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Complexity Theory
Lecture 8
Lecturer: Moni Naor
Recap
Last week:
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Randomized Reductions
Low memory verifiers
#P Completeness of Permanent
This Week:
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Toda’s Theorem: PH  P#P.
Program checking and hardness on the average of
the permanent
Interactive Proofs
Putting the Hierarchy in P#P
Toda’s Theorem: PH  P#P
Idea of the proof:
• Characterize PH in terms circuits
– a uniformly constructed constant depth circuit with exponential number of Æ
and Ç gates
• Consider circuits where the exponentially occurring gates are parity
(xor) ©
– This corresponds to P©P
– ©P the class of functions expressible as the number of accepting paths mod 2
in some NTM.
• Show how to approximate an Æ and an Ç gate using a © gate .
– This gives PH  RP©P
• Show RP©P  P#P
Tool:-biased probability spaces
Uniformly Direct Connect circuits
Let {Cn}n ¸ 1 be a family of circuits. We say that they are
Direct Connect Uniform family if there is a polynomial
time (in n and log the size of Cn) algorithm for the following
functions:
• TYPE(n,i) – providing the function gate i computes
– Can consider various bases. Fan-in of gates can vary
Example: Æ , Ç ,  , Input , Output
• IN(n,i,j) – providing k, the jth input into gate i (or none)
• OUT(n,i,j) – providing k, the jth output to which gate i
(or none) feeds
Characterization of PH in terms of DC
Theorem: a language L 2 PH iff it can be computed by a family {Cn}n ¸ 1
of circuits such that
• {Cn}n ¸ 1 is Direct Connect Uniform
• The gates used are: Æ , Ç , 
0(1)
• {Cn}n ¸ 1 has constant depth and 2n size
• The  gates appear only at the inputs
If only Ç gates have exponential fan-in, then this is a characterization of NP
key point: we can guess the value of the input as well and consider in the circuit only
(x,y) paths that accept.
Need to check whether input guess was correct
For the PH case, can guess the computation and input as well, need the
large fan-in Æ and Ç gates to simulate the alternation
Small bias probability spaces
Let  be a probability space with K random variables x1,
x2,… xK obtaining values in {0,1}. We say that it is biased if for any subset S µ {1…K}
|Pr[ ©i 2 S xi =1] - Pr[ ©i2 S xi =0]| · 
A probability space is 0-biased iff it is the uniform distribution
on x1, x2,… xK
– Size 2K
Much smaller spaces exists for >0
Description of a point can be O(log (K/))
Want an efficient way to compute xi from the representation of the
point in the sample space.
Should be polynomial in log i and the representation of the sample point
A construction for fixed 
Let K=2ℓ and H={h|h:{0,1}ℓ  {0,1}ℓ } be a family of
pairwise independent hash functions
Each point in the probability space is defined by
• a hash function h 2R H.
–
For 1 · j · 1 and 1 · i · K let
h_j(i)=1 iff first j bits of h(i) are `0’
and h_j(i)=0 otherwise
and
• a vector v1, v2, … vℓ 2R {0,1}ℓ
Each xi = ©1 · j · ℓ vj ¢ hj(i)
To describe a point in the
probability space:
Log |H| + log K bits
Computation:
log K + time to compute
h
Analysis of Construction
• Let S µ {1…K} and 2j-2 ≤ |S| ≤ 2j-1.
Event
AS = “there is exactly one i 2 S s.t. h_j(i)=1”
• We know from unique sat analysis that
Prh[AS ]¸ 1/8
Given that AS occurs, for any assignment to
v1,…, vj-1, vj+1 …, vℓ
since vj is undetermined we know
Pr[ ©i 2 S xi =1]=1/2.
•
Requirement: for all S µ {1…K}
|Pr[ ©i 2 S xi =1]
- Pr[ ©i2 S xi =0]| · 
Construction:
Each xi = ©1 · j · ℓ vj ¢ h_j(i)
Conclusion: 1/16 · Pr[ ©i 2 S xi =1] · 1/2 and
1/2 · Pr[ ©i 2 S xi =0] · 15/16 and hence
|Pr[ ©i 2 S xi =0] - Pr[ ©i2 S xi =0]| · 7/8
Can amplify by choosing a few independent constructions
and randomly Xoring the assignments
Replacing Ors with Xors
Consider an Or gate with K inputs y1, y2 … yK
• Choose K random variables x1, x2,… xK which are -biased
• Let zi = xi ¢ yi. Replace Çi=1K xi with ©i=1K zi
– If original is 0 with probability 1 new circuit is correct
– If original is 1 with probability ½ -  new circuit is correct
r1, r2, … rℓ
y1, y2, … yK
Ç
-bias generator
y1, y2, … yK
x1, x2, … xK
Can have several copies and
take the Or to reduce error
©
Replacing Ors with Xors
• By repeating the processc nc times can reduce the
probability of error to 2-n
– Total number of bits requires is still polynomial in n
• If there is a circuit with many Ors – can replace all of them
using the same set of random bits simultaneously.
– The probability of correct computation is still high
• Union bound over the bad events per gate
• What about And gates? Turn into Or gates using nots
Result: a circuit where only the © gates have exponentially
many inputs.
The  gates are not necessarily at the inputs
Computing DC Parity circuits
Theorem: A family {Cn}n ¸ 1 of circuits such that
• {Cn}n ¸ 1 is Direct Connect Uniform using:
© gates with exponential fan-in
Æ , Ç ,  gates with polynomial fan-in
• {Cn}n ¸ 1 has constant depth and 2n
can be computed in ©P
0(1)
size
Proof: need to construct a NTM where the parity of the
number of accepting paths equals the circuit value
Computing DC Parity circuits
NTM construction from DC Parity circuit
Procedure checkout:
At input:
on value `1’ return: yes
on value `0’ return: no
At And gate:
recursively check out all inputs
if they all return yes return: yes
At © gate:
Non-deterministically pick one of the inputs
iff it returns yes return: yes
At  gate:
Recall: Æ gates have polynomial fan-in
Due to constant depth process is
polynomial time
Since only the parity gates have
exponential fan-in, the subtree
chosen by the process is poly sized
Choose non-deterministically between { yes, recursive call to input}
NTM: Start at the output and check recursively. If returns yes then accept
PH  RP©P and beyond
• Given a language L 2 PH consider its DC
• Apply the transformation to parity circuits
• Apply the transformation from parity circuits to ©P and
make the call.
Derandomization: RP©P  P#P
Can consider all random assignments
Description is poly-length
Based on constructing a gadget that translates
Even number of accepting path to 0
Odd number of accepting paths to -1 mod 22m
EXP
The classes we discussed
PSPACE
#P
Time for new classes:
• IP
PH
Σ3P
• AM[2]
• …
Π3P
Δ 3P
Σ2P
Π2 P
Δ2P
NP
BPP
P
coNP
Hardness on the Average of the Permanent
• We saw that computing the permanent is #P-Complete
– True also for computing it mod M for sufficiently large M
• What about random matrices
– Can we argue that it is hard to compute per(A) correctly for a
random matrix A mod M ?
Theorem: if M ¸ n+2 is a prime and
• there is a polynomial time algorithm that computes
per(A) correctly for a random matrix A mod M with
probability at least 1-1/2n (over the choice of A),
• then there exists a probabilistic polynomial time algorithm
for computing per(A) for all matrices A
Can replace the 1-1/2n with 3/4
Hardness on the Average and Program
Correction of Polynomials
Let |F|>d+2 and
• f: Fn F be a function to which there is oracle access
– This is a program that has been implemented
• p: Fn F be a polynomial of degree d
– This is the function we are really interested in
We only have blackbox access to f
Sum of the degrees
in each monomial
• Suppose that f and p agree on a fraction of at least 1-1/(3d+3) of
their inputs
Then we can compute with high probability p(x) for any x 2 Fn
Connection to the permanent problem: per(A) is a polynomial of
degree n in the n2 variables corresponding to the entries
Can obtain a better result with better correction of errors
Reed-Solomon Codes
The randomized reduction
• On input x2 Fn: pick random y 2R Fn and consider the line
ℓ(t) = x + t y.
– Each point of ℓ(t) except ℓ(0) is uniformly distributed in Fn
• but they are not independent of each other
– q(t)=p(ℓ(t)) is a univariate polynomial in t of degree at most d
Therefore
Pr[p(ℓ(t) =f(ℓ(t)) for all t=1,2, … d+1]
=1- Pr[9t2 {1,2, … d+1}: p(ℓ(t) ≠ f(ℓ(t))]
¸ 1 –(d+1)/3(d+1) =2/3
If we know q(t)=p(ℓ(t)) at d+1 points, can interpolate to obtain
q(0)= p(ℓ(0))= p(x)
If we have a good correction procedure for polynomials, then sufficeint to be correct
on ¾ of the points.
Consequences
• Unlike other NP-Hard problems, cannot expect heuristics
that solve many instances
• Applications to cryptography?
– We are interested in hard on the average problems there, can we
use it?
– The problem is that these are not solved problems, that come
with certificates
• What about NP problems, are there such reductions for
them?
– The simple answer is no, unless the PH collapses
• This is a consequence of the classes we are to see next
Program Checking
• Let f be a program claiming to perform task T. A checker C for f is a
simple program with oracle/black-box access to f and where
– If f is good for T, i.e. 8y f(y)=T(y), then
8x Cf(x)=T(x) with probability at least 2/3
– If P fails on x, i.e. f(y)≠ T(x), then
Pr[Cf(x) accepts f(x)] is at most 1/3
What we just saw: a program checker for the permanent.
How about program checker for graph non-isomorphism?
How about program checker for SAT?
Self reducibility again helpful
How about program checker for non-SAT?
Proof systems
• What is a “proof”?
Complexity theoretic insight: at the minimum a proof
should be efficiently verified
Proof systems
For a language L, goal is to prove x  L
General requirements from a Proof system for L:
Defined by the verification algorithm V
– completeness: x  L   proof, V accepts (x, proof)
true assertions have proofs
– soundness: x  L   proof*, V rejects (x, proof*)
false assertions have no proofs
– efficiency:  x, proof, the machine running V(x, proof) us
efficient:
• runs in polynomial time in |x|
• ?
Classical Proofs
• Recall: L  NP iff expressible as
L = { x |  y, |y| < |x|k, (x, y)  R } and R  P.
• NP is the set of languages with classical proof systems (R
is the verifier)
We wish to extend the notion.
An extension we have already seen:
• two adversarial provers  and 
Interactive Proofs
• Two new ingredients:
– Randomness: verifier tosses coins
• Should err with some small probability
– Interaction: rather than simply “reading” the proof,
verifier interacts with prover
• Is the prover another TM?
• Framework captures the classical NP proof systems::
– prover sends proof.
– verifier runs algorithm for R
No use of randomness
Interactive Proofs
Interactive proof system for L is an interactive
protocol (P, V)
Random tape
Common input: x
Prover
New issue: who knows the
random tape
New resources:
• # of rounds
•Length of message
Verifier
.
.# rounds and length of
.messages is poly(|x|)
accept/
reject
Interactive Proofs
Definition: an interactive proof system for L is an
interactive protocol (P, V)
Perfect Completeness:
– completeness: x  L:
V accepts with Prob 1
Pr[V accepts in an execution of (P, V)(x)]  2/3
– soundness: x  L   P*
Pr[V accepts in an execution of (P*, V)(x)]  1/3
– efficiency: V is PPT machine
• Can we reduce the error to any ?
Error Reduction
• If we execute the protocol sequentially ℓ times let
Ij =1 if jth run is correct and 0 otherwise
The Ij’s are not necessarily independent of each other but, since can
tolerate any prover*
Pr[Ij =1 | any execution history] ¸ 2/3
If we compare to ℓ independent coins with probability 2/3 where we
take majority of answers
For any prover* the interactive proof stochastically dominates
• Can argue the same for ℓ parallel executions
Number of rounds is preserved
Things are not so simple when:
•More than one prover
•Prover is assumed to be efficient
Interactive Proofs
New complexity class:
IP = {L : L has an interactive proof system}
– Captures more broadly what it means to be convinced that a
statement is true
• But no certificate to store for future generations!
– Clearly NP µ IP. Potentially IP larger.
• How much larger?
– IP with perfect soundness and completeness is NP
• To go beyond NP randomness is essential
• Perfect soundness in itself implies NP power
Famous Example: Graph Isomorphism
Two graphs G0 = (V, E0) and G1 = (V, E1) are
isomorphic (G0  G1) if there exists a permutation
π:V  V
for which
(x, y)  E0  (π(x), π(y))  E1
Graph Isomorphism
• The problem GI = {(G0, G1) : G0  G1 }
– Is in NP
– But not known to be in P, or to be NP-complete
• One of Karp’s original open problems in famous NP-Completeness paper
• GNI = complement of GI
– not known to be in NP
Theorem: GNI  IP
– Was first indication IP may be more powerful than NP
GNI in IP
Interactive proof system for GNI:
Hidden Coins!
input: (G0, G1)
Prover
H = π(Gc)
if H  G0 r = 0
Else
r=1
Verifier
flip coin c R {0,1};
pick random πR S|V|
r
accept iff
r = c
GNI in IP
• Completeness:
– if G0 is not isomorphic to G1, then H is isomorphic to exactly
one of (G0, G1)
Perfect Completeness:
– prover will always choose correct r
V accepts with Prob 1
• Soundness:
– if G0  G1 then the distributions on H in case c = 0 and c = 1
are identical
– Hence: no information on c
• Any prover P* can succeed with probability exactly ½.
Hidden coins seem essential – but as we will see can obtain a
protocol with public coins only.
Lack of certificate
Bug or feature?
Disadvantages clear, but:
• Advantage: proof remains `property’ of prover and not
automatically shared with verifier
• Very important in cryptographic applications
– Zero-knowledge
• Many variants
• Can be used to transform any protocol designed to work with benign
players into one working with malicious ones
– The computational variant is useful for this purpose
• Can be used to obtain (plausible) deniability
Code Equivalence Problem
• For two k £ n matrices G1 and G2 over finite field
F we way that they are equivalent if there is
– A k £ k matrix S which is non singular over F
– An n £ n permutation matrix P
Such that G1 =SG2P
This means that if we think of the codewords generated by G1
and G2, then there is a 1-1 mapping after reordering the bit
positions
c=xG
Code word
Information word
Homework: Show that the non-equivalent of matrices
problem is in IP with constant number of rounds
The power of IP
• GNI  IP suggests IP more powerful than NP, since GNI
not known to be in NP
• GNI is in coNP
Today:
• coNP µ P#P µ IP
• IP µ PSPACE
Theorem: IP=PSPACE
IP µ PSPACE
Optimal strategy for prover:
• Strategy: for input x, at each step given the
interaction so determine the next message.
• Optimal strategy for x: the one yielding the best
probability of acceptance by V
Claim: Optimal strategy is computable in PSPACE
References
• Toda’s Theorem: Toda, FOCS 1989
• Progam Checking: Blum and Kannan, Blum, Luby and
Rubinfeld
• Average Hardness of permanent: Lipton 1990
– Polynomials – Beaver and Feigenbaum, 1990
• Interactive Proof system:
– Public coins version: Babai 1985 (Babai, Moran)
– Private Coins: Goldwasser Micali and Rackoff
• Proof system for GNI: Goldreich, Micali and Wigderson,
1986
• Private coins equals public coins: Goldwasser and Sipser,
1986