CS 151
Complexity Theory
Spring 2013
Final Solutions
Posted: June 10
Chris Umans
1. (a) The procedure that traverses a fan-in 2 depth O(logi n) circuit and outputs a formula
runs in Li this can be done by a recursive depth-rst traversal, which only requir
CS 151
Complexity Theory
Spring 2011
Solution Set 5
Posted: May 16
Chris Umans
1. We are given a Boolean circuit C on n variables x1 , x2 , . . . , xn with m , and gates. Our
3-CNF formula will have m auxiliary variables z1 , z2 , . . . , zm in addition t
CS 151
Complexity Theory
Spring 2011
Solution Set 4
Posted: April 28
Chris Umans
If you are turning in this problem set late, obviously you shouldnt consult these solutions.
1. Consider a language L ZPP decided by a machine M that runs in expected time nk
CS 151
Complexity Theory
Spring 2011
Solution Set 3
Posted: April 25
Chris Umans
1. (a) Note: it is most convenient to think of as the permutation k ( (k ) rather than
the more conventional k ( (k ) the two notions are equivalent by taking inverses;
howev
CS 151
Complexity Theory
Spring 2011
Solution Set 2
Posted: April 18
Chris Umans
1. Suppose L NP coNP. Then there exist languages R1 and R2 in P for which
L = cfw_x : y, |y | |x|k1 , (x, y ) R1
L = cfw_x : z, |z | |x|k2 , (x, z ) R2
On input x, our stro
CS 151
Complexity Theory
Spring 2011
Solution Set 1
Posted: April 11
Chris Umans
1. Let A be a language that is downward self-reducible. Given an input x, we simulate the
polynomial-time computation that (with queries) decides A, and recursively compute t
CS 151
Complexity Theory
Spring 2011
Midterm Solutions
Posted: May 5
Chris Umans
1. Consider a language L coNEXP. On an input of length n, the advice will be an exact
count of the number of inputs of length n not in the language. This is a number between
CS 151
Complexity Theory
Spring 2011
Problem Set 7
Out: May 19
Due: May 26
Reminder: you are encouraged to work in groups of two or three; however you must turn in your
own write-up and note with whom you worked. You may consult the course notes and the t
CS 151
Complexity Theory
Spring 2011
Problem Set 6
Out: May 12
Due: May 19
Reminder: you are encouraged to work in groups of two or three; however you must turn in your
own write-up and note with whom you worked. You may consult the course notes and the t
CS 151
Complexity Theory
Spring 2011
Problem Set 5
Out: May 5
Due: May 12
Reminder: you are encouraged to work in groups of two or three; however you must turn in your
own write-up and note with whom you worked. You may consult the course notes and the te
CS 151
Complexity Theory
Spring 2011
Problem Set 4
Out: April 21
Due: April 28
Reminder: you are encouraged to work in groups of two or three; however you must turn in your
own write-up and note with whom you worked. You may consult the course notes and t
CS 151
Complexity Theory
Spring 2011
Problem Set 3
Out: April 14
Due: April 21
Reminder: you are encouraged to work in groups of two or three; however you must turn in your
own write-up and note with whom you worked. You may consult the course notes and t
CS 151
Complexity Theory
Spring 2011
Problem Set 2
Out: April 7
Due: April 14
Reminder: you are encouraged to work in groups of two or three; however you must turn in your
own write-up and note with whom you worked. You may consult the course notes and th
CS 151
Complexity Theory
Spring 2011
Problem Set 1
Out: March 31
Due: April 7
Reminder: you are encouraged to work in groups of two or three; however you must turn in your
own write-up and note with whom you worked. You may consult the course notes and th
CS 151
Complexity Theory
Spring 2011
Midterm
Out: April 28
Due: May 5 at the beginning of class
This is a midterm. You may consult any of the course materials and the text (Papadimitriou), but
not any other source or person. Please attempt all problems, a
CS151
Complexity Theory
Course
Summary
Lecture 18
May 26, 2011
May 26, 2011
Course summary
Course summary
Time and space
Non-determinism
hierarchy theorems
FVAL in L
CVAL P-complete
QSAT PSPACE-complete
succinct CVAL EXP-complete
May 26, 2011
2
NT
CS 151
Complexity Theory
Spring 2011
Solution Set 6
Posted: May 23
Chris Umans
1. (a) We observe that the largest possible set shattered by a collection of 2m subsets is m,
since a set of size m + 1 has more than 2m distinct subsets. The VC dimension of a
CS 151
Complexity Theory
Spring 2011
Solution Set 7
Posted: May 26
Chris Umans
Obviously, if you have not yet turned in Problem Set 7, you shouldnt consult these solutions.
1. (a) We describe R separately for strings x of each length. Consider strings x o
Complexity Theory
Classify problems according to the
computational resources required
CS151
Complexity Theory
Lecture 1
April 2, 2013
running time
storage space
parallelism
randomness
rounds of interaction, communication, others
Attempt to answer: what is
Extended Church-Turing Thesis
consequence of extended Church-Turing
Thesis: all reasonable physically realizable
models of computation can be efficiently
simulated by a TM
CS151
Complexity Theory
Lecture 2
April 4, 2013
e.g. multi-tape vs. single tape T
Robust Time and Space Classes
Robust time and space classes:
CS151
Complexity Theory
L = SPACE(log n)
PSPACE = k SPACE(nk)
Lecture 3
April 9, 2013
P = k TIME(nk)
k
EXP = k TIME(2n )
April 9, 2013
Relationships between classes
A P-complete problem
So far
Ladners Theorem
Assuming P NP, what does the world
(inside NP) look like?
CS151
Complexity Theory
NP:
NPC
NPC
P
Lecture 4
April 11, 2013
NP:
P
April 11, 2013
Ladners Theorem
2
Ladners Theorem
Theorem (Ladner): If P NP, then there
exists L NP that is neit
CS 151
Complexity Theory
Spring 2013
Midterm Solutions
Posted: May 9
Chris Umans
1. Consider a language L coNEXP. On an input of length n, the advice will be an exact
count of the number of inputs of length n not in the language. This is a number between
CS 151
Complexity Theory
Spring 2013
Solution Set 1
Posted: April 11
Chris Umans
1. Let A be a language that is downward self-reducible. Given an input x, we simulate the
polynomial-time computation that (with queries) decides A, and recursively compute t
CS 151
Complexity Theory
Spring 2013
Solution Set 2
Posted: April 18
Chris Umans
1. Suppose L NP coNP. Then there exist languages R1 and R2 in P for which
L = cfw_x : y, |y | |x|k1 , (x, y ) R1
L = cfw_x : z, |z | |x|k2 , (x, z ) R2
On input x, our stro
CS 151
Complexity Theory
Spring 2013
Solution Set 3
Posted: April 25
Chris Umans
1. (a) Note: it is most convenient to think of as the permutation k ( (k ) rather than
the more conventional k ( (k ) the two notions are equivalent by taking inverses;
howev
CS 151
Complexity Theory
Spring 2013
Solution Set 4
Posted: May 2
Chris Umans
If you are turning in this problem set late, obviously you shouldnt consult these solutions.
1. Consider a language L ZPP decided by a machine M that runs in expected time nk fo
CS 151
Complexity Theory
Spring 2013
Solution Set 5
Posted: May 16
Chris Umans
1. We are given a Boolean circuit C on n variables x1 , x2 , . . . , xn with m , and gates. Our
3-CNF formula will have m auxiliary variables z1 , z2 , . . . , zm in addition t
CS 151
Complexity Theory
Spring 2013
Solution Set 6
Posted: May 23
Chris Umans
1. (a) We observe that the largest possible set shattered by a collection of 2m subsets is m,
since a set of size m + 1 has more than 2m distinct subsets. The VC dimension of a
CS 151
Complexity Theory
Spring 2013
Solution Set 7
Posted: May 30
Chris Umans
Obviously, if you have not yet turned in Problem Set 7, you shouldnt consult these solutions.
1. (a) Let pi = Pry [f (x + y ) f (y ) = i]. The probability two random voters dis