Due Wednesday August 28

0. Assume Alice and Bob follow some communication protocol that is
overheard by that pesky eavesdropper Eve. Alice and Bob would like
that their communication is "cryptographically secure" in that Eve
doesn't learn anything from what she hears. Do you best to come up
with a FORMAL definition of "cryptographically secure". 

You may NOT look in the text, or use any outside sources. The point of this exercise is
not for you to find the "right" definition, but to understand the
difficulties of coming up with such a definition by going through the
thought process yourself. You proposed definition will not be graded
on correctness. If you can't think of a definition, write a paragraph
or two on things that you thought about. Everyone that turns in something
will get full credit here. 

Due Friday August 30

1. Show from first principles that there is no finite state machine
that accepts the language of formulas of the form x+y=z, where
natural numbers x, y and z encoded in binary in the standard way,
and where x plus y is indeed equal to z.
So for example the input "10100+111=11011" is in the language.
So intuitively this shows that finite state machines can
not "add" in the most obvious interpretation of "add".

2. Show that there is a finite state machine that can accept the language over
a string of 8 symbols, that intuitively mean

[000]
[001]
[010]
[011]
[100]
[101]
[110]
[111]

where the first column plus the second column is equal to the third 
column. So to reuse our example from problem 1, the 5 symbol input

[101][001][110][011][011] 

is in the language because 10100+111=11011. 
For simplicity, you may assume that you can read the input tape in 
either direction (though this assumption isn't strictly necessary).
So intuitively this says that finite state machines can "add"
for some reasonable intepretation of "add".


Due Monday September 9

3. Problem 1.7 from the text. Basically you want to show how to simulate a 
Turing Machine with a 2 dimensional tape by a Turing machine with a one 
dimensional tape. 

The purpose of assigning this problem is to gain some some familiarity 
with simulation arguments. 

Suggestion: Use the proof of claim 1.6 in the back of the book both to get an idea of
how to solve this problem, and as a model as to the level of 
detail that you should have in your write-up.


Note: This is problem 1.11 in the online preliminary version
of the book at http://www.cs.princeton.edu/theory/complexity/modelchap.pdf
In general problem numberings will be from the final version of
the text, not the online preliminary version


Due Wednesday September 11


4. Consider a programming language mini-Java that only has one type of
loop, and the number of iterations of the loop must be
determined when the loop is first encountered. So a
loop statement might look like "Repeat n times", and
the variable n is evaluated when the statement is reached.
You can assume that the program has a variable Size that
is instantiated to the input size when the program starts running
(otherwise, you couldn't even read the input).
So all mini-Java programs must halt on all inputs. Show by diagonalization
that there is a language accepted by a Java program that
is not accepted by any mini-Java program. Use Diagonalization.
Give a concrete example of a language that is not acceptable by any mini-Java
program, but that is accepted by a Java program.


The purpose of assigning this problem is to gain some familiarity 
with diagonalization arguments. 

Due Friday September 13

5. Consider the following decision problem. The input consists of a collection
of types of unit squares, four special colors that we will
call L, R, T, and B, and a collection of color rules. 
A UNIT SQUARE TYPE is an ordered 4-tuple of colors.
There is no a priori bound on the number of possible colors,
and you can use as many unit squares as you like that
have these four colors in clockwise order around the outside.
For example, a unit square type might be (red, red, blue, green).
A color rule is a pair of colors meaning that square sides of these
colors are allowed to touch. For example, a color rule might be
(red, blue). Each of the four finite automata take as input 
a string of colors, and either reject or accept the string.
The decision is to determine whether there is a rectangle
filled with available types of unit squares, placed
to obey the color rules, 
where the colors on the top are all T, 
the colors on the left are all L,
the colors on the right are all R, and
the colors on the bottom are all B.
Show that there is no algorithm to solve this problem.

Hint: Recall the proof that the term rewriting problem 
is not computable. Think of row i of the rectangle as being  
the configuration of a  Turing machine after step i. 
Think of a sequence of rows as being a computation history.
Think of many of the colors encoding either some small
string of tape symbols, or a position of the tape head,
the state in the finite state control, and some tape symbols.
The only issue that is not straightforward is how to guarantee
consistency when the tape head moves. To handle this, think
about there being some overlap in information between consecutive
colors on the bottom/top of a row.

Further hint: Note that there is no algorithm to determine whether
a Turing Machine halts when given the empty tape as input. You
might prove this as a warmup, or just look in the class notes.


The purpose of the problem is to get used to the concepts of
configuration and computation history, and proving hardness
via a reduction. 


Due Monday September 16

6. The goal of this problem is to finish up
the proof in class of a version of Goedel's incompleteness
theorem that there is no complete sound axiomatization of standard arithmetic.
Consider an string S that represents the computation history
of a particular Turing machine T, e.g.
S = S1#S2# .... #SK
for a computation that takes K steps. Here Si is the state
on the ith step. The number of symbols J in each Si is the
number of states in T plus the size of the tape alphabet for
T plus one symbol to represent the position of the tape
head. Consider S as a base J integer. Explain how
to express the informal sentence "S represents a valid
computation for T" as a first order logical sentence
using standard arithmetic operators such as plus, minus,
times, division, mod, floor, ceiling, equals, less than, etc
and standard logical operators: and, or, not, etc.
Note that this is required for the proof of that there
is no complete axiomatization of standard arithmetic.


The purpose of this problem is to reinforce the proof of Godel's 
incompleteness theorem, and get comfortable with the arithmetization 
of computation.

Due Wednesday September  25 (email to mpn1@pitt.edu by noon)

7. Consider first order logical sentences of arithmetic where your
are allowed to use = and +. Without loss of generality we may 
assume that the only logical operators are AND and NOT, and
that all quantifiers appear first. So you might have a formula
like: 

THEREEXISTS x FORALL y FORALL z  THEREEXISTS w such that (x+y=z) AND NOT (y+z=w+x)

Our goal here is to show that there is an algorithm to accept
exactly the language of true formula using the following strategy.

Recall from problem 2 that we know how to build a finite state machine
that accept tuples (x,y,z,w) properly encoded 
that satisfy (x+y=z) and a finite state machine
that accepts tuples (x, y, z, w) properly encode that satisfy (y+z=w+x). Explain why if you have
finite state machines for (x+y=z) and (y+z=w+x), you can get a finite
state machine to tuples (x, y, z, w) properly encode that satisfy (x+y=z) AND NOT (y+z=w+x).
Now consider the quantifiers from
the inside out. 
Explain how to construct a finite state machine
that accepts tuples (x, y, z) properly encoded with the property that
THEREEXISTS w such that (x+y=z) AND NOT (y+z=w+x)
Explain how to construct a finite state machine
that accepts tuples (x, y) properly encoded with the property that
FORALL z  THEREEXISTS w such that (x+y=z) AND NOT (y+z=w+x)
Extend this to the whole formula. Then explain how to use the
final finite state machine to get your final answer. 

It is OK for your write up to use this example, but it should also
explain how the algorithm would work on a generic instance.
Concentrate on high level ideas, not low level details.


The purpose of the problem is to increase your comfort with first order logic. 

Due Friday September 27 (email to mpn1@pitt.edu by noon)

8. Recall the definition of entropy of a probability distribution X over a set U:
H(X) = Sum_{x in U}  prob(x) lg (1/prob(x))

Now define the conditional entropy of not necessarily independent 
probability distributions X and Y to be:
H(X|Y) = Sum_{y} prob(y) Sum_{x} prob(x|y) lg 1/prob(x|y)

Subproblem A) Prove that H(X|Y) = Sum_{x,y} prob(x,y) lg 1/ prob(x|y)

Subproblem B)) Explain why intuitively H(X|Y) tells you how much information one gains from
seeing the outcome of X, given that one has already seen the outcome of Y.

Subproblem C) Prove that  H(X) - H(X|Y )= H(Y)-H(Y|X).

Subproblem D) Consider the process of drawing cards without replacement from a deck, and then
looking at the cards in some order. What does the above statement say about how
the order that you look at the cards affects the amount of information that you receive
after each time you look at the cards?


The purpose of this problem is to get more confortable with both the intuition and formal definitions
of entropy. 

Due Monday September 30

9. Assume a log-space reduction from a language A to a language B. 

So more precisely, there is a Turing machine T with three tapes, a read only input tape, 
a read/write work tape, and a write-only output tape. T only uses log of the input size many cells 
on the read/write work tape.  Further T never backs up the tape head on the write only output tape,
so the tape head on the write only tape either stays in position or moves to the right. 
The machine T has the property that a string x in in A iff the contents of the write tape when T ends
computation on input x is in B. 

Now show that if there is a log space Turing machine S that accepts B, then there is a log space
Turing Machine U that accepts A. Note that this is not trivial because U will have enough work tape
to write down the output of T. This is a good problem to show how to make use of the fact that
space is reusable.

Due Wednesday October 2

10. Consider the following game played by two players A and B on a directed graph G 
with a designated start vertex s. On the first move, player A picks an edge (s, v) leaving s.
Then s and v are designated as visited.  On each even numbered move, player B picks an edge (v, w)
from the current vertex v to a vertex w hasn't been visited before.  If no such vertex w exists,
then player B loses. Vertex w then is designated as visited and becomes the current vertex. 
On each odd numbered move, player A picks an edge (v, w)
from the current vertex v to a vertex w hasn't been visited before.  If no such vertex w exists,
then player A loses. Vertex w then is designated as visited and becomes the current vertex. 
So the players A and B taking turns picking edges in a directed path P starting at s, with a player
losing if he/she can't extend the path. Prove that determining for a particular G and s, whether
the first player has a winning strategy is PSPACE-complete under polynomial time reductions. 
You can prove hardness by reduction from the problem of determining whether a quantified Boolean formula F is true. 

Hint: Conceptually the graph G will have two parts, G_1 and G_2, that P will traverse in that order.
In some sense G_1 represents the quantifiers in F. So there will one vertex in G_1 for each quantifier
in F. And for each such vertex, there will be two exiting edges corresponding to true and false. 
Then G_2 needs to verify that there are choices for the first player (corresponding to the existential quantifications
in F) in G_1, such that for all choices of the second player (corresponding to the universal quantifications)
in G_1, the first player will win iff F is true. 


Due Friday October 4

11. Show that if SAT is polynomial-time many-to-one reducible to the complement of SAT that the
polynomial time hierarchy is equal to NP. This is problem 5.3 from the text. The purpose of the
problem is to get comfortable with the definition of the polynomial time hierarchy. The problem
is almost trivial if one was comfortable with this definition.

Due Monday October 7 (email to Michael mpn1@pitt.edu by noon)


12.  The time hierarchy theorem leaves open the possibility that Time(n) = Time(n log log n).
Show that it is unlikely that one can show that  Time(n) = Time(n log log n) by simulation
by showing that there is an language B such that Time(n)^B is not equal Time(n log log n)^B.

Hint: Try to mimic the proof that there is a language B such that P^B is not equal NP^B.
First ask yourself what question about B can a Time(n log log n) machine can easily answer that a 
TIME(n) machine can not answer. 

Obviously the purpose of this problem is to get more comfortable with the Baker, Gill and Soloway result.

Due Friday October 11 (email to Michael mpn1@pitt.edu by noon)

13. Show that VC-Dimension is Sigma_3^p-complete. This is exercise 5.13 in the hard copy fo the book.
And exercise 11 in the online preliminary copy. 

Hint: The online copy of the text has a suggestion.  I'm not sure how hard this problem is (so it may be hard). 
The purpose of the problem is to gain comfort with the concept of completeness and the definition
of the polynomial time hierarchy. 

Due Monday October 14

14. Show that NC^1 is a subset of LOGSPACE. 

Hint: This of the NC^1 computation as a circuit. Show how to compute the value of the circuit using log space. DFS.


Due Wednesday October 16

15.  Problems 6.6 and 6.7 from the text. They are related and quite easy.

	6.6: Show that for every k>0 that Sigma_2^p has languages whose circuit complexity is Omega(n^k).
	The circuit complexity is the size of the minimum sized circuit that decides the language.

	6.7: Show that if P=NP, then there is a language in exponential time that requires circuits of size Omega(2^n/n). 
	Hint: The answer for 6.6 should be useful.

Duw Friday October 18

16. Prove that ZPP = RP intersect co-RP

Due Monday October 21

17. Problem 7.7 from the text. This is problem 5 in chapter 7 in the onnline preliminary version of the text. 


Due Wednesday October 23

18.  This homework is to be done idividually, not in your group.
Read the history of the IP=PSPACE result given in http://www.cs.pitt.edu/~kirk/cs2110/BabaiIPStory.pdf
Write a paragraph explaining what you think was the right way to handle the publication issue.
That is, "what papers should have come out of this"? and "who should the authors have been"?

Due Monday October 28

19. Problem 8.1 from the text. This is the same as problem 1, minus part c, in Chapter 8 in the online 
preliminary version of the text.

Due Wednesday October 30

20.  Show that MAM is a subset of AM. Note that this is a special case of 8.7 in the
text, and that there is a hint in the book.


Due Monday November 4 (email to the TA by noon)

21. Problem 9.4 from the text. I strongly encourage you to first do the proof for when n=1. 

Due Friday Nobember 8 (email to the TA by noon)

22. 9.5 from the text. 

Due Wednesday November 13

23. Problem 9.15 from the text

Due Friday November 15

24. Read and understand the protocol in problem exercise 9.17 part b 
from the text. Write a one paragraph summary of the protocol
in your own words. Then write a paragraph explaining intuitively
what it means for an interactive proof to be computationally zero knowledge.
Then write a paragraph to explain why it is intuitively believable
why this protocol is computationally zero knowledge. 


25.  Recall the 1/2 silvered mirror experiment, which was essentially
a Hadamard gate, followed by a "not", followed by a Hadamard gate.
In this experiment the state of the bit coming out of the Hadamard gate
was the same as the state of the bit entering if the entering bit
was not in superposition. Show by basic calculations that this remains true
if the entering bit is in superposition. 

Due Monday November 18


26.  
part a) Show that if you have a 2-bit system in state
a|00> + b | 01> + c |10> + d |11>. Consider two experiments,
in one you measure the first qubit, and then measure the second qubit,
and in the other,  
you measure the second qubit, and then measure the first qubit.
Show that the probability distributions that the classical bits
that you observe in the end is independent the order that you look
at the bits. This is very easy, but and is just a warm up problem
to get your brain flowing.
part b)Work out in detail the probability that Alice and Bob
observe a=b in the case that x=y=1 in the EPR experiment
in the book. Show every step of your calculations, and give
lots of explanation.

Due Wednesday November 20

27.  Consider the same set up as the parity game. Alice and
Bob split two entangled bits. Alice and Bob are then
split up. Alice is then given 2 classical bits x and y. 
Depending on the value of x, and y, Alice sends Bob 1
qubit. From this single qubit, Bob determines with certainty
the two classical bits x and y from his entangled bit
and the sent qubit. Explain how to accomplish this. 
You need to explain Alice's encoding strategy, Bob's decoding
strategy, and why these strategies work. Note that you can
extend this to allow for the transmission of 2n classical 
bits using only n quantum bits.

Hint: The 4 possible states of the sent qubit will be
+ 1/sqrt(2) |0> +   1/sqrt(2) |1>
+ 1/sqrt(2) |0> -   1/sqrt(2) |1>
- 1/sqrt(2) |0> +   1/sqrt(2) |1>
- 1/sqrt(2) |0> -   1/sqrt(2) |1>



Due Friday November 22

28. The goal of this problem is to find a way to transmit information
about a qubit by sending two classical bits. Alice and Bob split up
entangled bits |ab| in state |00> + |11>. Assume that now Alice is given 
qubit x. So x is in some unknown superposition between states |0> and |1>.
Alice now performs the following reversible operation on x, if a=1 then 
negate x. Alice then runs qubit a through a Hadamard gate. Alice now measures
the current values of a and x, and sends these two classical bits to Bob. 
Explain what the state of all the particles a, b, and x is after each of
Alice's operations. Then explain how Bob can use the two classical particles
to change the state of b to the original state of x.
Note that when Alice measures a and x, then she can no longer recover
the original state of qubit x.


Due Monday November 25


29. Problem 11.2 from the text. This is very easy.

30. Problem 11.7 from the text. This should be pretty straightforward.
Just convert the interactive proof for the permanent from chapter 8
as the interactive proof for graph non-isomorphism was converted 
in example 11.7.

Due Wednesday November 27 (There is no class on this day. Email to 
the TA by 10AM)

31. Problem 11.16 from the text. Note that the problem allows
for rational solutions.

Hint: There is a simple reduction from MAXSAT.


Due Monday Monday December 2  (There is no class on this day. Email to 
the TA by 10AM)


32. Problem 13.19 from the text