1

Approximate Satisfiability and Equivalence

Michel de Rougemont

University Paris II & LRI

Joint work with E. Fischer, Technion,

F. Magniez, LRI ,

LICS 2006

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1. Tester on a class K: approximation of decision problems

• Equality between two strings (trees) : (ε2, ε) Tolerant tester , additive approximation of the Edit Distance with Moves.

• Membership: is w in L ?

3. Equivalence tester between two regular properties: polynomial time algorithm (Exact Equivalence is PSPACE complete)

4. Generalizations: regular trees, context-free languages, infinite words,

5. Current research: probabilistic systems.

Plan

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• Decisions on noisy inputs (distance to a language)

• Model Checking: can we approximate hard problems? Bounded MC, Abstraction, …..

• Black-Box Checking: does B satisfies P ?

Motivations

B

L

€

AGCTAGGA.....ACT

€

x

€

y

€

Learn B'

€

Decide B' ≈ε B and B'⊆ε P

4

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

F is testable if there is a probabilistic algorithm A such that

• A is an ε -tester for all ε • Time(A) is independent of n=size(U).

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

Goldwasser, D. Ron, 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 fromfar -ε

ε

Approximate Satisfiability and Equivalence

GF ε≡

G

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1. Classical Edit Distance:Insertions, Deletions, Modifications

2. Edit Distance with moves : dist(w,w’)

0111000011110011001

0111011110000011001

3. Edit Distance with Moves generalizes to Ordered Trees

Edit Distances with Moves

{ }'( , ') ; ( , ) ( , ')

W Ldist W W dist W L Min dist W W∈=

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Tester for equality: 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, shingles

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 −

1ε=k

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Tester for equality

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

Uniform statistics ( ): W=001010101110

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

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’)/n.

Tester 1: If |Y(w)-Y(w’)| <ε. accept, else reject.

1)(

...1∑=

=Ni

iXNwY

0...010

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=iX

2441

111)(. ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛=Wstatu

1)'(

...1∑=

=Ni

iXNwY

1ε=k

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Soundness: ε-close strings have close statistics

Robustness: ε-far strings have far statistics

We prove:1. b.stat is robust 2. u.stat is sound3. u.stat is robust (harder)

Theorem 1 :Soundness and Robustness

.)',( nwwdist ε≤

.)',( nwwdist ε≥

).)'(.)(. .21()',( nwstatbwstatbwwdist ε+−≤

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

dist w wd u stat w u stat w c u stat w u stat w

nε − ≤ ≤ −

hard

.6)'(.)(. .)',( 2 εε ≤−⇒≤ wstatuwstatunwwdist

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Robustness of b.stat

Robustness of b-stat: ).)'(.)(. .21()',( nwstatbwstatbwwdist ε+−≤

.)',( then )'(.)(. If nwwdistwstatbwstatb ε≤=

)'()''( 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 εε ≤−⇒≤ wstatuwstatunwwdist

ε.2

12)'(.)(.

nknkwstatuwstatu ≤+−

≤−

.6

1)1(3.2)'(.)(. εnkn

kwstatuwstatu ≤+−−≤−

.6)'(.)(. ε≤− 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

1ε=k

)(. ii vstatbX =)(. 11 vstatbX =

1v iv

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

vstatbEvstatbEnKwstatbu

Kniiti== ∑

=

. 2kcK=

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Uniform Statistics

ABKnkbu −=−− )1).(1( : by missed k length of subwords#

., onsdistributi uniform twoand ALet : Lemma BA μμB⊆A

BB

AB−=− .2. Then BA μμ

).

()(.)(. 4

/2

nOwstatuwstatbu ε

εΣ=−

εε

/2

3. ,1 with lemma previous Apply the

Σ≈+−=

nKknB

.)(. )(. w 4

/2

nwstatuwstatbu ε

εΣ≤−∀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 Σ≤×≥−0]

2)(.)(.Pr[

2. tandn enough largeFor k >≤−⇒

Σ= εε wstatbuvstatb

€

∀w∃v bu.stat(w) − b.stat(v) ≤ε

2 and dist(v,w) ≤ cε

Lemma 1:

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Robustness of the uniform Statistics

Lemma 2:

Lemma 1:

.5,6)'(. )(. .5)',( εε ≥−⇒≥ wstatuwstatunwwdist

2)(.)(. vw ε≤−∃∀ vstatbwstatbu

.)(. )(. w 4

/2

nwstatuwstatbu ε

εΣ≤−∀

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 εε ≥≥+−

.5)'( )( ifAccept ),O(cN εε ≤−= wYwY

(Probabilistic method)

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1. Membership: decide if

2. Inclusion and Equivalence

Equivalence tester

3. Testers for Membership and Equivalence

1 2 1 2 if v (except finitely) v is close to L L L v Lε ε⊆ ∀ ∈ ⇒ −

122121 and if LLLLLL εεε ⊆⊆≡

accepts then If 2121 ) ,rA(rLL =

32y probabilit with rejects then ) ( If 2121 ≥≡¬ ),r(A rLL ε

, of tionsrepresenta finite 2121 LL,rr

or is far from w L w Lε∈ −

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Automata for Regular languages

A: automaton with m states on Σ, Ak automaton with m states on Σk.Basic property:

Proposition:

Caratheodory’s theorem: in dimension d, convex hull of N pointscan be decomposed into in the union of convex hulls of d+1 points.

Large loops can be decomposed. Small loops (less than m=|A|) suffice.

))(.),...,(.Hull(Convex-Let 1

0 t,,...v1

t

v

vstatbvstatbt

U≥

=Η

, where..... to is 1 mvuvvuclosewLw il ≤−⇒∈ ε

€

v1 ....v l { } is a multi - set of Ak - compatible loops

))(.),...,(.Hull(Convex- 1

1, t,,...vk

1

t

mvv

vstatbvstatbit

U≤+Σ=

=Η

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Approximate Parikh mapping

Lemma: For every X in H, w of size n s. t.

δ≤)(.-X wstatb

X .

b-stat(w)

w

nn

mOw,L )).

.

.(

2()dist(

/1

εεδ

εΣ++≤

H is a fair representation of L

€

Lemma : If w ∈ L, then b.stat(w)∈ H

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Example

Y(w)

5.0 ,2 *1*)10(*)01(*1*0*)010( ==+= εkr

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

03/13/13/1

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

0001

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

1000

0

0

1

0

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

0

1

0

0

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

H={stat(w) : w in r } is a union of polytopes.

2 Polytopes for r.

H

Membership Tester:

Compute ( ). Accept if ( ( ) ) , else reject.Y w dist Y w ,H ε≤

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Construction of H in polynomial time

k.mΣ

€

Pt = b − stat(i →t j) { }

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Enumeration of all m loops:

Number of b-stat of words of length m on Σk is less than : Some loops have same b-stat: ABBA and BBAA

Construct H by matrix iteration:

k.Σm

11 tt PPP o=+

1,...mfor t , and between length t of word: =→ jiji t

of loops compatibleA ofb.stat 1 of hullsconvex Consider k k+Σ

€

size less than m .

2 size has H then and step with discretize We

)(kO

k

Σ

Σεε

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Membership 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 in

Remark 1: Time complexity of previous testers was exponential in m.

Remark 2: The same method works for L context-free.

O(k)Σ

O(k).Σm

2O(k).Σ

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Equivalence Tester for regular properties

1 2Tester for inclusion : r r⊆

1 2 ?H Hε ε⊆ε1H

ε2H

1 2Equivalence Tester for : r rε≡

1 2 2 1 and ?H H H Hε ε ε ε⊆ ⊆

Time polynomial in m=Max(|A |, |B |): O(k).Σm

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3. Generalization: Trees

(1,(1,(1,.),1),.)=c

(1,.)=c

T: Ordered (extended) Tree of rank 2 T’: squeleton

W: word with labels. Apply u.stat on W and define u.stat(T).

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Infinite words

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.

Approximate Model-Checking

€

limsupn →∞ dist(w(n),w'(n)) ≤ ε

w(n))(. nstatb

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Other Logics

Equivalence of Context-free grammars is undecidable, Approximate Equivalence in exponential time.

Consider formulas in different Logics (LFP, m-calculus,…..). Can Equivalence, Implication be approximated on a definable class K with a distance?

Definability and approximation: can first-order definable classes of trees testable with the Edit distance?

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4. Probabilistic Systems

Probabilistic Automata: Ma is a stochastic matrix for letter a. If w=a1 a1 …. an then Mw =Ma1 …. Man

PM Probabilistic Membership: Is ut.Mw.v> λ ?

APM: Approximate Probabilistic Membership: Let P= ut.Mw.v> λ

•Decide if w satisfies P or if w is ε -far from P.

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Approximations for Probabilistic Automata

1. Approximate probabilities• Introduce ε around λ

• Approximate membership

• Approximate Equivalence (Tzeng 92) is harder than Equivalence.

2. Approximate distances between states• Generalization of bisimulation

• Desharnais et al., Van Breugel-Worell

3. Our approach: Approximation on the input

http://www.lri.fr/~mdr/verap

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Basic Decompositions in H

s1=abcs2=ba

s3=bc

s4=aa s3=ccc

H1 H2

W=aabaaaaababcabcabcabcabcabcbc close to

W’=(aa)3(ba)2(abc)6

N Samples approximate ustat(W) close to :λ1.ustat((aa)*)+ λ2.ustat((ba)*) + λ3.ustat((abc)*)

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Basic Decompositions in H1

For each summit s, basic loop in A, let h(s,n)=Probability to follow s after n iterations of s

Analyze all loops mutliple of s: h(s,n)= rn for n large enough.

Analyze all possible decompositions of ustat(w) in H:

s1=abcs2=ab

s3

s4=aa

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Claim Hypothesis: all simple loops are distinct.

Input W of length n

Claim: Upper bound for ut.Mw’.v for W’ close to W.

λ1.ustat((aa)*)+ λ2.ustat((ba)*) + λ3.ustat((abc)*) indicates densities λ1, λ2, λ3 to follow aa, ba, abc on Hi.

We need to connect loops aa, ba, abc by some inputs: there are finitely many possibilities. Let Ci the best probability.

31 2 .. .. .

nn ni aa ba abcB r r r λλ λ=

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Tester for APM in O(1) Input W of length n

Tester(W,k)• Sample W with N(k) samples,

• Select H such that ustat is close,

• Decompose ustat on possible subpolytopes Hi with at most d+1 summits, and obtain a bound Bi,

• Consider all possible links on Hi, let Ci the optimal bound,

• Let D=Maxi Ci.Di

If λ< D, Accept else Reject.

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Non determinism and Probabilistic

Can we combine both non determinism and probabilistic behaviors?

Stationary distributions for a given scheduler are distributions on the states, for which there is also a polytope representation. Classical results exist about positional schedulers.

Problem: does the projection of these distributions on ustat vectors keep the distances? Thesis of Mathieu Tracol.

For large scale systems, evolutionary games also provide a statistical representation of the states. Can we predict approximate properties of the Equilibria?

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Conclusion

1. Tolerant Tester for Equality on strings under the Edit Distance with Moves

• Additive approximation in O(1) of the EDM

2. Equivalence tester for automata• Polynomial time approximate algorithm (PSPACE-complete)• Generalization to Buchi automata : approximate Model-

Checking• Context-Free Languages: exponential algorithm (exact problem

is undecidable)

3. Generalization to trees, infinite words4. Probabilistic systems.