Example solution: 1.4
Avery Musbach
February 9, 2010
(a) We claim that a language L cfw_0, 1 is in roNTIME(f (n) i there
exists a polynomial p : N N, a natural number c N and a f (nc )-time
TM M (which I shall call the verier for L) such that for every x
CSci 5403, Spring 2010
Homework 1
due: Feb 2, 2010
1. Constructibility. Recall that a function t : N N is time-constructible if there is
a Turing Machine that computes the binary representation of t(n) given 1n as input, in
time O(t(n). Suppose f and g ar
CSci 5403, Spring 2010
Brian Berzins
Derived with: Stefan Nelson-Lindall, Hannah Jaber
Homework 1.3 Solution
(a) Show that quadeq is NP-complete.
First: We show that quadeq NP by demonstrating that there is a
polynomial time verier for it. One such verier
CSci 5403, Spring 2010
Homework 3
due: March 2, 2010
1. PH lower bounds. Show that for every k > 0, PH contains languages whose circuit
complexity is (nk ).
Hint: Recall that by the size hierarchy theorem, there exist languages with circuit
complexity (nk
CSci 5403, Spring 2010
Homework 4
due: March 23, 2010
1. Randomness does not help in PSPACE. In lecture, it was stated without proof
that BPPSPACE = PSPACE. In this problem, well prove an even stronger version of this
result. Dene the class PPSPACE to be
Ted Kaminski
CSci 5403 - Computational Complexity Homework 3 Problem 1 Solution
1
Problem statement
Show that for every k > 0, PH contains languages whose circuit complexity is (nk ).
Hint: Recall that by the size hierarchy theorem, there exist languages
COMMUNICATION COMPLEXITY
CSci 5403
COMPLEXITY THEORY
LECTURE XXIV:
COMMUNICATION COMPLEXITY
y 2 cfw_0,1n
x 2 cfw_0,1n
(x,y) ?
How many bits do Alice and Bob need to send?
Example. (x,y) = ixi + iyi mod 2 can be
computed with two bits:
ixi mod 2
ixi + iyi
CSci 5403
COMPLEXITY THEORY
LECTURE XXIII:
PCPs AND HARDNESS OF APPROXIMATION
Definition. A gap problem is a promise problem
(Y,N) derived from optimization problem (R,val):
Y = cfw_ x : OPT(x) c , N = cfw_x : OPT(x) < c/ .
Claim. If there is a reduction
CSci 5403
COMPLEXITY THEORY
LECTURE XXII:
APPROXIMATION ALGORITHMS
Definition. Let A be an algorithm such that for all x,
A(x) F(x). Then A has approximation ratio for:
- the minimization problem (R,cost) if
x, cost(x,A(x) OPT(x)
- the maximization proble
PSEUDORANDOM NUMBERS
CSci 5403
COMPLEXITY THEORY
LECTURE XXI:
FORMAL FOUNDATIONS OF
PSEUDORANDOMNESS
Where do the random bits in a PPT come from?
In C we call rand() to get pseudo-random bits.
static int x;
int rand(void) cfw_
xi+1 = axi + b mod p;
return
RESULTS OF A PROOF
CSci 5403
By interacting with a traditional prover the
verifier gains access to new information:
G HAM: v1,v2,vn
COMPLEXITY THEORY
LECTURE XX:
ZERO KNOWLEDGE
Prover
Quite So.
Verifier
What about an interactive prover?
x has a square roo
INTERACTIVE PROOFS
CSci 5403
COMPLEXITY THEORY
LECTURE XIX:
INTERACTIVE PROOFS III
Definition. An interactive proof system has two
players:
1 = V(x;r)
2 = P(x,1)
Arbitrary
Prover P :
* *
k-1= V(x,1,2,k-2;r)
k= P(x,1,2,k-1)
k+1=accept/reject
PPT
Verifier V
WHAT IS A PROOF?
CSci 5403
G HAM: v1,v2,vn
COMPLEXITY THEORY
Quite So.
Prover
LECTURE XVII:
INTERACTIVE PROOFS I
Verifier
A traditional proof is an efficiently verifiable static
sequence of symbols that establishes that xL.
NP = cfw_ languages with short
CSci 5403
COMPLEXITY THEORY
LECTURE XVI:
COUNTING PROBLEMS
AND
RANDOMIZED REDUCTIONS
Definition. #P is the class of functions such that
there exists a polytime TM M where
(x) = |cfw_ y : M(x,y) accepts |
Example. #SAT() = # satisfying assignments to .
Def
Bounds on circuit computation so far:
CSci 5403
COMPLEXITY THEORY
LECTURE XXV:
CIRCUIT LOWER BOUNDS
Size Hierarchy Theorem: For every (n) = o(2n/n)
there exists a function with complexity (n)
Shannon bounds: most functions have
complexity (2n/n)
Relations
Bounds on circuit computation so far:
CSci 5403
COMPLEXITY THEORY
LECTURE XXVI:
CIRCUIT LOWER BOUNDS
Size Hierarchy Theorem: For every (n) = o(2n/n)
there exists a function with complexity (n)
Shannon bounds: most functions have
complexity (2n/n)
Relation
CSci 5403, Spring 2010
Exam 1
due: Feb 19, 2010
For each of the following languages, prove the tightest upper and lower bounds you can
on its complexity. For our purposes, an upper bound is a proof that the language resides
in some complexity class, and a
Izaksonas-Smith, Evan
Problem 1.2
Approximate Cuts
Problem
The max-cut problem is dened as follows. Given an undirected graph G = (V, E ), a cut is a set of
vertices S V . The weight of the cut is the number of edges that cross between S and V \ S , i.e.
CSci 5403, Spring 2010
Homework 1
due: Feb 2, 2010
Chung-I Lin
Ruoyu Sun
Yun Zhang
1. Question 1
Given f (n) and g (n) are time constructible, there are two Turing Machine M 1 and M 2 that
compute f (n) and g (n) on input 1n in time O(f (n) and O(g (n) re
CSci 5403, Spring 2010
Homework 8
due: May 6, 2010
1. A calculation. Let X1 , . . . , Xn be boolean independent random variables such that for
all i, Pr[Xi = 1] = , and let X = n=1 Xi (mod 2). Prove that Pr[X = 1] = 1/2+ 1 (12 )n .
i
2
2. Not-so-pseudo. S
CSci 5403, Spring 2010
Homework 7
due: April 27, 2010
1. Fool me once. . . Dene the function fMAJ : cfw_0, 1n cfw_0, 1n cfw_0, 1 to return 1 i at
least n + 1 bits of its input are 1. Give a fooling set of size n for fMAJ .
2. Rank my matrix Let fIP (x, y
CSci 5403, Spring 2010
Homework 6
due: April 15, 2010
1. Isomorphic to Nothing? Prove that the interactive proof system for Graph Nonisomorphism given in section 8.3 is honest-verier perfect zero-knowledge.
2. Predicting versus distinguishing Finish the p
CSci 5403, Spring 2010
Homework 5
due: April 1, 2010
1. Randomized reduction or proof. In Section 7.6, Arora & Barak dene the class
BP NP to be the set of all languages having a randomized reduction to a language in NP.
(A randomized reduction from L1 to
CSci 5403, Spring 2010
Homework 2
due: Feb 16, 2010
1. Games. Prove that the following problems are PSPACE-complete:
(a) A Stochastic Satisability instance is a quantied boolean formula, only the universal
quantier is replaced by the random R quantier (m
CSci 5403, Spring 2010
Exam 3
due: April 21, 2010
You may use the textbook and the class notes and example solutions, but no other sources
to complete this exam.
Graph Coloring. Recall that a graph G = (V, E ) is k -colorable if there exists an assignment
CSci 5403, Spring 2010
Exam 3
due: March 24, 2010
You may use the textbook and the class notes and example solutions, but no other sources
to complete this exam. In particular, you may assume without proof that for every i 0
there exists a eld F2i with 2i
So far we have seen problems of canonical form:
CSci 5403
COMPLEXITY THEORY
LECTURE XV:
COUNTING PROBLEMS
(x)?
9x. (x), 2 2cnf
C(x)?
L
NL
P
9 x. (x)
NP
9 x 8 y (x,y)
PH
(Courtesy Steven Rudich)
#P
PSPACE
Next two lectures:
How many x satisfy (x)?
PRONUNCI
How do the probabilistic complexity classes:
BPP
CSci 5403
COMPLEXITY THEORY
LECTURE XIV:
HOW DOES RANDOMNESS FIT IN?
coRP ZPP RP
Fit in our picture of efficient deterministic classes:
P/poly
PSPACE
PH
coNP
P NC NL
L
NP
Theorem. BPP P/poly.
Proposition. P
CSci 5403
COMPLEXITY THEORY
LECTURE XIII:
WHY CONSTANTS STILL DONT MATTER
RP = cfw_ A | exists polytime TM M:
x 2 A ) Prr[M(x,r) accepts]
x A ) Prr[M(x,r) accepts] = 0.
coRP = cfw_ A | exists polytime TM M:
x2 A ) Prr[M(x,r) accepts] = 1
x A ) Prr[M(x,r