the art of reduction the story of one reduction (or, depth through breadth) avi wigderson institute...
TRANSCRIPT
![Page 1: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/1.jpg)
The art of reduction
The story of one reduction
(or, Depth through Breadth)
Avi WigdersonInstitute for Advanced
Study Princeton
![Page 2: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/2.jpg)
How difficult are the problems we want to solve?
Problem A ? Problem B ?
Reduction: An efficient algorithm for A implies an efficient algorithm for B
If A is easy then B is easy
If B is hard then A is hard
![Page 3: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/3.jpg)
Constraint Satisfaction I
AI, Operating Systems, Compilers, Verification, Scheduling, …
3-SAT: Find a Boolean assignment to
the xi which satisfies all constraints
(x1x3 x7) (x1x2 x4) (x5x3 x6) …
3-COLORING: Find a vertex coloringusing which satisfies all constraints
![Page 4: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/4.jpg)
Classic reductions Thm[Cook, Levin ’71] 3-SAT is NP-complete
Thm[Karp 72]: 3-COLORING is NP-completeProof: If 3-COLORING easy then 3-SAT easy
formula graph
satisfying legalassignment coloring
Claim: In every legal 3-coloring, z = x y
F
y x
z OR-gadget
T
x y
![Page 5: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/5.jpg)
NP - completeness
Papadimutriou: “The largest intellectual export of CS to Math & Science.”
“2000 titles containing ‘NP-complete’in Physics and Chemistry alone”
“We use it to argue computational difficulty. They to argue structural nastiness”
Most NP-completeness reductions are “gadget” reductions.
![Page 6: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/6.jpg)
Constraint Satisfaction IIAI, Programming languages, Compilers,
Verification, Scheduling, … 3-SAT: Find a Boolean assignment to the
xi which satisfies most constraints
(x1x3 x7) (x1x2 x4) (x5x3 x6) …
Trivial to satisfy 7/8 of all constraints!(simply choose a random assignment)How much better can we do efficiently?
![Page 7: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/7.jpg)
Hardness of approximation
Thm[Hastad 01] Satisfying 7/8+ fraction of all constraints in 3-SAT is NP-hard
Proof: The PCP theorem [Arora-Safra, Arora-Lund-Motwani-Sudan-Szegedy]
& much more [Feige-Goldwasser-Lovasz-Safra-Szegedy, Hastad, Raz…]
Nontrivial approx exact solution of 3-SAT of 3-SAT
Interactive proofs
![Page 8: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/8.jpg)
3-SAT is 7/8+ hard
3-SAT is .99 hard
NP3-SAT is hard
Influencesin Booleanfunctions
DistributedComputing
2 DecadesDecade
![Page 9: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/9.jpg)
How to minimize players’ influence
Public Information Model [BenOr-Linial] :Joint random coin flipping / voting Every good player flips a coin, then combine using function f
Function InfluenceParity 1
Thm [Kahn—Kalai—Linial] : For every Boolean function
on n bits, some player has influence (log n)/n(uses Functional & Harmonic Analysis)
P parity
Mmajority
M M M
M
Majority 1/7Iterated Majority 1/8
![Page 10: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/10.jpg)
3-SAT is 7/8+ hard
3-SAT is .99 hard
IP NP
3-SAT hard
Optimization Approx Algorithms
ArithmetizationProgramChecking
2IP
PCP
Cryptography Zero-knowledge
IP=PSPACE
2IP=NEXP
PCP =NP
ProbabilisticInteractiveProof systems
IP = Interactive Proofs2IP = 2-prover Interactive ProofsPCP = Probabilistically Checkable Proofs
![Page 11: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/11.jpg)
IP = (Probabilistic) Interactive Proofs[Babai, Goldwasser-Micali-Rackoff]
ProverAccept (x,w)iff xL
Accept (x,q,a)whp iff xL
Prover
Verifier
w
q1
a1……qr
ar
Wiles Referee
Socrates
Verifier (probabilistic)
Student
L NP
L IP
x
![Page 12: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/12.jpg)
ZK IP: Zero-Knowledge Interactive Proofs[Goldwasser-Micali-Rackoff]
Accept (x,q,a)whp iff xL
Prover q1
a1……qr
arSocrates
Verifier
Student
L ZK IP
x
Gets no otherInformation !
Thm [Goldreich-Micali-Wigderson] :If 1-way functions exist, NP ZK IP(Every proof can be made into a ZK proof
!)Thm [Ostrovsky-Wigderson] ZK 1WF
![Page 13: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/13.jpg)
2IP: 2-Prover Interactive Proofs[BenOr-Goldwasser-Kilian-Wigderson]
Verif. accept (x,q,a,p,b) whp iff xL
Prover1 q1
a1……qr
arSocrates
Verifier
Student
L2IP:
x
p1
b1……pr
br
Prover2
Plato
Theorem [BGKW] : NP ZK 2IP
![Page 14: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/14.jpg)
What is the power of Randomness and Interaction in Proofs?
NP
IP
2IPTrivial inclusions IP PSPACE 2IP NEXP
Few nontrivial examplesGraph non isomorphism
……Few years of stalemate
Polynomial SpaceNondeterministicExponential Time
![Page 15: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/15.jpg)
Meanwhile, in a different galaxy…Self testing & correcting programs[Blum-Kannan, Blum-Luby-Rubinfeld]
Given a program P for a function g : Fn Fand input x, compute g(x) correctlywhen P errs on (unknown) 1/8 of Fn
Easy case g is linear: g(w+z)=g(w)+g(z)Given x, Pick y at random, and compute P(x+y)-P(y) x Proby [P(x+y)-P(y) g(x)]
1/4
Other functions?
Fn
P errsx+y y
xP used as Black-Box
![Page 16: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/16.jpg)
Determinant & Permanent
X=(xij) is an nxn matrix over F
Two polynomials of degree n:
Determinant: d(X)= Sn sgn() i[n] Xi(i)
Permanent: g(X)= Sn i[n] Xi(i)
Thm[Gauss] Determinant is easyThm[Valiant] Permanent is hard
![Page 17: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/17.jpg)
The Permanent is self-correctable[Beaver-Feigenbaum, Lipton]
Given X, pick Y at random, and computeP(X+1Y),P(X+2Y),… P(X+(n+1)Y) WHP g(X+1Y),g(X+2Y),… g(X+(n+1)Y)
But g(X+uY)=h(u) is apolynomial of deg n.Interpolate to geth(0)=g(X)
g(X) = Sn i[n] Xi(i)
Fnxn
P errs
x+3yx+2y
xx+y
|| ||||
P errs on 1/(8n)
![Page 18: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/18.jpg)
Hardness amplification
If the Permanent can be efficiently computed
for most inputs, then it can for all inputs !
If the Permanent is hard in the worst-case, then it is also hard on average
Worst-case Average case reductionWorks for any low degree polynomial.Arithmetization: Boolean
functionspolynomials
![Page 19: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/19.jpg)
Back to Interactive Proofs
Thm [Nisan] Permanent 2IP
Thm [Lund-Fortnow-Karloff-Nisan] coNP IP
Thm [Shamir] IP = PSPACE
Thm [Babai-Fortnow-Lund] 2IP = NEXP
Thm [Arora-Safra, Arora-Lund-Motwani-Sudan-Szegedy] PCP = NP
![Page 20: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/20.jpg)
Meaning
Thm [Lund-Fortnow-Karloff-Nisan] coNP IP
- Tautologies have short, efficient proofs- can be replaced by (and prob interaction)
Thm [Shamir] IP = PSPACE
- Optimal strategies in games have efficient pfs- Optimal strategy of one player can simply be a
random strategy
Thm [Babai-Fortnow-Lund] 2IP = NEXP
Thm [Feige-Goldwasser-Lovasz-Safra-Szegedy]2IP=NEXP CLIQUE is hard to approximate
![Page 21: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/21.jpg)
PCP: Probabilistically Checkable Proofs
Prover Verifier
w
Wiles Referee
x
NP verifier: Can read all of wPCP verifier: Can (randomly) access
only a constant number of bits in
wPCP Thm [AS,ALMSS]: NP = PCPProofs robust proofs reductionPCP Thm [Dinur ‘06]: gap
amplification
WHPAccept (x,w)iff xLAccept (x,wi,wj,wk)iff xL
![Page 22: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/22.jpg)
3-SAT is 7/8+ hard
3-SAT is .99 hard
3-SAT is .995 hard
3-SAT is 1-1/n hard
IP NP3-SAT hard
2IP
PCP
3-SAT is 1-2/n hard
3-SAT is 1-4/n hard
InapproxGapamplification
SL=L SpaceComplexity
Algebraic
Com
bin
ato
rial
SL=L : Graph Connectivity LogSpace
![Page 23: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/23.jpg)
Getting out of mazes / Navigating unknown terrains (without map & memory)
Theseus
Ariadne
Thm [Aleliunas-Karp-Lipton-Lovasz-Rackoff ’80] A random walk will visit every vertex in n2 steps (with probability >99% )
Only a local view (logspace)n–vertex maze/graph
Thm [Reingold ‘05] : SL=L:A deterministic walk, computable in Logspace, will visit every vertex. Uses ZigZag expanders [Reingold-Vadhan-Wigderson]
Crete1000BC
Mars2006
![Page 24: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/24.jpg)
3-SAT is 1-4/n hard
3-SAT is .995 hard
3-SAT is .99 hard
3-SAT is 1-2/n hard
3-SAT is 1-1/n hard
3-SAT is hard
< 1-4/n
<.995
Expanders < .99
< 1-2/n
< 1-1/n
Connected graphs
Dinur’s proof ofPCP thm ‘06log n phases:Inapprox gapAmplificationPhase:Graph poweringand composition
(of the graphof constraints)
Reingold’s proofof SL=L ‘05log n phases:Spectral gapamplificationPhase:Graph poweringand composition
Expander graphs Comm Networks Error correction Complexity, Math …
Zigzagproduct
![Page 25: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/25.jpg)
Conclusions &Open Problems
![Page 26: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/26.jpg)
Computational definitions and reductionsgive new insights to millennia-old concepts,
e.g.Learning, Proof, Randomness, Knowledge,
Secret
Theorem(s): If FACTORING is hard then- Learning simple functions is impossible- Natural Proofs of hardness are impossible- Randomness does not speed-up
algorithms- Zero-Knowledge proofs are possible- Cryptography & E-commerce are possible
![Page 27: The art of reduction The story of one reduction (or, Depth through Breadth) Avi Wigderson Institute for Advanced Study Princeton](https://reader035.vdocuments.site/reader035/viewer/2022081516/5697bfdd1a28abf838cb15f4/html5/thumbnails/27.jpg)
Can we base cryptography on PNP?Is 3-SAT hard on average?
Thm [Feigenbaum-Fortnow, Bogdanov-Trevisan]:
Not with Black-Box reductions, unless… The polynomial time hierarchy collapses.
Explore non Black-Box reductions!What can be gleaned from program code
?
Open Problems