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Property testing and learning on strings and trees

Michel de Rougemont

University Paris II & LRI

Joint work with E. Fischer, Technion,

F. Magniez, LRI

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1. Testers and Correctors on a class K

2. Tester for regular words and regular trees with the Edit Distance with Moves

3. Detailed proof of a key result (u.stat captures the distance)

4. Application to learning regular properties

Property testing

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Let F be a property on a class K of structures U

An ε -tester for F is a probabilistic algorithm A such that:• If U |= F, A accepts• If U is ε far from F, A rejects with high probability • Time(A) independent of n.

Robust characterizations of polynomials, R. Rubinfeld, M. Sudan, 1994O. Goldreich, S. Goldwasser and D. Ron, Property Testing and its connection to Learning and

Approximation, 1996.

Tester usually implies a linear time corrector.

(ε1, ε2)-Tolerant Tester

1. Testers on a class K

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1. Satisfiability : T |= F

2. Approximate Satisfiability T |= F

3. Approximate Equivalence

Image on a class K of trees

F F F

F defar -

Approximate Satisfiability and Equivalence

GF

G

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History of Testers

Self-testers and correctors for Linear Algebra ,Blum & Kanan 1989

Robust characterizations of polynomials, R. Rubinfeld, M. Sudan, 1994

Testers for graph properties : k-colorability, Goldreich and al. 1996

Regular languages have testers, Alon et al. 2000s

Testers for Regular tree languages , Mdr and Magniez, 2004

Charaterization of testable properties on graphs, Alon et al. 2005

New areas: Sublinear algorithms, Approximation of decision problems

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1. Distance d’Edition: Insertions, Effacements, Modifications

2. Distance Edition avec déplacements:

0111000011110011001

0111011110000011001

3. Distance Edition avec déplacements se généralise aux arbres ordonnés

2. Edit Distance with moves

'( , ') ; ( , ) ( , ')

W Ldist W W dist W L Min dist W W

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Uniform Statistics

W=001010101110 longueur n, n-k+1 blocs de longueur k=1/ε

1

1.#....

#)(.

2

1

knn

nWstatu

k

...."00...1" ofnumber #"00...0" ofnumber #

2

1

nn

"11...1" ofnumber #

....2kn

Pour k=2, n-k+1=11

1

4 1. ( ) . . ( )

4 11

2

u stat W u stat W

( , ') . ( ) . ( ') ,

dist W W u stat W u stat W proche,longueur desont mots les lorsque

Distance de mots: • NP-complet• Testable, O(1): échantillonner N sous-mots de longueur k: Y(W) et Y(W’) Si |Y(w)-Y(w’)| <ε. accepter, sinon rejeter

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Tester for a regular language

W: 0000000000111111111111Y: 000001000011111101111Z: 1111111111110000000000

T: 01001010001011000111010101

a b

0

1

1H A

0.5 / 2

/ 2. ( ) . ( ) . ( )

/ 2

0.5 / 2

u stat W u stat Z u stat Y

0001

1000

)(.25,025,025,025.0

Tstatu

T YW

Z

Automate A définit L, et un polytope H dans l’espace des u.stats

Testeur x dans L: • Testable, O(1): calculer Y(W),

• Si dist(Y(w),H) <ε. accepter, sinon rejeter Remarque: robustesse au bruit.

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Pair (A,H)

Blocs, k=2, m=4, | Σ |=4, | Σ| k +1=17:

Boucles de taille 1 bloc: {(aa,ca:1),(bb,2),(cc,ac:3),(dd:4)}

1 2

3 4

a

b

b

ca

cd

d

aa ca

H A

ac cc

bb

dd

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Corrector of a regular language

Y: 000001000011111101111 est ε -proche de L(A)

Correction déterministe:1. Décomposition en sous-mots admissibles

000001000011111101111 000001 000111111 1111 2. Décomposition en composantes connexes

000001 000 111111 11113. Recomposition (déplacements)

000 000001 111111 1111 distance 3 de Y

a b

0

1

1

A

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Corrector of an ordered tree

2 moves, dist=2

Automate d’arbre ou DTD: t: l,r r: l,r

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XML Corrector: http://www.lri.fr/~mdr/xml/

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Applications

Testers: • Estimate the distance between two XML files,• Décide if an XML F is ε-valid,• Décide if two DTDs are close.

Correctors: If an XML file F is ε-close from a DTD,• Find a valid F’ ε-close to F; • Rank XML files for a set of DTD’s (supervised learning)

Program Verification:• Decide if two automata are ε-close in polynomial time.• Approximate Model-Checking: http://www.lri.fr/~mdr/vera/

• Specification language• Model • Distance

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3. Block and Uniform statistics

W=001010101110 length n, b.stat: consecutive subwords of length k, n/k blocksu.stat: any subwords of length k, n-k+1 blocks

1401

61)(.

Wstatb

#....

#

/1)(.

2

1

kn

n

knWstatb ....

"00...1" ofnumber #"00...0" ofnumber #

2

1

nn

"11...1" ofnumber #

....2kn

For k=2, n/k=6 2

441

111)(.

Wstatu

1)'(.)(. :studyMain WstatuWstatu

1k

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Tester for equality of strings

Edit distance with moves. NP-complete problem, but approximable in constant time with additive error.

Uniform statistics ( ): W=001010101110

Theorem 1. |u.stat(w)-ustat(w’)| approximates dist(w,w’) .

Sample N subwords of length k, compute Y(w) and Y(w’):

Lemma (Chernoff). Y(w) approximates u.stat(w).

Corollary. |Y(w)-Y(w’)| approximates dist(w,w’) .

Tester: If |Y(w)-Y(w’)| <ε. accept, else reject.

1)(

...1

Ni

iXN

wY

0...010

iX

2441

111)(.

Wstatu

1)'(

...1

Ni

iXN

wY

1k

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Let F be a property on strings.

Soundness: ε-close strings have close statistics

Robustness: ε-far strings have far statistics

F is Equality on pairs of strings.For theorem 1, we prove:

1. b.stat is robust2. u.stat is sound3. u.stat is robust

Soundness and Robustness

.)',( nwwdist

.)',( nwwdist

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Robustness of b.stat

Robustness of b-stat: ).)'(.)(. .21()',( nwstatbwstatbwwdist

.)',( then )'(.)(. If nwwdistwstatbwstatb

)'()''( t.s. 'w'construct then )'(.)(. If wstatbwstatbwstatbwstatb

1401

61)(.

Wstatb

1302

61)'(.

Wstatb

in W' 3 andin W 4 "10" #but in W' 2 andin W 1"00"#

: Example on w. onssubstituti )'(.)(.2

most at after wstatbwstatb.n

"10" intoit change andin W "00" ofblock one take:'W'

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Soundness of u.stat

Soundness of u-stat:

Simple edit:

Move w=A.B.C.D, w’=A.C.B.D:

Hence, for ε2.n operations,

Remark: b.stat is not sound.Problem: robustness of u.stat ? Harder! We need an auxiliary distribution and two key lemmas.

.6)'(.)(. .)',( 2 wstatuwstatunwwdist

.2

12)'(.)(.

nknkwstatuwstatu

.6

1)1(3.2)'(.)(. nkn

kwstatuwstatu

.6)'(.)(. wstatuwstatu

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Statistics on words

k

k

Kt k-t

Block statistics: b.stat

Uniform statistics: u.stat

Block Uniform statistics: bu.stat

1k

)(. ii vstatbX )(. 11 vstatbX

1v iv

))(.())(.()(./,...1

vstatbEvstatbEnKwstatbu

Kniiti

. 2kcK

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Uniform Statistics

ABKnkbu )1).(1( : by missedk length of subwords#

., onsdistributi uniform twoand ALet : Lemma BA BA

BB

AB .2.Then BA

).

()(.)(. 4

/2

nOwstatbuvstatu

/2

3. ,1 with lemma previous Apply the

nKknB

.)(. )(. w 4

/2

nwstatuwstatbu

Lemma 2:

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Block Uniform Statistics

))(.())(.()(./,...1

vstatbEvstatbEnKwstatbu

Kniiti

1][0 ],)[(.][ ),(. uXuvstatbuXvstatbX iiiii

])[(. is on Average t.independen is ][Each uwstatbui uXi

2Kn-8

e]])[(.])[(.])[(.Pr[ : Bound Chernofft

uwstatbutuwstatbuuvstatb 2

Kn-8k

.e])(.)(.)(.Pr[ : BoundUnion t

wstatbutwstatbuvstatb 0]

2)(.)(.Pr[

2. tandn enough largeFor k

wstatbuvstatb

cw)dist(v, and 2

)(.)(. vw vstatbwstatbuLemma 1:

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Robustness of the uniform Statistics

Robustness of u-stat:

By Lemma 1:

By Lemma 2:

.5,6)'(. )(. .5)',( wstatuwstatunwwdist

2)(.)(. vw vstatbwstatbu

.)(. )(. w 4

/2

nwstatuwstatbu

w' w,from close v'Get v,

stat.u- of robustness impliesstat -b of Robustness

Tolerant tester:

Theorem: for two words w and w’ large enough, the tester:1. Accepts if w=w’ with probability 1 2. Accepts if w,w’ are ε2-close with probability 2/33. Rejects if w,w’ are ε-far with probability 2/3

..5)',( ).)'(.)(. .21( :bstat of Robustness nwwdistnwstatbwstatb

.5)'( )( ifAccept ),O(cN wYwY

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Membership and Equivalence tester

Membership Tester for w in L (regular):1. Construction of the tester: Precompute Hε 2. Tester: Compute Y(w) (approx. b.stat(w)). Accept iff Y(w) is at distance less than ε to Hε

Construction: Time is Tester: query complexity in time complexity inRemark 1: Time complexity of previous testers was exponential in m.Remark 2: The same method works for L context-free.

Tester of 1. Compute Hε,A and Hε,B

2. Reject if Hε,A and Hε,B are different.

Time polynomial in m=Max(|A |, |B |):

BA

O(k).

m

O(k)

O(k).

m

2O(k).

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4. Application to learning

Model: take random words according to a distribution D:

U.stat representation:

Negative examples could include the distance.

Learning algorithm: convex hulls of positive examples.

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PAC learning

The regular language is a polytope for u.stat.

Polytopes have a finite VC dimension. Hence they are PAC learnable.

Problem: the learnt concept may be ε-far from the language L.

For special distributions D, it may be ε-close. Example: D is uniform and the polytopes are « large ».

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Conclusion

1. Tester for the Edit Distance with Moves 2. Tester for membership to a regular set3. Equivalence tester for automata

• Polynomial time approximate algorithm (PSPACE-complete)• Generalization to Buchi automata : approximate Model-

Checking• Context-Free Languages: exponential algorithm (undecidable

problem)

4. PAC learning versus dist-Learning

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Generalizations

Buchi Automata. Distance on infinite words:Two words are ε-close if

A word is ε-close to a language L if there exists w’ in L s. t. W and w’ are ε-close.

Statistics: set of accumulation points of

H: compatible loops of connected components of accepting states

Tester for Buchi Automata: Compute HA and HB

Reject if HA and HB are different.

Equivalence of CF grammars is undecidable, Approximate equivalence in exponential.

(n))w'dist(w(n), lim sup n

w(n))(. nstatb