CMU 15-251
Spring 2015
Homework 2 Writing session on Wednesday January 28
NOTE: On this homework, you may only use facts about regular languages that we proved in
class. More precisely: You may use the fact that if L1 and L2 are regular then so is L1 L2 .
15-251: Great Theoretical Ideas In Computer Science
Homework 8: Kingdome Maths
(due 23:59 Wednesday, March 19)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle task is t
CMU 15-251
Spring 2015
Midterm 2 Wednesday March 18
Name:
Andrew ID:
Section:
Instructions:
You have 180 minutes.
You may hand in your midterm and leave early, but you cannot leave within the last 10
minutes.
You are not allowed to use books, notes, or
CMU 15-251
Spring 2015
Homework 4 Writing session on Wednesday February 18
The notation log means log2 .
1. (SOLO) Prove the following:
(a) n2 is not (n3 )
Let n0 , c be given.
1
Choose n > n0 and n > .
c
Then, note that 1 < nc.
Multiplying by n2 gives us
CMU 15-251
Spring 2015
Homework 5 Writing session on Wednesday February 25
1. Critique the Proof. (SOLO.)
(a) Consider the following claim and its proof.
Claim: Any graph with n vertices and n 1 edges is a tree.
Proof: We prove the claim by induction. The
CMU 15-251
Spring 2015
Homework 1 Writing session on Wednesday January 21
0. (SOLO) Carefully read the course policies at http:/www.cs.cmu.edu/~15251
Pls read.
1. (SOLO) Prove (by structural induction) that in any propositional formula, the number of
vari
CMU 15-251
Spring 2015
Homework 2 Writing session on Wednesday January 28
NOTE: On this homework, you may only use facts about regular languages that we proved in
class. More precisely: You may use the fact that if L1 and L2 are regular then so is L1 L2 .
CMU 15-251
Spring 2015
Homework 3 Writing session on Wednesday February 4
On this first page of the homework, we make some definitions that may be referred to in the
subsequent problems. Let M be a Turing machine where Q is the state set, t is the blank s
CMU 15-251
Spring 2015
Homework 10 Writing session on Wednesday April 22
1. Powers of Transition Matrix. (SOLO) Let K be the transition matrix of a Markov
chain with states cfw_1, 2, . . . , n. Let 0 denote some initial probability distribution on the
sta
CMU 15-251
Spring 2015
Homework 5 Writing session on Wednesday February 25
1. Critique the Proof. (SOLO.)
(a) Consider the following claim and its proof.
Claim: Any graph with n vertices and n 1 edges is a tree.
Proof: We prove the claim by induction. The
CMU 15-251
Spring 2015
Homework 3 Writing session on Wednesday February 4
On this first page of the homework, we make some definitions that may be referred to in the
subsequent problems. Let M be a Turing machine where Q is the state set, t is the blank s
CMU 15-251
Spring 2015
Homework 9 Writing session on Wednesday April 8
1. Exercise with RSA. (SOLO)
(a) Recall the RSA crypto system. Well encode and decode a simple message using RSA.
The message is allowed to be any string containing only lower case let
CMU 15-251
Spring 2015
Homework 8 Writing session on Wednesday April 1
1. Number theory practice (SOLO).
(a) Compute g = GCD(179, 138), and express g = k 179 + ` 138 for some integers k and `.
g = 1 = 37 179 + 48 138
(b) Show that if p and q are distinct
CMU 15-251
Spring 2015
Homework 7 Writing session on Wednesday March 25
1. Americas Favorite Binary Tree (SOLO). 64 teams compete in a single-elimination
basketball tournament. If you dont know what single-elimination tournament means,
please look it up o
CMU 15-251
Spring 2015
Homework 1 Writing session on Wednesday January 21
0. (SOLO) Carefully read the course policies at http:/www.cs.cmu.edu/~15251
1. (SOLO) Prove (by structural induction) that in any propositional formula, the number of
variable occur
CMU 15-251
Spring 2015
Midterm 3 Wednesday April 15
Name:
Andrew ID:
Section:
Instructions:
You have 180 minutes.
You may hand in your midterm and leave early, but you cannot leave within the last 10
minutes.
You are not allowed to use books, notes, or
CMU 15-251
Spring 2015
Midterm 1 Wednesday February 11
Name:
Andrew ID:
Section:
Instructions:
You have 180 minutes. The exam is out of 70 points.
You may hand in your midterm and leave early, but you cannot leave within the last 10
minutes.
You are no
CMU 15-251
Spring 2015
Homework 6 Writing session on Wednesday March 4
For this homework, you may assume that the following problems are NP-complete:
CIRCUIT-SAT: Given a Boolean circuit with AND, OR, and NOT gates, is there an assignment to the input var
15-251: Great Theoretical Ideas In Computer Science
Homework 12 (due 23:59 Wednesday, April 23)
Directions:Write up carefully argued solutions to the following problems. The rst task is to be complete
and correct. The more subtle task is to keep it simple
15-251: Great Theoretical Ideas In Computer Science
Homework 13 (due 23:59 Wednesday, April 30)
Directions:Write up carefully argued solutions to the following problems. The rst task is to be complete
and correct. The more subtle task is to keep it simple
15-251: Great Theoretical Ideas In Computer Science
Homework 4 (due 23:59 Wednesday, February 11)
Directions:Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle task is to keep it simp
15-251: Great Theoretical Ideas In Computer Science
Test 2. April 2, 2014. Sleator & Sutner
Name:
Andrew ID:
Section:
INSTRUCTIONS:
WRITE YOUR NAME and ANDREW ID ABOVE.
This is a closed book test. You may not use notes. You may not use a calculator.
If
15-251: Great Theoretical Ideas In Computer Science
Homework 5: DragonBall Graph Theory
(due 23:59 Monday, February 17)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle
15-251: Great Theoretical Ideas In Computer Science
Homework 7 (due 23:59 Wednesday, March 5)
Directions:Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle task is to keep it simple a
15-251: Great Theoretical Ideas In Computer Science
Homework 6 (due 23:59 Wednesday, February 26)
Directions:Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle task is to keep it simp
15-251 Assignment 2
Page 1 of 7
15-251 : Great Theoretical Ideas In Computer Science
Spring 2014
Assignment 2: In the Game of Proofs, You Prove or You Fail
Due: Wednesday, Jan. 29, 2013 11:59 PM
Question:
1
2
3
4
5
6
Total
Points:
10
10
10
30
20
20
100
Sc
15-251: Great Theoretical Ideas In Computer Science
Homework 9 (due 23:59 Wednesday, March 26)
Directions:Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle task is to keep it simple
15-251: Great Theoretical Ideas In Computer Science
Homework 3 (due 23:59 Wednesday, February 5)
Directions: Write up carefully argued solutions to the following problems. The rst task is to be complete and correct. The more subtle task is to keep it simp