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\author{Ada Lovelace}
\collaborators{Charlie Babbage, Mike Faraday}

\begin{document}


\begin{center}
{\Large CS 535: Complexity Theory, Fall 2020}

\bigskip

{\Large Homework 7}

\smallskip

Due: 2:00AM, Saturday, November 7, 2020.

\end{center}

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	\noindent Name: \@author \\
	Collaborators: \@collaborators
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\paragraph{Reminder.}
Homework must be typeset with \LaTeX\ preferred. Make sure you understand the course collaboration and honesty policy before beginning this assignment. Collaboration is permitted, but you must write the solutions {\em by
	yourself without assistance}. You must also identify your
collaborators. Assignments missing a collaboration statement will not be accepted. Getting solutions from outside sources such as the
Web or students not enrolled in the class is strictly forbidden.

\bigskip

\setcounter{problem}{-1}
\begin{problem}[Term Paper] Your term paper topic and partner (if applicable) are due on Gradescope at the same time this homework assignment is. Instructions for the term paper are here: \url{https://cs-people.bu.edu/mbun/courses/535_F20/handouts/term_paper.pdf} and a list of suggested topics is here: \url{https://piazza.com/class/keda2wyieyz10e?cid=277}.
\end{problem}

\medskip

\begin{problem}[Circuit Lower Bounds for $\class{PH}$]
	In this problem, you will prove that $\class{PH}$ can compute languages with high circuit complexity. Specifically, you will show that for every integer $k \ge 1$, there is a language in $\class{\Sigma}_2^p$ that cannot be computed by circuits of size at most $n^k$.
	\begin{enumerate}[(a)]
		
		
		\item Show that for every input length $n$, there exists a function $f: \zo^n \to \zo$ that is computed by a circuit of size at most $20n^k$, but not computed by any circuit of size at most $n^k$. Hint: Use the nonuniform hierarchy theorem (Theorem 6.22 in Arora-Barak). (2 points)
		
		\begin{solution}
			Your solution here.
		\end{solution}
	
		\item Let $C, C'$ be circuits, both on $n$-bit inputs. Say that $C'$ comes lexicographically before $C$, written $C' <_{\text{lex}} C$, if the string encoding $C'$ precedes the string encoding $C$ in the lexicographic ordering. Define the language $L$ to consist of all strings $x$ such that $C(x) = 1$, where $C$ is the lexicographically first circuit of size at most $20|x|^{k}$ that is not computed by any circuit of size at most $|x|^k$. Show that $L \notin \class{SIZE}(n^k)$.  (1 point)
		
		\begin{solution}
			Your solution here.
		\end{solution}
	
		\item Show that the language $L \in \class{\Sigma}_4^p$.  Conclude that $\class{\Sigma}_4^p \not\subseteq \class{SIZE}(n^k)$. (6 points)
		
		\smallskip
		
		Hint: $C$ is the lexicographically first circuit of size at most $20n^k$ that is not computed by any circuit of size at most $n^k$ if: $|C| \le 20n^k$ and for all $C' <_{\text{lex}} C$ where $|C'| \le 20n^k$, there exists a smaller circuit $C''$ of size $\le n^k$ such that $C'' \equiv C'$.
		
		\begin{solution}
			Your solution here.
		\end{solution}
	
		\item Combine part (c) with the Karp-Lipton Theorem ($\NP \subseteq \Ppoly \implies \class{PH} = \class{\Sigma}_2^p$) to show that $\class{\Sigma}_2^p \not\subseteq \class{SIZE}(n^k)$. (3 points)
		
		\begin{solution}
			Your solution here.
		\end{solution}
	
		\item Does part (d) imply $\class{\Sigma}_2^p \not\subseteq \Ppoly$?  Explain your answer. (3 points)
	
		\begin{solution}
		Your solution here.
		\end{solution}
	\end{enumerate}
\end{problem}


\bigskip


\begin{problem}[$\ZPP$ vs. $\RP \cap \coRP$]
	Let $L \in \RP \cap \coRP$ be decided by an $\RP$ algorithm $M_0$ and a $\coRP$ algorithm $M_1$, each running in time $p(n)$ for some polynomial $p$. Show that the following is a zero-error randomized algorithm deciding $L$ in expected polynomial time, and thus $\RP\cap \coRP\subseteq \ZPP$:
	\begin{quote}
		On input $x$: \\
		Repeat the following indefinitely:
		\begin{enumerate}
			\item Run $M_0$ on input $x$. If it accepts, accept; else, continue.
			\item Run $M_1$ on input $x$. If it rejects, reject; else, continue.
		\end{enumerate}
	\end{quote}
	We already showed in class that $\ZPP \subseteq \RP\cap \coRP$, so this completes the proof that $\ZPP = \RP \cap \coRP$.	(5 points)
\end{problem}

\begin{solution}
	Your solution here.
\end{solution}


\end{document}






