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Probabilistically Checkable Proofs Probabilistically Checkable Proofs

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Probabilistically Checkable Proofs - PPT Presentation

Madhu Sudan MIT CSAIL 09232009 1 Probabilistic Checking of Proofs TexPoint fonts used in EMF Read the TexPoint manual before you delete this box A A A A Can Proofs Be Checked Efficiently ID: 373314

probabilistic proofs checking 2009 proofs probabilistic 2009 checking proof error theorem coloring pcp verifier theorems length small verify pcps shallow constant probabilistically

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Slide1

Probabilistically Checkable Proofs

Madhu Sudan MIT CSAIL

09/23/2009

1

Probabilistic Checking of Proofs

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.:

A

A

A

ASlide2

Can Proofs Be Checked Efficiently?

The Riemann Hypothesis is true (12

th

Revision)

By

Ayror Sappen

# Pages to follow: 15783

09/23/2009

2

Probabilistic Checking of ProofsSlide3

Proofs and TheoremsConventional belief: Proofs need to be read carefully to be verified.

Modern constraint: Don’t have the time (to do anything, leave alone) read proofs.This talk:New format for writing proofs.Efficiently verifiable probabilistically, with small error probability.Not much longer than conventional proofs.09/23/2009

3

Probabilistic Checking of ProofsSlide4

Outline of talkQuick primer on the Computational perspective on

theorems and proofs (proofs can look very different than you’d think).Definition of Probabilistically Checkable Proofs (PCPs).Some overview of “ancient” (15 year old) and “modern” (3 year old) PCP constructions.

09/23/2009

4

Probabilistic Checking of ProofsSlide5

Theorems: Deep and ShallowA Deep Theorem: Proof: (too long to fit in this section).

A Shallow Theorem:The number 3190966795047991905432 has a divisor between 25800000000 and 25900000000.Proof: 25846840632.

8

n ¸ 3,

xn + y

n

z

n

09/23/2009

5

Probabilistic Checking of ProofsSlide6

Computational PerspectiveTheory of NP-completeness:Every (deep) theorem reduces to shallow one.

Given theorem T and bound n on length of proof, there exist integers A,B,C between 0 and 2

nO(1) such that

A has a divisor between B and

C iff

T

has a proof of length

n

.

Shallow theorem easy to compute from deep

.A,B,C computable from

T in time poly(n)

.Shallow proofs are not much longer.

09/23/2009

6

Probabilistic Checking of Proofs

[

Kilian

]Slide7

P & NPP = Easy Computational Problems

Solvable in polynomial time(E.g., Verifying correctness of proofs)NP = Problems whose solution is easy to verify(E.g., Finding proofs of mathematical theorems)NP-Complete = Hardest problems in NP

Is P = NP?Is finding a solution as easy as specifying its properties?

Can we replace every mathematician by a computer?Wishing = Working!

09/23/2009

7

Probabilistic Checking of ProofsSlide8

More Broadly: New formats for proofsNew format for proof of

T: Divisor D (A,B,C don’t have to be specified since they are known to (computable by) verifier.)Theory of Computation replete with examples of such “alternate” lifestyles for mathematicians (formats for proofs).

Equivalence: (1) new theorem can be computed from old one efficiently, and (2) new proof is not much longer than old one.

Question: Why seek new formats? What

benefits can they offer?

Can they help

?

09/23/2009

8

Probabilistic Checking of ProofsSlide9

Probabilistically Checkable ProofsHow do we formalize “formats”?

Answer: Formalize the Verifier instead. “Format” now corresponds to whatever the verifier accepts.Will define PCP verifier (probabilistic, errs with small probability, reads few bits of proof) next.

09/23/2009

9

Probabilistic Checking of ProofsSlide10

V

0

1

0

T

P

PCP Verifier

Reads Theorem

2. Tosses coins

3. Reads few bits of proof

4. Accepts/Rejects.

010010100101010101010

HTHTTH

09/23/2009

10

Probabilistic Checking of Proofs

T

invalid

)

8

P, V

accepts

w.p

.

·

1/2

T

valid

)

9

P

s.t

.

V

accepts

w.p

. 1Slide11

Features of interestNumber of bits of proof

queried must be small (constant?). Length of PCP proof must be small (linear?, quadratic?) compared to conventional proofs.Optionally: Classical proof can be converted

to PCP proof efficiently. (Rarely required in Logic.)

Do such verifiers exist?

PCP Theorem [

Arora

, Lund,

Motwani

, S.,

Szegedy

, 1992]:

They do; with constant queries and

polynomial PCP length.

[2006]

– New construction due to

Dinur

.

09/23/2009

11

Probabilistic Checking of ProofsSlide12

Part II

– Ingredients of PCPs09/23/200912

Probabilistic Checking of ProofsSlide13

Essential Ingredients of PCPsLocality of error:If theorem is wrong (

and so “proof” has an error), then error in proof can be pinpointed locally (found by verifier that reads only few bits of proof).Abundance of error:Errors in proof are abundant

(easily seen in random probes of proof).

How do we construct a proof system with these features?

09/23/200913

Probabilistic Checking of ProofsSlide14

3-Coloring

g

d

e

f

c

b

a

Locality: From NP-completeness

Color vertices

s.t

. endpoints of edge have

different colors.

is NP-complete:

T

P

a

c

d

b

e

f

g

09/23/2009

14

Probabilistic Checking of ProofsSlide15

To verify Verifier constructs

Expects as proof.To verify: Picks an edge and verifies endpoints distinctly colored.Error: Monochromatic edge = 2 pieces of proof.Local! But errors not frequent.

3-Coloring Verifier:

T

09/23/2009

15

Probabilistic Checking of ProofsSlide16

Amplifying Error: Algebraic ApproachGraph = E: V

£ V  {0,1}Place V in finite field F

(V

µ F)

Extend E to a polynomial E

(

x,y

)

E

:

F

£ F

F

E

|

V

£

V

= E

Coloring =

Â

: V

{0,1,2}

Extend to

Â

:

F

F

3-colors:

8

x

2

V

,

Â

(x)

¢

(

Â

(x)-1)

¢

(

Â

(x)-2) = 0

Validity:

8

x,y

2

V

,

E

(

x,y

)

¢

j 2

{-2

,-1,1,2} (Â(x)-Â(y)-j) = 009/23/2009Probabilistic Checking of Proofs

16Slide17

Algebraic theorems and proofsAlgebraic Theorems:

For V µ F, operators A, B, C;

deg. bound d

,

9 Â

of degree

d

s.t

.

A

(

Â), B

(

Â

),

C

(

Â

)

zero on

V

.

Algebraic Proofs:

Evaluations

of

Â

,

A

(

Â

),

B

(

Â

),

C

(

Â

).

Additional stuff, e.g., to prove zero

on

V

Verification?

Low-degree testing (Verify degrees)

~ “

Discrete rigidity phenomena

”?

Test consistency

~

Error-correcting codes

!

09/23/2009

17

Probabilistic Checking of ProofsSlide18

Some detailsE.g. to show Â

¢ (Â – 1) ¢ (Â – 2) = 0 on V.09/23/2009

Probabilistic Checking of Proofs

18

Checks:

Verify degree of

Consistency of

Â

,

¡

,

¢Slide19

More detailsExtends also to bivariate polynomials

(E(x,y))Immediately yields PCP with O(log n) queries, where n is (classical) proof size.

But leads to improvement by “composition”:Verifier’s task has reduced in complexity from verifying statements of length

n, to verifying statements of length log n

. Recurse.

(includes some cheating)

09/23/2009

Probabilistic Checking of Proofs

19Slide20

Amplifying Error: Graphically

Dinur

Transformation:

There exists a linear-time algorithm

A

:

A(G)

3

-colorable if

G

is

3

-colorable.

Fraction of monochromatic edges in

A(G)

(in any coloring) is twice (minimum) fraction in

G

(unless fraction in

G

is

¸

²

0

)

A

09/23/2009

20

Probabilistic Checking of ProofsSlide21

Graphical amplificationSeries of applications of A

:Increases error to absolute constant Yield PCPAchieve A in two steps:Step 1: Increase error-detection prob. By converting to (generalized) K-coloring Random walks, expanders, spectral analysis of graphs.

Step 2: Convert K

-coloring back to 3-coloring, losing only a small constant in error-detection.

Testing (~ “Discrete rigidity phenomenon

again

)

09/23/2009

21

Probabilistic Checking of ProofsSlide22

Step 2: (same as in earlier PCPs)Step 1: 3-coloring ! K-coloring

coloringÂ(u)

¥

(U) = {Âu

(N(u))}

!

u~v

edges

U~X (if

u~v~w~x

)

!

vertices

u

U (= u

[

N(u))

!

Some details

09/23/2009

Probabilistic Checking of Proofs

22

constraints

Â

(u)

Â

(v)

If

u~v~w~x

then

Â

u

(v

)

Â

x

(w

)

!

G

A(G)

!Slide23

ConclusionProof verification by rapid checks is possible.

Does not imply math. journals will change requirements!But not because it is not possible!Logic is not inherently fragile!

PCPs build on and lead to rich mathematical techniques.

Huge implications to combinatorial optimization

(“inapproximability”)

Practical use?

Automated verification of “data integrity”

Needs better size tradeoffs

… and for practice to catch up with theory.

09/23/2009

23

Probabilistic Checking of ProofsSlide24

Thank You!09/23/2009

24Probabilistic Checking of Proofs