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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Probabilistic Checking of ProofsSlide24
Thank You!09/23/2009
24Probabilistic Checking of Proofs