\documentclass[11pt]{amsart}

\usepackage{amsmath,amsthm,amssymb}
\usepackage{stmaryrd}

\usepackage{fancyhdr} % Allows customization of verbatim environments

\usepackage{xcolor}

%\usepackage[textheight=8.5in, margin=1.6in]{geometry}
\usepackage[
  hmarginratio={1:1},     % equal left and right margins
%  vmarginratio={1:1},     % equal top and bottom margins
  %textwidth=400pt,        % new text width
  heightrounded,          % always useful
  %bindingcorrection=5mm,  % binding correction
  total={6in,8in}
]{geometry}
\headheight=13pt
\footskip=13pt

\usepackage{needspace}  % <-- to prevent bad pagebreak over environment titles
% Use:  \Needspace{3\baselineskip} before opening an environment

\usepackage{graphicx}

% Tikz stuff.
\usepackage{tikz}
%\usepackage{tkz-graph}
\usepackage{pgfplots}
%\usepackage{tikz-3dplot}
\pgfplotsset{compat=newest}
%\usepgfplotslibrary{fillbetween}
%\usetikzlibrary{cd}
%\usetikzlibrary{positioning}
%\usetikzlibrary{decorations.pathmorphing}    
%\usetikzlibrary{decorations.pathreplacing}
%\usetikzlibrary{arrows.meta,calc}
%\usetikzlibrary{bending}
%\usetikzlibrary{decorations.markings,shapes.geometric}
%\tikzset{->-/.style={decoration={markings, mark=at position #1 with
%  {\arrow{>}}},postaction={decorate}}}
%\tikzset{-|-/.style={decoration={markings, mark=at position #1 with
%  {\arrow{stealth}}},postaction={decorate}}}
%\tikzset{movearrow/.style 2 args ={
%        decoration={markings,
%	  mark= at position {#1} with {\arrow{#2}} ,
%        },
%        postaction={decorate}
%    }
%}
%\tikzset{<--/.style={decoration={markings, mark=at position #1 with
%  {\arrow{<}}},postaction={decorate}}}
\tikzstyle{ball} = [circle,shading=ball, ball color=black,
    minimum size=1mm,inner sep=1.3pt]
%\tikzstyle{bball} = [circle,shading=ball, ball color=blue,
%    minimum size=1mm,inner sep=1.3pt]
%\tikzstyle{miniball} = [circle,shading=ball, ball color=black,
%    minimum size=1mm,inner sep=0.5pt]
%\tikzstyle{redminiball} = [circle,shading=ball, ball color=red,
%    minimum size=1mm,inner sep=0.5pt]
%\tikzstyle{mminiball} = [circle,shading=ball, ball color=black,
%    minimum size=0.6mm,inner sep=0.1pt]
%\tikzstyle{mycircle} = [draw,circle, minimum size=1mm,inner sep=1.3pt]
%\newcommand*{\DashedArrow}[1][]{\mathbin{\tikz [baseline=-0.25ex,-latex, dashed,#1] \draw [#1] (0pt,0.5ex) -- (1.3em,0.5ex);}}%

%%%% tikz style for finite automata
\usetikzlibrary{automata, positioning, arrows}
\tikzset{fa/.style={
    ->, % makes the edges directed
    >=stealth, % makes the arrow heads bold
    node distance=2cm, % specifies the minimum distance between two nodes. Change if necessary.
    every state/.style={thick, fill=gray!10}, % sets the properties for each ’state’ node
    initial text=$ $, % sets the text that appears on the start arrow
}}
%%%%

\usepackage{needspace}  % <-- to prevent bad pagebreak over environment titles
% Use:  \Needspace{3\baselineskip} before opening an environment

\usepackage[shortlabels]{enumitem}
\setlist[enumerate]{itemsep=4pt,leftmargin=*,label={(\alph*)},topsep=4pt}
%\setlist[enumerate]{itemsep=2pt,label={\arabic*.}}

\usepackage[T1]{fontenc}
\renewcommand{\labelitemi}{\guillemotright}

\allowdisplaybreaks

\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\W}{\mathbb{W}}
\newcommand{\KK}{\Bbbk}
\newcommand{\ve}{\varepsilon}
\newcommand{\scribe}{
  \begin{center}%
    Scribe~$=$ \rule{7cm}{0.15mm}%
  \end{center}%
  \medskip%
}
\DeclareMathOperator{\im}{\mathrm{im}}
\DeclareMathOperator{\id}{\mathrm{id}}
\DeclareMathOperator{\glb}{\mathrm{glb}}
\DeclareMathOperator{\lub}{\mathrm{lub}}

\newcounter{problem}
\renewcommand \thesection {\theproblem}
\newenvironment{prob}[1][]
{
\refstepcounter{problem}\par\medskip
\def\myArg{#1}
  \ifx\myArg\empty
    \noindent \textsc{Problem~\theproblem.} \rmfamily
  \else
    \noindent \textsc{Problem~\theproblem\ (\textnormal{#1}).} \rmfamily
  \fi
}
{
\medskip

}

\newenvironment{sol}
{\emph{Solution.}
}
{
  \medskip

}

\parskip = 0.05in
\parindent = 0.0in
\raggedbottom

%\pagestyle{empty}
%\pagenumbering{gobble}

\fancypagestyle{firststyle}
{
\fancyhead[L,LO]{
\textit{\small Math 387 Group problems, Friday Week 12}
}
\renewcommand{\headrulewidth}{0pt}
}

\begin{document}
\thispagestyle{firststyle}
\bigskip

\textsc{Review}
\bigskip

\begin{prob}
  Create a DFA for the language
  \[
    A=\{w: \text{$w$ is a binary representation of a natural number that is a
    multiple of~$4$}\}.
  \]
  We take \emph{binary representation} to mean that~$w$ will have leftmost digit equal
  to $1$ unless $w=0$.
\end{prob}

\begin{prob}
  Convert the following NFA into a DFA using the subset construction.  Use only
  the nodes that are necessary.
  \begin{center}
    \begin{tikzpicture}[fa,baseline={([yshift=-1em] current bounding box.north)}]
      \node[state, initial] (0) {$q_{0}$};
      \node[state, right of=0] (1) {$q_{1}$};
      \node[state, accepting, right of=1] (2) {$q_{2}$};
      \draw (0) edge[loop below,below] node{$0$} (0);
      \draw (0) edge[above] node{$0$} (1);
      \draw (1) edge[above] node{$1$} (2);
      \draw (2) edge[loop below,below] node{$1$} (2);
      \draw (2) edge[below, bend left] node{$0,1$} (0);
    \end{tikzpicture}
  \end{center}
\end{prob}

\begin{prob}
   Consider the language
   \[
     E_{\mathrm{CFG}} = \{\langle G\rangle: \text{$G$ is a CFG and
     $L(G)=\emptyset$}\}.
   \]
   Prove that~$E_{\mathrm{CFG}}$ is decidable.
\end{prob}

\begin{prob}
  Is the subset of a context free language necessarily context free?  Give a
  proof or a counter-example.
\end{prob}

\begin{prob}
   Consider the language
   \[
     B = \{\langle M\rangle: \text{$M$ is a DFA that accepts at least one
     (nonempty) palindrome}\}.
   \]
   Prove that $B$ is decidable. (Hint: context free languages are not closed
   under intersection, but \ldots)
\end{prob}

\begin{prob}
  Let $\Sigma=\{a,b,c\}$, and consider the following language:
  \[
    L = \{a^{m}b^{n}c^{r}: \text{$m\leq n$ and $m,n,r\geq0$}\}.
  \]
  Is $L$ regular?  Prove your answer is correct.
\end{prob}

\begin{prob}
  Let~$M$ be a Turing machine.  Is it possible
  that~$L(M)=\left\{\varepsilon\right\}$?  If so, is it possible for $\varepsilon$
  to be accepted with the TM taking no steps?
\end{prob}

\begin{prob}
Let $E_{\mathrm{TM}}=\{\langle M\rangle: \text{$M$ is a TM and
$L(M)=\emptyset$}\}$. Is~$\overline{E_{\mathrm{TM}}}$ , the complement
of~$E_{\mathrm{TM}}$, Turing-recognizable?  Prove your answer.
\end{prob}
\end{document}
