Thread Subject: Why is the “P vs. NP” question still open?

Arithmetic Hardness vs. Randomness
Hardness = Randomness

Boolean circuit lower bounds,
Derandomization of Boolean algorithms via PRG
[Nisan&Wigderson, Babai,Fortnow,Nisan,Wigderson,
Impagliazzo&Wigderson, Sudan,Trevisan,Vadhan, Shaltiel&Umans]
    Are circuit lower bounds necessary for showing BPP = P ?
Boolean Circuit Complexity
Boolean Circuit Approximation: Given a Boolean circuit C, approximate
Pr [ C(x) = 1 ] § 0.1
Thm [Impagliazzo,Kabanets,Wigderson]
If Boolean Circuit Approximation in P, then Nondeterministic EXPTIME
contains languages of superpolynomial circuit complexity
Arithmetic Circuit Complexity
Polynomial Identity Testing (PIT) : Given an arithmetic circuit F
(over integers), decide if F ´ 0 ( syntactically !!!)
… easy to solve in random polytime by sampling
Thm[Kabanets&Impagliazzo] PIT 2 P ) superpolynomial circuit lower
bounds : Boolean for NEXPTIME, or arithmetic for Permanent
Derandomizing PIT
Arithmetic circuit lower bounds )
Derandomization of Polynomial Identity Testing
Thm [Kabanets&Impagliazzo]
Permanent requires 2(n) size arithmetic circuits
    ) Polynomial Identity Testing in npolylog n time
             (for poly n - degree polynomials)
Derandomizing PIT is essentially equivalent to proving arithmetic
circuit lower bound
Arithmetic circuit lower bounds
)
Derandomization of PIT
Arithmetic Circuits
x1, x2, …, xn
arithmetic circuit F
F has + and * gates
F computes a polynomial in x1, …, xn of degree < 2|F|
(over integers)
F(x1,x2,…,xn)
 Generating pseudorandom tuples of integers
Generator
random seed
pseudorandom x1, …, xn
circuit F
 0 ?
Nisan-Wigderson Generator
p
p
p
x1
x2
xn
“almost disjoint”
generator
seed
Combinatorial design:
Given set U = { 1, …, m } ,
pick subsets S1, …, Sn of U
so that 8 i,j
|Si| = k and
|Si Å Sj| < log n
U
S1
S2
1 2 3 4 5 6 7 8 9
Nisan-Wigderson Generator
p
p
p
x1
x2
xn
“almost disjoint”
generator
seed r
3 12 4 7 11 1 0 3 5
12 4 1 3
3 11 1 5
seed r
x1 = r|S1
x2 = r|S2
Assume: p is a k-variate polynomial of arithmetic circuit complexity 2
(k)
Using NW Generator
p
p
p
x1
x2
xn
“almost disjoint”
generator
seed r
F
F (p(x1), p(x2), …,p(xn))
Security of NW Generator
F ( y1, y2, …, yn )
F ( p(x1), p(x2), …, p(xn) )
F ( p(x1), y2, …, yn )
F ( p(x1), p(x2),…, yn )
. . .
pseudorandom
random
Security of NW Generator
F ( y1, y2, …, yn )
F ( p(x1), p(x2), …, p(xn) )
F ( p(x1), y2, …, yn )
F ( p(x1), p(x2),…, yn )
. . .
Suppose F is not identically 0,
but F = 0 on all outputs of NW Generator
´ 0
 0
 0
´ 0
à first time
Security of NW Generator
F ( p(x1), y2, …, yn )
F ( p(x1), p(x2),…, yn )
 0
´ 0
Can fix the seed r outside S2, and fix y3, …, yn
so that we get from F a new polynomial H
H ( z1,…, zk, y )
H ( z1,…, zk, p(z1,…,zk) )
 0
´ 0
p(z1,…,zk) is a y-root of H (z1,…, zk, y) !
So, y - p(z1,…,zk) is a factor of H (z1,…, zk, y)
Security of NW Generator
y - p(z1,…, zk) is a factor of H (z1,…, zk, y)
Thm [Kaltofen]: If a degree d multivariate polynomial g over rationals
has arithmetic circuit complexity < s, then all factors of g have
arithmetic circuit complexity < poly(d,s)
By construction, H has degree poly(deg(F),deg(p)) and circuit
complexity < poly(|F|).
So, p(z1,…,zk) has “small” arithmetic circuit complexity. A
contradiction,
Ahrithmetic hardness-randomness tradeoff
Thm [Kabanets&Impagliazzo]:
Suppose Permanent over rationals requires arithmetic circuit size s.
Let F be an arithmetic circuit over integers computing a polynomial of
degree poly(|F|). Then testing if F ´ 0 can be done in deterministic
time
2n for any  > 0, if s is superpolynomial,
npolylog n, if s is exponential.
Derandomization of PIT
)
Circuit lower bounds
Polytime algorithm for PIT implies circuit lower bounds
Thm [Kabanets&Impagliazzo]:
Suppose PIT 2 P over integers. ( That is, given an arithmetic circuit
F over integers, can decide in time poly(|F|) if F ´ 0. )
Then one of the following is true:
NEXP has superpolynomial Boolean circuit complexity, or
Permanent over integers has superpolynomial arithmetic circuit
complexity.
No Derandomization of PIT
without
Circuit lower bounds
Similar Results
[Karp&Lipton, Babai,Fortnow&Lund]:
NP = P ) EXP not in PolySize
[Impagliazzo,Kabanets&Wigderson]:
Boolean Circuit Approximation in P ) NEXP not in PolySize
Proof Method
[Karp&Lipton, Babai,Fortnow&Lund]:
EXP µ PolySize ) EXP = 2 = MA
So, NP = P ) EXP not in PolySize
[Impagliazzo,Kabanets,Wigderson]:
NEXP µ PolySize ) NEXP = MA =NPCircuit Approx
So, Circuit Approx in P ) NEXP not in PolySize
[Kabanets&Impagliazzo]:
NEXP µ PolySize & Perm in arithm PolySize ) NEXP = MA = NPPIT
So, PIT in P ) circuit lower bound for NEXP or Perm
Main Tools
[Valiant]:
Perm is #P-complete
[Toda]:
2 µ P#P
[Impagliazzo,Kabanets&Wigderson]:
   NEXP µ PolySize ) NEXP = 2
Self-Reducibility of Perm
Time Hierarchy Theorems
Main Theorem
The following assumptions are inconsistent:
NEXP in PolySize
Perm in arithmetic PolySize
PIT in P
Proof:
NEXP in PolySize ) NEXP = 2 = P#P = PPerm
                                     [IKW + Toda + Valiant]
Main Lemma: Perm in arithmetic PolySize ) Perm in NPPIT
Proof of Main Theorem
The following assumptions are inconsistent:
NEXP in PolySize ) NEXP = PPerm
Perm in arithmetic PolySize ) Perm in NPPIT
PIT in P
) NEXP in NP
Contradicts Time Hierarchy Theorem !
Properties of Permanent
Definition: For an integer n x n matrix A = (ai,j),
Perm (A) =  i a i,(i)
where  ranges over all permutations of {1,…, n}
Downward Self-Reducibility of Perm :
               Perm (A) = j a1,j * Perm (Aj),
where Aj is the jth minor of A along the 1st row
Polynomial Identities for Permanent
Lemma:
Let pi, 1· i· n, be a polynomial on i2 variables.
Then pi = Perm iff
                       p1(x) ´ x,
and, for all 1 < i · n,
   pi (X) ´ j x1,j * pi-1(Xj),
where X is an i x i matrix of variables, and Xj is the jth minor
of X along the 1st row
Proof by induction; use downward self-reducibility of Perm
Main Lemma:
Perm in arithm PolySize ) Perm in NPPolyIdTest
Proof: To compute Perm (M) for nxn matrix M,
Nondeterministically guess polysize arithm circuits Fi on i2 inputs,
for 1· i· n
Using oracle access to PolyIdTest, check that Fi’s satisfy the
Identities of Permanent. If the check fails, abort the computation.
Output Fn(M)
Special Case of Polynomial Identity Testing
Symbolic Determinant Zero Testing: Given a matrix M of integer
constants and variables, decide if Det(M) ´ 0
… easy to solve in random polytime (even in random parallel time)
Thm[Kabanets&Impagliazzo] If Symbolic Determinant Zero Testing is in
P, then either NEXP not in PolySize, or Perm not in arithmetic formula
PolySize.
Hardness of Determinant Zero Testing
Thm[Kabanets&Impagliazzo] If Symbolic Determinant Zero Testing is in
P, then either NEXP not in PolySize, or Perm not in arithmetic formula
PolySize.
Proof Idea:
Use [Valiant] to simulate an arithmetic formula of size s by a
symbolic determinant of k x k matrix, for k = O(s).
Get polytime algorithm to test if F ´ 0 for arithmetic formulas F …
Universality of Determinant
Thm[Valiant] Every arithmetic formula F of size s can be mapped to a
symbolic matrix MF of size at most 3s x 3s so that F ´ Det (MF).
Proof:
Basis: x !
Let  ! , and  !
Universality of Determinant
Basis: x !
Let  ! , and  !
Then
 *  ! , and  +  !
Conclusions and Open Problems
Derandomizing Polynomial Identity Testing is essentially equivalent to
proving circuit lower bound for NEXP
Poly Identity Testing is the “hardest” problem in BPP
Deterministic Poly Identity Testing for restricted classes of
arithmetic circuits (e.g., depth-3 circuits) ? partial results in
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