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\newcommand{\np}{\mathop{\rm NP}}
\newcommand{\conp}{\mathop{\rm co-NP}}
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\newcommand{\faux}{{f_{\rm aux}}}
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\begin{document}

\noindent {\large Advanced Complexity Theory \hfill 
Madhu Sudan\\[-.1in]
6.841/18.405J \hfill  \mbox{}\\[-.1in]
Due:  Wednesday, May 8, 2002\hfill \mbox{}\\[.4in]}
{\LARGE \centering Problem Set 4 \\[.4in] \par}


\section*{Problems}


% Amplification of AM, IP & MIP.


\begin{enumerate}
% Permanent is in IP.
\item
\begin{enumerate}
\item Show that if co-SAT is in AM (one-round interactive
proofs with public coins), then the PH collapses.
\item Show that AMAM (the class of languages with two
round interactive proofs with public coins) is contained
in AM.
\item Show that AM with two-sided error is contained in AM with
one-sided error.
\end{enumerate}

\item 
\begin{enumerate}
\item In the lecture on showing PSPACE $\subseteq$ IP, we
showed that evaluation of straightline programs of polynomials 
with width $w = 2$ is in IP. (I.e., the $i$th
polynomial can be computed with two oracle calls to the
$(i-1)$th polynomial.) Generalize this to the case of 
straightline programs of polynomials with arbitrary 
(polynomial) width.
\item Using Part (a), give a direct proof that the permanent
is in IP.
\end{enumerate}

\item 
A $p$-prover 1-round proof system consists of a probabilistic
polynomial time verifier $V$ interacting with $p$ provers
$P_1,\ldots,P_p$.
On input $x$ of length $n$, the verifier tosses $r(n)$ coins,
generates queries $q_1,\ldots,q_p$ and send $q_i$ to $P_i$
who responds with $a_i$, a string of length $a(n)$. 
The verifier then determines whether
to accept or not based on $x$, the random string and the answers
$a_1,\ldots,a_p$. Completeness and soundness are defined as usual.

Show that SAT has a 3-prover 1-round proof system with perfect
completeness, soundness bounded away from $1$, where the verifier
tosses $O(\log n)$ coins and the answers are $\poly log n$ bits
long.

You may use the following version of the low-degree 
test. 

Let $\L_m$ be the space of lines in $\F^m$ and let
$\F^{(d)}[x]$ denote the set of univariate polynomials of 
degree at most $d$.
Given a function $f:\F^m\to\F$, let $\faux: \L_m \to \F^{(d)}[x]$
be a function that maps lines in $\F^m$ to degree $d$ polynomials.
Consider the test that picks a random line $\ell \in \L_m$ and
a point $x \in \ell$ at random, and then verifies if 
$(\faux(\ell))(x)$ agrees with $f(x)$. Then this test has the 
following properties:
\begin{description}
\item[Completeness] If $f$ is a degree $d$ polynomial, then 
there exists a function $\faux$ such that the test accepts
with probability $1$.
\item[Soundness] There exists $\delta_0 > 0$ such that
the following is true:
If $f$ differs from every degree $d$ polynomial in at least
$\delta$-fraction of the places, then for every $\faux$
the test rejects
with probability at least $\min\{\delta_0, \delta/2\}$.
\end{description}

\item Show that the error of an $r(n)$-round interactive 
proof system can be amplified while preserving the number
of rounds. Specifically suppose $\omega$ is the maximum
(over all provers) probability that $V$ accepts an input
$x$. Let $V^2$ be the verifier that picks two independent
random strings $R_1$ and $R_2$ according to the distribution used
by $V$ and carries out two instances of the interactive
protocol $V \leftrightarrow P$ in parallel with a prover
$P^2$; and accepts if both instances accept. Show that
the maximum probability with which $V^2$ accepts is 
exactly $\omega^2$.

Does the error of a 2-prover 1-round interactive proof system
also get amplified similarly under parallel repetition?
Give your guess on the answer. For extra credit, prove your answer!
\end{enumerate}

\noindent {\bf Instructions:}
\begin{itemize}
\item Usual rules on collaboration.
\item You may consult any material whatsoever! However cite all 
sources.
\item Turn in the solutions to the above problems by 11am on
Wednesday, May 8, 2002.
\end{itemize}

\end{document}
