rule S; • using an existing rule; • creating a new rule; • and deleting a rule.

Sequitur options

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Results

On text:– 2.82 bpc– compress 3.46 bpc– gzip 3.25 bpc– PPMC 2.52 bpc

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Algorithms

RPNI

K-Reversible

GRIDS

SEQUITUR

L*

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4.6 Using a simplicity bias

(Langley & Stromsten, 00)Based on algorithm GRIDS

(Wolff, 82)

Main characteristics:– MDL principle;– Not characterizable;– Not tested on large benchmarks.

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Two learning operatorsCreation of non terminals and rules

NP ART ADJ NOUNNP ART ADJ ADJ NOUN

NP ART AP1NP ART ADJ AP1AP1 ADJ NOUN

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Merging two non terminals

NP ART AP1NP ART AP2AP1 ADJ NOUNAP2 ADJ AP1

NP ART AP1AP1 ADJ NOUNAP1 ADJ AP1

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• Scoring function: MDL

principle: G+wT d(w)

• Algorithm: – find best merge that improves current grammar

– if no such merge exists, find best creation

– halt when no improvement

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Results

• On subsets of English grammars (15 rules, 8 non terminals, 9 terminals): 120 sentences to converge

• on (ab)*: all (15) strings of length 30

• on Dyck1: all (65) strings of length 12

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Algorithms

RPNI

K-Reversible

GRIDS

SEQUITUR

L*

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5 Open questions and conclusions

• dealing with noise• classes of languages that adequately mix Chomsky’s hierarchy with edit distance compacity

• stochastic context-free grammars• polynomial learning from text• learning POMDPs• fast algorithms

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ERNESTO SÁBATO, EL TÚNEL

Intuí que había caído en una trampa y quise huir. Hice un enorme esfuerzo, pero era tarde: mi cuerpo ya no me obedecía. Me resigné a presenciar lo que iba a pasar, como si fuera un acontecimiento ajeno a mi persona. El hombre aquel comenzó a transformarme en pájaro, en un pájaro de tamaño humano. Empezó por los pies: vi cómo se convenían poco a poco en unas patas de gallo o algo así. Después siguió la transformación de todo el cuerpo, hacia arriba, como sube el agua en un estanque. Mi única esperanza estaba ahora en los amigos, que inexplicablemente no habían llegado. Cuando por fin llegaron, sucedió algo que me horrorizó: no notaron mi transformación. Me trataron como siempre, lo que probaba que me veían como siempre. Pensando que el mago los ilusionaba de modo que me vieran como una persona normal, decidí referir lo que me había hecho. Aunque mi propósito era referir el fenómeno con tranquilidad, para no agravar la situación irritando al mago con una reacción demasiado violenta (lo que podría inducirlo a hacer algo todavía peor), comencé a contar todo a gritos. Entonces observé dos hechos asombrosos: la frase que quería pronunciar salió convertida en un áspero chillido de pájaro, un chillido desesperado y extraño, quizá por lo que encerraba de humano; y, lo que era infinitamente peor, mis amigos no oyeron ese chillido, como no habían visto mi cuerpo de gran pájaro; por el contrario, parecían oír mi voz habitual diciendo cosas habituales, porque en ningún momento mostraron el menor asombro. Me callé, espantado. El dueño de casa me miró entonces con un sarcástico brillo en sus ojos, casi imperceptible y en todo caso sólo advertido por mí. Entonces comprendí que nadie, nunca, sabría que yo había sido transformado en pájaro. Estaba perdido para siempre y el secreto iría conmigo a la tumba.

1 erice 2005, the analysis of patterns. grammatical inference 1 colin de la higuera grammatical...

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