Data checking and verification
Randomization and coding
A guide to PCP, an approach towards
classification of NP problems

P versus NP
Have you heard about P vs NP problem?
• Millenium prize problems by Cray institutehttp://www.claymath.org/millennium/• P versus NP is not only million dollar problem,
Its value is trillion dollars.

Class P and class NP• Consider a computational problem M of input size n• P: class of polynomial time solvable problems in usualmodel of computation– M is in P if there is an algorithm to solve it in polynomialtime• Example: Sorting, Searching, Graph connectivity• Shortest path, text pattern matching• NP: class of polynomial time solvable problems in
nondeterministic model of computation– M is in NP if there is a nondeterministic algorithm to solve
it in polynomial time (difficult to understand, isn’t it?)• P is a subclass of NP

More intuitive description• Verification is often easier than solution– Solving an equation is more difficult than verification• NP: If we know the solution, we can show it suchthat verification is done in polynomial time
– 1: Hamilton path
– 2: Graph coloring,– 3: Does this SUDOK puzzle have a solution?

• A problem (probably) not in NP– 1: Will black win this Chess game ?
– 2: SOKOBAN game
– 3: #3SAT (number 3SAT)• Intuitively, a problem that we are convinced withcorrectness of solution easily if it is shown.

NP and co-NP• It is notable that negation of an NP problem
might not be in NP– Is a given graph G is colored by using k colors?
• We give the actual coloring to veryfy. Thus NP problem– Is a given graph G has no coloring with k colors?
• If so, how to convince it?

• Co-NP problem: negation of an NP problem• It is believed NP and co-NP are different• It might be true that NP ∩ coNP = P

P vs NP
• Is it true that verification is easier than solution ?
– Consider an equation f(x) = x5 - 3x4 +3x2 + 4 = 0– Ask “ is this equation has an integer solution less than n (say,n=100)”.
– Verification: f(2) = 32-48+12+4 = 0– Solution???

• P vs NP: Is there any problem where verification is easy (Ptime) but solution is difficult (not in P-time)?
• Philosophical question– Is it easy to learn than solve by yourself?
• Is it true that a difficult problem remains to be difficult even if you aresuggested a solution (you must make sure it is true).

– In real life, we must solve many NP problems(or even moredifficult problems). Human can solve them by training (likeSUDOK) for most of instances. Why??

Implication of P=NP
• If P=NP, we can solve many problems– Many-body problem in physics– Protein folding problem in biology– Optimal scheduling in manufacturing– Optimal traffic control– Many problems in computational chemistry• If P=NP, we have serious inconvenience– Current cryptology assumes P is not NP– Information security system is destroyed• Most of researchers believes P is not NP, and the above
situation only occurs in Scientific Fiction– But no one knows the truth• The biggest mathematical challenge in 21st century

Several approaches towards P vs NP
http://people.cs.uchicago.edu/~fortnow/papers/pnp-cacm.pdf
Lance Fortnow’s article• NP- complete theory (S.Cook and R.Karp)– Almost all NP problems are either in P or NP-complete( one ofthe most difficult problems in NP)• Circuit complexity– Show NP needs exponential size of circuits• Algebraic/group theoretic method– Relation to generalized Riemann hypothesis (Mulmuley)• From mathematical logic• Relation to randomness• Interactive Proof and Checkable Proof

Textbooks/papers• Randomized Algorithms (Motwani-Raghavan)• Efficient Checking of Polynomials and Proofs
and the Hardness of Approximation Problems(M. Sudan, ACM Distinguished Thesis, 1995)• Proof Verification and the Hardness ofApproximation Problems (S. Arora, C. Lund, R.
Motwani, M. Sudan, M. Szegedy), J. ACM, Vol45(3), 1998, pp. 501-555

Rajeev Motwani (from wikipedia)•••••

Motwani joined Stanford soon after U.C. Berkeley. Motwani was one of the co-authors (with LarryPage and Sergey Brin, and Terry Winograd) of an influential early paper on the PageRank algorithm,
the basis for Google's search techniques. He also co-authored another seminal search paper WhatCan You Do With A Web In Your Pocket with those same authors.[3]He was also an author of two widely-used theoretical computer science textbooks, RandomizedAlgorithms (Cambridge University Press 1995, ISBN 978-0521474658, with Prabhakar Raghavan)
and Introduction to Automata Theory, Languages, and Computation (2nd ed., Addison-Wesley, 2000,
with John Hopcroft and Jeffrey Ullman).
Prior to his involvement with Google, Motwani founded the Mining Data at Stanford project (MIDAS),an umbrella organization for several groups looking into new and innovative data management
concepts. His research included data privacy, web search, robotics, and computational drug design.
He was an avid angel investor and had funded a number of successful startups to emerge fromStanford. He sat on the boards of Google, Kaboodle, Mimosa Systems, Adchemy, Baynote, Vuclip,
NeoPath Networks (acquired by Cisco Systems in 2007), Tapulous and Stanford Student
Enterprises among others. He was also active in the Business Association of Stanford Entrepeneurial
Students (BASES).[4][5][6]He was a winner of the Gödel Prize in 2001 for his work on the PCP theorem and its applications
to hardness of approximation.[7][8]Verification
Algebraic methods to verify
information

Verification in real life
• Given two data A and B, we want to verify A=B– We want to do it without reading A and B explicitly• If A and B are “persons”– How we can “read” data of persons completely?
• Impossible!

– Name, blood type, color of hair/eye,……
– ID numbers, secret keywords,– Finger print, DNA identification• If A and B are twins………

Fingerprinting• We recall the data checking problem we
discussed some weeks ago– Given a set of n data {a(1),a(2),…,a(n)}, and wesuspect one (or more) data is modified.
– How should we check efficiently?
• The sum of all data
• Use hash function, and sum of h(a(i))– This is an example of “fingerprint”

Fingerprinting for matrix multiplication• Consider a prime p• Given n by n matrices A, B, and C, we want toverify AB = C mod p– Computation in the field GF(p) (or Zp )• Can you do it in O(n2) time?

Pattern matching via verification• Given a text T of length n (bit string)• For each query pattern P of length m, we wantto find location of occurrence of P in T
• Both m and n are long (say, m = 100000, n =10000000)• KMP algorithm, BM algorithm: optimalO(n+m), but not much practical• Can we apply verification idea?

Verifying an identity• X = (x1,x2,..,xd)• We want to verify a polynomial identity
F1(X)=F2(X) of degree nVerifying an identity• X = (x1,x2,..,xd)• We want to verify a polynomial identity
F1(X)=F2(X) of degree n– Or more cruel identities

Prover and Proof
Prover is stronger than proof, since
we can ask questions instead ofreading the proof

Interactive Proof• Suppose that you can ask a god (or a powerfulsupercomputer ) to solve a problem• If the answer is “YES” (or “NO”), do you believe it blindly?
– In real life, we blindly believe the weather forecast which acomputer software reports– In ancient Greek, people believed “oracle of Apollo”.
– If you are given a program/software , how you believe it?
• Or how you write a program convincing others its correctness?

– Even in a university, most of students (and professors) believeWikipedia blindly ????

• We want to request a proof or an evidence.
– For a NP problem, we can ask for a proof if the answer is Yes– But if answer is NO, can we do something?

Graph non-isomorphism• You have two graphs G=(V,E) and G’=(V’,E’)• You suspect that they are isomorphic– There is a one-to-one map f of vertices of G such that(x,y) is an edge of E if and only if (f(x), f(y)) is in E’• You ask your professor who says he is always honest
and can solve the problem for any pair of graphs.
– If he answers “yes”, you can ask him to show the map f.
– Can you believe him if he says “NO”?

An interactive proof• You (verifier) have G and G’, and ask yourprofessor (prover) whether G=G’
• Professor answers “NO”, but you suspect thathe tells a lie.
• You ask some more questions to the professor
to reveal whether he is honest– You can flip a coin, and the random choice is notknown to the professor

How IP is strong• I will show that #3SAT is solved by usinginteractive proof system– #3SAT: Find the number of solutions of a logicalequation (given in a certain form)• This implies #P is in IP, and co-NP is in IP– do not worry about such terminology• A. Shamir showed that IP = PSPACE– PSPACE is considered to be larger than NP– 2 player’s game like GO and Chess are in PSPACE

3SAT and #3SAT• 3SAT: Is a logic equation F(X(1),X(2),..,X(n)) = 1 in 3CNF formula has a solution?
• #3SAT: How many solutions F(X)=1 has?
• 3SAT is an NP-complete problem– Any NP problem can be transformed into a 3SAT problem
in polynomial time.
– If 3SAT is in P, then NP = P– To show 3SAT is NP-complete• SAT is NP-complete (Cook’s theorem)• SAT is transformed into 3SAT• #3SAT is more difficult (called #P complete problem)– Toda’s theory (1989, Goedel award)

From SAT to 3SAT• Show that SAT is P-time soluble if 3SAT is.
• Given an instance (U,C) of SAT, we show atransformation of it into (U’, C’) of 3SAT
– U: set of variables, C:set of clauses• U={X(1),X(2),..,X(n)}, C={C(1),C(2),..,C(m)}• c( j )  l1, j  l2, j  ...  lk , j  {l1, j , l2, j ,.., lk , j }• For each clause C of C of length k, we consider
k-3 new variables, and transform it into a setof clauses each of which has length 3• Thus, we transform into 3SAT input with at
most nm variables and mn clauses

Transforming a clause• C = {z(1) , z(2) ,.., z(k)}– z(i) is either X(i) or its negation• We define new variables y(1),y(2), y(k-3)– These variables are only used to transform C• The clause c is transformed into
– S(c) = {{z(1),z(2),y(1)}, {y(1) , z(3), y(2)}, {y(2),z(4),y(3)},..,{y(k-3),z(k-1),z(k)}}– c is satisfied if and only if all clauses in S(c) aresatisfiedIntaractive proof for #3SAT
Step 1: Arithmetization• F(X(1),X(2),…X(n)) = a logical function in 3CNF
• C(i) = L(i,1) ∨L(i,2)∨L(i,3)• F = C(1) ∧C(2)∧..∧C(m)• Transfrom F into a real function f• For each literal L = L(i , j), we define l(i,j) = 1-x(k)if L=X(k) and l(i,j) = x(k) if L=X(k)• c(i) = 1 - l(i,1)l(i,2)l(i,3)• Observation: C(i) is satisfied if and only if c(i)=1• F = c(1)c(2)…c(m): a polynomial in x(1),..x(n)Number of solutions• If F= (X(1) ∨X(2)∨ X(3))∧（X(1)∨X(3)∨X(4)),• f= (1-(1-x(1))(1-x(2))x(3)) (1-x(1)x(3)(1-x(4)))• Number of solutions of F=1 is 8 + 6 = 14.
111• Define # f    ...  f ( x(1), x(2),.., x(n))
x (1) 0 x ( 2 ) 0x ( n ) 0

• Then, # of solutions of F=1 equals #f• Computation is difficult, but your professor who says
he can compute #f for any f.
• Ask the professor “What is #f”, and he answers that“its value is s ”. (say, “its value is 2487000”)• Verify whether he tells a truth.

Key idea
1#f 11

 ...  f ( x(1), x(2),.., x(n))

x (1) 0 x ( 2 ) 0x ( n ) 0

1f i ( x(1), x(2),.., x(i))  1

 ...  f ( x(1), x(2),.., x(n))

x ( i 1) 0 x ( i  2 ) 0Lemma•f0 = #f
• f ( x(1))  1
111x ( n ) 01  ...  f ( x(1), x(2),.., x(n))

x ( 2 )  0 x ( 3)  0x ( n ) 0•fn(x(1),..x(n)) = f(x(1),x(2),..,x(n))•fj-1(x(1),…,x(j-1)) = fj(x(1),..,x(j-1), 0 ) + fj(x(1),..,x(j-1), 1)

Basic (failing) strategy 1• Question 1: What is the value of f0?
– Professor answers : 247800• Question 2: What are f1(0) and f1(1)– Professor answers: 4000 and 243800– You check f0 = f1(0) + f1(1)– If check fails, professor tells a lie: The end.
– Else, guess which of two values is wrong…..

• Question 3: What are f2(0,1) and f2(0,0)?

Basic (failing) strategy 2• Question 1: What is the value of f0?
– Professor answers : 247800• Question 2: What is the function f1(z)?
– Professor answers: 222800z6 + 12000 z5 + 4000.
– You check f0 = f1(0) + f1(1)– If fail, professor tells a lie: The end. Else, continue• Question 3: What is the function f2 (x(1),x(2))?
• Proceed this processSuccessful strategy• Question 1: What is the value of f0?
– Professor answers : 247800• Question 2: What is the function f1(z)?

– Professor answers: g1(z) = 222800z6 + 12000 z5 + 4000.
– You check f0 = g1(0) + g1(1)– If fail, professor tells a lie: The end. Else, continue• Select a random value r, and compute g1(r)– say, r = 367• Question 3: What is the function f2(367, z)– Professor answers: g2 (z) = 34800 z5 + 34900 z2 + 403000– You check g1(r) = g2(0) + g2(1)• Next, select another random value r’, and compute g2(r)• CONTINUEAnalysis• Al l functions are considered in GF(p) for a primep > 2n• What is the probability that gi(z) is not fi(z) butgi(r) = fi(r) ?
– In other words, algorithm does not detect the lie inthe i-th step

• Error probability is at most 3nm/p, VERY SMALL• So if the professor tells a lie, the system detects itwith high probability.

PCP

PCP (probabilistic checkable proof)• Instead of god, we give a written proof.
• For a NP problem, we have a proof of length n– But, verifier wants to save time to verify– You prepare a proof such that verifier can easily
verify the correctness• This is just like database query!
• Like a database, we prepare the proof in a nicestructure.
– We need help of randomness and error correctingcode

A puzzle
Captain cook hided a great treasure, but he need tohide for a long time to prevent from arrested.
He will send letters to his 20 pirates to inform thelocation of the treasure, but they are only
reliable if they watch each other.
So, he want to encode the secret key so that it isrevealed if and only if 11 or more meet.
How he should do?

Popular error correcting codes• Reed solomon code
– Your CD is encoded by using it– Use a polynomial on a field F• F= GF(2q) in practical implementation• Here, we use GF(p) for a prime p• Hadamard code
– Use randomness– We can correct very large error.

Reed Solomon code
• a(1)a(2),..,a(k) :the key we want to send
– Each a(0) is a member of F, thus a large number.

• Let F(x) = a(k)xk+ a(k-1)xk-1+..+ a(1)x + a(0)• We randomly select m>k values x(i) and lety(i)=F(x(i)).
• As for captain Cook, k=10, m=20
• Send (x(i), y(i)) to the i-th pirates for i=1,2,..,20