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\begin{center}
	Honors Theory, Spring 2002, Yap\\
	Homework 4\\
\end{center}

Exercises on machines models (Turing, Pointer).
Note that you do not need to hand in problems that has
0 points. 

This is due on Monday March 18.  But for any problem that
you hand in BEFORE the midterm, I will give you the solution
to that problem.

\begin{enumerate}

\item (0 Points)
In the lecture notes and in class, we show that
for any $c\in\NN$ and any $k$-tape $M$, we can
construct a $(k+1)$-tape $M'$ with the following property:
(i) each tape symbol of $M'$ encodes $c$ consecutive tape symbols of $M$, 
(ii) $M'$ begins by converting the input word
into this compressed from on its extra tape (so that it can now
read $c$ input symbols at a time), and
(iii) $M'$ simulates $c$ steps of $N$ using only $6$ steps.

Can you reduce the number ``$6$''?
\\ (i) Improve the number $6$ in case $k=1$.
\\ (ii) When $k>1$, can you improve $6$ to $5$?

\ans{}{
}

\item (5+15 Points)
Construct a pointer machine PM
that adds two dyadic numbers.
\\
(i) Describe the dyadic algorithm for addition.
Is it simpler or more complex than binary addition?
\ans{}{ We need to consider 7 possibilities:

	0+0=1, 0+1=00, 1+1=01,
	0+0+0=00, 0+0+1=01, 0+1+1=10, 1+1+1=11
}
\\
(ii) We need some I/O conventions for pointer machines (PM's).
Let $\Delta$ be the \dt{alphabet} for the PM, and $G$ be the
computation graph of the PM.
If $w\in \Delta^*$, let $[w]_G$ denote the node in $G$ represented by $w$.
For instance, the empty string $\eps$ represents the center $[\eps]$.
Assume a special symbol $\beta\in\Delta$ that denotes
the \dt{bit value} of any node $w\in\Delta^*$ in the following sense:
if $[w.\beta] = [\eps]$, then we say that the node $[w]$ stores
the $1$ bit, else it stores tbe $0$ bit.
How do we represent a list of nodes?
Assume a special ``next symbol'' $\nu\in\Delta$ such
that $[w.\nu]$ represents the next node after $[w]$,
provided $[w,\nu]\neq [\eps]$; otherwise we say
there is no next node.  Define the list headed by $w$ to be the sequence
	$$L(w) \as ([w], [w\nu], [w\nu^2]\dd [w\nu^{n-1}]),$$
terminated by the first $n$ such that $[w\nu^n]=[\eps]$.
The list $L(w)$ may be infinite (in the case of a loop).
The binary string represented
by $w$ is simply the sequence of bit string stored in $L(w)$.
Input are indicated by two linear lists $L(\alpha)$
and $L(\alpha^2)$, where $\alpha\in\Delta$ is a special ``input symbol''.
When the computation ends, we want
the output string to stored in the list $L(\alpha^3)$.

\item (30 Points)
Verify the Polynomial Simulation Thesis (L), for
Turing machines and Pointer machines.  We want to verify two
main cases, depending on the mode $\mu$ and resource $\rho$:
we want you to verify:
\\ (i) $(\mu,\rho)$ = (Fundamental Mode, Time).  More
precisely, $\dtime(n^{O(1)}) = \dtime_{PM}(n^{O(1)})$.
The latter class is the class of languages accepted by PM's
in deterministic polynomial time.
\\ (ii) $(\mu,\rho)$ = (Nondeterministic Mode, Space).
That is $\nspace(n^{O(1)}) = \nspace_{PM}(n^{O(1)})$.

Please only sketch the simulations,
at a fairly high level (but you should give
details when necessary).  Try not to write more
than 4 pages for this problem.
[Remember the words of the wise speaker,
``if I have more time, I would have given a shorter speech''.]

\item (10x3 Points)
% Problem 2.16
There are three possible responses to the
following statements: True, False or Perhaps.
``Perhaps'' reflects the uncertainty that only
assume the results in chapter 2.
If your answer is True or False, you must give
brief reasons in order to obtain full credit.
% 
\\ (a) T/F/P: $\nspace(n)\ib \dtime(O(1)^n)$.
% ANS: T: $NSPACE(n)\ib DTIME(n^3 O(1)^n)=DTIME(O(1)^n)$ 
\\ (b) T/F/P: $\ntime(n)\ib \dtime(2^n)$.   
% ANS: P: unknown.    
\\ (c) T/F/P: $\nrev(n)\ib \drev(O(1)^n)$. 
% ANS: F: $DREV(2)=RE$.
%


\end{enumerate}

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