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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 4}

\smallskip

Due: 8:00PM, Friday, October 9, 2020.

\end{center}

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	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.

\medskip

\begin{problem}[``Mid''-Semester Feedback Form]
	Please fill out the feedback form here: \url{https://forms.gle/YdQ2tqHsfbM6VcxH9}. (It's anonymous, so we can't check whether you did it, but we value your voice!)
\end{problem}

\medskip

\begin{problem}[Time vs. Space] As usual, you can quote anything stated in the main body of Arora-Barak or during the lectures to solve these problems. (Which may give some of them very short solutions.)
	 \begin{enumerate}[(a)]
	 	\item  Show that $\NL \subseteq \P$. (2 points)
		\item Define $\class{PolyL} = \cup_{c = 1}^\infty \SPACE(\log^c n)$. Show that $\NL \subseteq \class{PolyL}$. (2 points)
		\item Prove that there are no $\class{PolyL}$-complete problems (with respect to logspace reductions). Hint: Assume that such a complete problem were to exist. Show that it would contradict the space hierarchy theorem. (4 points)
		\item Define the class $\class{SC}$ (``Steve's Class,'' in honor of Stephen Cook) to consist of all languages $A$ for which there exists a deterministic TM $M$ deciding $A$ that runs in time $\poly(n)$ and uses space $\poly\log(n)$. It is an open question whether $\NL \subseteq \class{SC}$. Explain why parts (a) and (b) together do not resolve this open question. (2 points)
	 \end{enumerate}
\end{problem}

\medskip

\begin{problem}[Consequences of $\NL = \coNL$] \hfill
	\begin{enumerate}[(a)]
		\item Let $S(n) \ge \log n$ be a space-constructible function. Show that $\NSPACE(S(n)) = \class{coNSPACE}(S(n))$. (4 points)
		\item Define the language $\mathprob{DLEN}$ to consist of all tuples $\langle G, s, t, d\rangle$ where $G$ is an directed graph, and the distance from $s$ to $t$ in $G$ is exactly $d$. (That is, there exists a path from $s$ to $t$ of length $d$  \textbf{and there is no shorter path}.) Prove that $\mathprob{DLEN}$ is $\NL$-complete. (6 points)
	\end{enumerate}
\end{problem}

\medskip

\begin{problem}[*Bonus Problem* Implication SAT]
	An ``implication CNF'' is a CNF where every clause has at most one positive variable. For example, $(x_1 \lor \overline{x_2} \lor \overline{x_3}) \land (x_4 \lor \overline{x_1} \lor \overline{x_2})$ is an implication CNF, but $(x_1 \lor x_2 \lor \overline{x_3}) \land (x_4 \lor \overline{x_1} \lor \overline{x_2})$ is not. The name comes from the (useful) fact that the logical expression $x_1 \lor \overline{x_2} \lor \dots \lor \overline{x_k}$ is equivalent to $(x_2 \land \dots \land x_k) \to x_1$.
	
	\begin{enumerate}[(a)]
		\item Define the language $\mathprob{IMPSAT} = \{\varphi \mid \varphi \text{ is a satisfiable implication CNF}\}$. Prove that $\mathprob{IMPSAT}$ is $\P$-complete under logspace reductions.
		
		\item What goes wrong in your proof if you use it to try to show that $\mathprob{IMPSAT}$ is $\NP$-complete?
	\end{enumerate}
	Hint: You can make the same kinds of simplifying assumptions that we made in our proof sketch of the Cook-Levin Theorem from class---single-tape TM, binary tape alphabet, input padded with trailing zeroes. Focus more on the main ideas than on the details.
\end{problem}
\end{document}






