\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}}


\begin{document}

\handout{ps10}{Homework Assignment 10: the LAST one!!}{April 24, 1992}
\due{Friday, May 1}

\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



Remember to write up problems 1 through 5 in such a way that each of
these problems can be turned in separately.  Put your name on each
problem, with your last name clearly written in CAPITAL LETTERS, and
with your section and the NAME of your recitation instructor (e.g.
RIVEST/LYNCH, W1) also in capital letters. Try to always provide clear
(this is not in contradiction with short!) justifications to your
constructions.  

\paragraph{Problems.} 

The first four problems are from Martin.

\begin{enumerate} 

\item[1.] (25 points) 23.15 (15 pts) and 24.5 (10 pts)

\item[2.] (20 points) 23.16 

\item[3.] (15 points) 24.3


\item[4.] (20 points) 24.9 (a)
({\em Hint:} Reduce the three-dimensional problem to it. Use the defn.\
of three-dimensional matching from Martin p.446.)


\item[5.] (20 points) 
Show that the language $A$ is in $\npoly$ iff there is a language $B$
in $\poly$ and a polynomial $P$ such that:
$$A=\{x; \mbox{ there exists } y\mbox{ such that } |y|\leq P(|x|) \mbox{ and } (x,y)\in B\}.$$




\end{enumerate} 

\end{document}
