%\documentstyle[10pt,twoside]{article}
\documentstyle[twoside, 11pt]{article}
\setlength{\oddsidemargin}{0.25 in}
\setlength{\evensidemargin}{-0.25 in}
\setlength{\topmargin}{-0.6 in}
\setlength{\textwidth}{6.5 in}
\setlength{\textheight}{8.5 in}
\setlength{\headsep}{0.75 in}
\setlength{\parindent}{0 in}
\setlength{\parskip}{0.1 in}
%
% The following commands sets up the lecnum (lecture number)
% counter and make various numbering schemes work relative
% to the lecture number.
%
\newcounter{lecnum}
\renewcommand{\thepage}{\thelecnum-\arabic{page}}
\renewcommand{\thesection}{\thelecnum.\arabic{section}}
\renewcommand{\theequation}{\thelecnum.\arabic{equation}}
\renewcommand{\thefigure}{\thelecnum.\arabic{figure}}
\renewcommand{\thetable}{\thelecnum.\arabic{table}}
%
% The following macro is used to generate the header.
%
\newcommand{\problemset}[3]{
\pagestyle{myheadings}
\thispagestyle{plain}
\newpage
\setcounter{lecnum}{#1}
\setcounter{page}{1}
\noindent
\begin{center}
\framebox{
\vbox{\vspace{2mm}
\hbox to 6.28in { {\bf CS~151~~~Complexity Theory
\hfill Spring 2021} }
\vspace{4mm}
\hbox to 6.28in { {\Large \hfill Problem Set #1 \hfill} }
\vspace{2mm}
\hbox to 6.28in { {\it Out: #2 \hfill Due: #3} }
\vspace{2mm}}
}
\end{center}
\vspace*{4mm}
}
%
% Convention for citations is authors' initials followed by the year.
% For example, to cite a paper by Leighton and Maggs you would type
% \cite{LM89}, and to cite a paper by Strassen you would type \cite{S69}.
% (To avoid bibliography problems, for now we redefine the \cite command.)
%
\renewcommand{\cite}[1]{[#1]}
\input{epsf}
%Use this command for a figure; it puts a figure in wherever you want it.
%usage: \fig{NUMBER}{FIGURE-SIZE}{CAPTION}{FILENAME}
\newcommand{\fig}[4]{
\vspace{0.2 in}
\setlength{\epsfxsize}{#2}
\centerline{\epsfbox{#4}}
\begin{center}
Figure \thelecnum.#1:~#3
\end{center}
}
% Use these for theorems, lemmas, proofs, etc.
\newtheorem{theorem}{Theorem}[lecnum]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newenvironment{proof}{{\bf Proof:}}{\hfill\rule{2mm}{2mm}}
% Some useful equation alignment commands, borrowed from TeX
\makeatletter
\def\eqalign#1{\,\vcenter{\openup\jot\m@th
\ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil
\crcr#1\crcr}}\,}
\def\eqalignno#1{\displ@y \tabskip\@centering
\halign to\displaywidth{\hfil$\displaystyle{##}$\tabskip\z@skip
&$\displaystyle{{}##}$\hfil\tabskip\@centering
&\llap{$##$}\tabskip\z@skip\crcr
#1\crcr}}
\def\leqalignno#1{\displ@y \tabskip\@centering
\halign to\displaywidth{\hfil$\displaystyle{##}$\tabskip\z@skip
&$\displaystyle{{}##}$\hfil\tabskip\@centering
&\kern-\displaywidth\rlap{$##$}\tabskip\displaywidth\crcr
#1\crcr}}
\makeatother
% **** IF YOU WANT TO DEFINE ADDITIONAL MACROS FOR YOURSELF, PUT THEM HERE:
\begin{document}
\problemset{1}{April 1}{{\bf April 8}}
Reminder: you are encouraged to work in groups of two or three;
however you must turn in your own write-up and note with whom you
worked. You may consult the course notes and the optional text (Papadimitriou). The
full honor code guidelines and collaboration policy can be found in the course syllabus.
Please attempt all problems. {\bf Please turn in your solutions via
Gradescope, by 1pm Los Angeles time on the due date.}
\begin{enumerate}
\item Downward self-reducibility. For a language $A$, define
\[A_{< n} = \{x \in A : \; |x| < n\}.\]
Language $A$ is said to be {\em downward self-reducible} if it is
possible to determine in polynomial time if $x \in A$ using the
results of queries of the form ``is $y \in A_{<
|x|}$?'' The queries may be {\em adaptive}, meaning that the polynomial time procedure may choose later queries depending on the results of earlier ones. Show that every downward self-reducible language is in
{\bf PSPACE}.
\item Show that one of the following inequalities must hold:
$\mbox{\bf L} \ne \mbox{\bf P}$ or $\mbox{\bf P} \ne \mbox{\bf
PSPACE}$. Note that both are believed to be true, and no one knows
how to prove either one is true.
\item Show that logspace reductions are closed under composition.
Then use the same ideas to prove that if language $A$ is $\mbox{\bf P}$-complete,
then $A \in \mbox{\bf L}$ implies $\mbox{\bf L} = \mbox{\bf P}$.
\item Use a padding argument to show that if $\mbox{\bf L} =
\mbox{\bf P}$ then $\mbox{\bf PSPACE} = \mbox{\bf EXP}$.
\item Prove that $\mbox{\bf SPACE($O(n)$)} \ne \mbox{\bf P}$.
(Note that while this is an interesting result, it doesn't seem to
shed any light on the major open questions $\mbox{\bf
L}\stackrel{?}{=}\mbox{\bf P}$ and $\mbox{\bf
P}\stackrel{?}{=}\mbox{\bf PSPACE}$). Hint: consider a language
$A$ and a padded version of $A$. How are the two languages related
with respect to space? How are they related with respect to time?
\end{enumerate}
\end{document}