CSE596 Fall 2011: Solution to Homework 1
September 20, 2011
A formula F is a tautology every assignment to its variables satises it
no assignment to its variables satisfy F
1.
F is not satisable
2. First of all, note that every variable (or its negation
CSE596 Assignment 2 Answer Key Fall 2014
[Make sure you read the machine diagrams in a fixed-width font.]
(A) For finite automata M, if delta(q,c) is undefined, then (even if q is in F) make a new "dead state" q_d and define delta(q,c) = q_d. Or put anoth
CSE596 Assignment 1 Answer Key Fall 2014
Answers to the study-guide questions come first. They are both brief and full.
(A) The main difference is that a test of the predicate G(n) = "2n+2 is the sum of two primes" will always halt within a known amount o
CSE596, Fall 2014
Problem Set 9
Due Fri. Nov. 14
Reading for next week. Lectures through next Wednesday will still be in Homer-Selman chapter 6
and the long section 4 of the corresponding ALR material (Chapter 28: Reducibility and Completeness) on NP-comp
CSE596, Fall 2014
Problem Set 10
Due Fri. Dec. 5
Reading for next week. The snow day has compressed the nal week of class, but the plan is
the same as what was said before: Homer-Selman chapter 7, sections 7.2, 7.3.1. (We have already
covered 7.5.1, and 7
CSE596, Fall 2014
Problem Set 8
Due Wed. Nov. 5
Reading for next week. Homer-Selman chapter 6 all (skimming the long proof of the
Cook-Levin theorem), in-tandem with reading the long section 4 of the corresponding ALR
material (Chapter 28: Reducibility an
CSE596, Fall 2014
Problem Set 7
Due Wed. Oct. 29
Lectures and Reading. Friday through Monday will be in section 5.4. Savitchs Theorem
(5.13 in the text) will be postponed until after we cover Chapter 6 as stated before, and the
Immerman-Szelepcsnyi Theore
CSE596, Fall 2014
Problem Set 6
Due Wed. Oct. 22
Lectures and Reading. Text, chapter 5, sections 5.15.4, and the ALR Complexity
Classes chapter up through section 2.5. The most imp[ortant theorem is labeled Theorem 2.3 (a-c) in the ALR notes, and correspo
CSE596, Fall 2014
Problem Set 4
Due Wed. Oct. 1
Lectures and Reading. We are still in sections up through 3.5 next week. Please also (for
Monday; there will be a preview in Fridays lecture) read the handout on diagonalization on
the course webpage, and re
CSE596, Fall 2014
Problem Set 5
Due Wed. Oct. 8
Lectures and Reading. I have decided to cover the S-m-n Theorem in section 3.6 but skip the
Recursion Theorem in sections 3.7 and 3.10. Hence please read the rst half of the handout on
the course webpage tha
CSE596, Fall 2014
Problem Set 3
Due Wed. Sept. 24
Lectures and Reading. For next week, tackle Chapter 3 up through section 3.5. I prefer to
cover Theorem 3.5 in-tandem with section 3.2, before details from sections 3.3 and 3.4. Put
another way, I prioriti
CSE596, Fall 2014
Problem Set 1
Due Fri. Sept. 12
Lectures and Reading. This weeks lectures are covering formal languages (section 1.1
plus aspects of 1.5) and regular expressions (notes on webpage), then moving on to Turing
Machines (sections 2.12.3). Se
CSE596 Problem Set 3 Answer Key Fall 2014
(A) Over alphabet cfw_0,1, take A = cfw_x: #0(x) != #1(x). Then A is nonregular (its complement is a basic example), but A.A is regular. At first I thought A.A = cfw_0,1^*, but there are some strings like 01010 th
CSE596 Problem Set 4 Answer Key Fall 2014
(A) If there were one clear best definition, everyone would use it. (Oops-that's 11 words.:-)
With \pi_L, however, 1q_cfw_acc is the only possible halting configuration, so there is nothing that compares "illogica
CSE596
Problem Set 5 Answer Key
Fall 2014
(A) Show that D many-one reduces to EQ = cfw_ e#j : L(Me ) = L(Mj ) . This means that given
any machine Mi we want to compute f (i) = (e, j) such that L(Me ) = L(Mj ) if and only if Mi
does not accept i. As before
CSE596 Fall 2011: Solution To Homework 2
September 25, 2011
1. (a) A one-taped Turing machine that recognizes cfw_0n 1n 0n |n 0 works
as follows :
i. It rst scans the input to check if it is in the form of 0 1 0 .
If not, REJECT.
ii. It scans the input fr
CSE596
Problem Set 2 Answer Key
Fall 2012
(1) Prove that the language cfw_ xay : y = xR is non-regular. Use the Myhill-Nerode technique
from notes and lecture. (12 pts.)
Answer: Take S = b a. Clearly S is innite. Let any u, v S with u = v be given.
Then
CSE596
Problem Set 3 Answer Key
Fall 2012
(1) Prove that the Myhill-Nerode dimension of the language P of palindromes over the
binary alphabet = cfw_ a, b is 2m (which incidentally is the maximum possible value).
Proof: Take S = m . Then |S | = 2m , so i
CSE596
Problem Set 4 Answer Key
Fall 2012
(1) Create a language L such that L is decidable, but Prefs (L) is undecidable. Prove your
answer. (18 pts.)
Answer: Take L to be the language V of valid accepting computations c of a universal Turing
machine MU .
CSE596
Problem Set 5 AnswerKey
Fall 2012
(1) Prove that the following decision problem Crash is undecidable, by reduction from the
Acceptance Problem or Halting Problem:
Instance: A Turing machine M with a single one-way-innite tape, as in the text;
Quest
CSE596, Fall 2012
Prelim I Answer Key
Oct. 10, 2012
Open book, open notes, closed neighbors, 48 minutes. The exam totals 80 pts., subdivided as
shown. Do all three problems in the exam booklets provided. Show your work this may
help for partial credit.
Pr
CSE596
Problem Set 9 Answer Key
Fall 2014
(A) Show that NP = co-NP if and only if there is a language A NP co-NP such that A
is NP-complete under p . (3 study-guide points)
m
Answer: If NP = co-NP then taking A = SAT gives an NP-complete language in NPco-
CSE596, Fall 2014
Problem Set 10
Due Fri. Dec. 5
(A) Show that NP co-NP equals the set of oracle languages B such that NPB = NP. (3 pts.)
First suppose NPB = NP. Since B and B always belong to PB , they belong to NPB , hence to
NP. But B, B NP is equivale
CSE596
Problem Set 7 Answer Key
Fall 2014
(A) What is the aw in the apparent proof by diagonalization that DLBA = DLBA? It is
that the DLBAs Me being simulated can have arbitrarily large alphabets e , but the machine M
trying to accept D = cfw_ e : e L(Me
CSE596
Problem Set 8 Answer Key
Fall 2014
(A) Text, page 122, Homeworks 5.14 (which was close to covered in lecture regarding padding
and translation) and 5.16. The answers can be brief; note that the texts LBA is what I called
NLBA and equals NSPACE[n].
CSE596
Problem Set 6 Answer Key
Fall 2014
(A) Instance: A one-tape Turing machine M ; Question: Does there exist an input
x such that M (x) overwrites a blank cell by a non-blank character? This changes the exam
problem (3a) by quantifying x rather than x
CSE596, Fall 2014
Problem Set 2
Due Fri. Sept. 19
Lectures and Reading. After coverage of the Myhill-Nerode Theorem on Monday continuing into Wednesday, next week will transit back to the text in Chapter 2. Hence read the
rest of Chapter 2. In Section 2.5