\documentstyle[12pt,twoside,amssymb]{article}
\input{defs-6045}

\newcommand{\poly}{{\Bbb P}}
\newcommand{\npoly}{{\Bbb NP}}
\newcommand{\RP}{{\Bbb RP}}
\newcommand{\nat}{{\Bbb N}}
\newcommand{\fa}{\forall}
\newcommand{\sig}{\Sigma}
\newcommand{\arr}{\rightarrow}
\newcommand{\pless}{\leq_{\poly}}
\newcommand{\logsp}{{\Bbb L}}
\newcommand{\Nlog}{{\Bbb NL}}



\begin{document}

\handout{spaceprobs}{Sample Space Problems}{May 4, 1992}

\medskip

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Here are some exercises on space complexity that are similar to those
we would have given out as homework problems.  Exercises on parallel
computation can be found in the parallel computation notes (Handout 54).


\paragraph{Problems.} 

\begin{enumerate} 

\item Review the following theorems to make sure you understand the proofs.
\begin{enumerate}
\item $SPACE(S(n)) \subseteq TIME(2^{cS(n)})$.
\item (Savitch's Theorem) $NSPACE(S(n)) \subseteq SPACE(S^2(n))$.
\item If $A \leq_L B$ and $B \leq_L C$ then $A \leq_L C$.
\item $\Nlog \subseteq \poly$.
\end{enumerate}

\item Show that $\{x~rev(x) | x \in \{0,1\}^*\} \in \logsp$.

\item Let $PATH = \{<G,a,b> | $ There is a path from $a$ to $b$ in the directed
graph $G \}$.
\begin{enumerate}
\item Show that $PATH \in \Nlog$.
\item Using the definitions of NP-completeness and P-completeness as examples, define what it means for a problem to be NL-complete with respect to log-space reductions.
\item Sketch a proof that $PATH$ is NL-complete.  (Hint:  Look at the proof of Savitch's Theorem; it shows how nondeterministic space problems can be modelled as a path problem).
\end{enumerate}

\item In our proof of Savitch's theorem, we used a program that called itself
recursively.  Sketch a proof of Savitch's theorem that does not use
recursive calls.  (Hint: ``Unwind'' the recursion in Savitch's theorem
to see what gets left on the ``stack''.  Then write a program that emulates
the stack behavior.)

\item Let $K_1^S = \{M : M$ accepts $e(M)$ in $\leq S(n)$ space $(n = |e(M)|) \}$.  We will say that a function $S(n)$ is {\it fully space-constructible}
if there is a TM that on any input of length $n$ marks off exactly $S(n)$
tape squares.  Show that if $S(n)$ is fully-space constructible and 
$S(n) \geq \log n$, then $K_1^S \not\in SPACE(S(n))$.  (Hint: Review the 
proof that $K_1^T \not\in TIME(T(n))$.  There is an additional problem
with $SPACE$ that we didn't have with $TIME$.  Suppose the machine $M$ uses
$\leq S(n)$ space on input $e(M)$, but never halts?  What can we do with such
machines?).


\end{enumerate} 

\end{document}
