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% LaTeX Problem Set Template originally designed
% by former CS103 TA Sachin Padmanabhan, with updates,
% edits, and simplifications by the lovely folks below.
%
% Updated for Fall 2018 by Michael Zhu
% Updated for Fall 2019 by Joshua Spayd
% Updated for Fall 2020 by Lucy Lu
% Updated for Winter/Spring 2021 by Cynthia Bailey Lee
% Updated for Fall 2021 by Grant McClearn
% Commands slimmed down and simplified in Fall 2022 by Keith Schwarz

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\title{CS 103: Mathematical Foundations of Computing\\Problem Set \#6}
\author{[TODO: Replace this with your name(s)]}
\date{\today}

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\lhead{\runauthor}
\chead{Problem Set 6}
\rhead{\today}
\lfoot{}
\cfoot{CS 103: Mathematical Foundations of Computing --- Fall 2025}
\rfoot{\thepage}

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\begin{document}

\maketitle

\begin{center}
  \emph{Due Friday, November 7, at 1:00 PM Pacific}
\end{center}

Do not put your answers to Problem 1 on Problem 2 in this file. You'll submit those separately on Gradescope.

Here's a quick reference of symbols you may want to use in this problem set.

\begin{itemize}
    \item Alphabets are written as $\Sigma$.
    \item The set of all strings over $\Sigma$ is denoted $\Sigma^\star$
    \item The empty string is written as $\varepsilon$. The "var" here refers to a "variant" of the letter epsilon; that's the one we use in this class.
    \item Subscripts are done as is $q_{137}$; superscripts are done as $a^{137}$.
    \item You can make text render \texttt{like a typewriter} in text mode or in $\mathtt{math}$ mode.
    \item You can write language complements as $\overline{L}$.
    \item You can write inequality signs as <, >, $\ge$, $\le$.
\end{itemize}

\pagebreak

\section*{Problem Three: $\powerset{\Sigma^*}$}
\begin{shaded}
Write your answer to Problem Three here.
\end{shaded}

\newpage

\section*{Problem Four: Much Ado About Nothing, Part II}
    i.
    \begin{shaded}
    Write your answer to Problem Four, part i. here.
    \end{shaded}

    \newpage
    
    ii.
    \begin{shaded}
    Write your answer to Problem Four, part ii. here.
    \end{shaded}

    \newpage
    
    iii.
    \begin{shaded}
    Write your answer to Problem Four, part iii. here.
    \end{shaded}

    \newpage
    
    iv.
    \begin{shaded}
    Write your answer to Problem Four, part iv. here.
    \end{shaded}

    \newpage
    
    v.
    \begin{shaded}
    Write your answer to Problem Four, part v. here.
    \end{shaded}

    \newpage
    
    vi.
    \begin{shaded}
    Write your answer to Problem Four, part vi. here.
    \end{shaded}

\newpage

\section*{Problem Five: Error-Correcting Codes}

i.
    \begin{shaded}
    Write your answer to Problem Five, part i. here.
    \end{shaded}

    \newpage
    
    ii.
    \begin{shaded}
    Write your answer to Problem Five, part ii. here.
    \end{shaded}

    \newpage
    
    iii.
    \begin{shaded}
    Write your answer to Problem Five, part iii. here.
    \end{shaded}

    \newpage
    
    iv.
    \begin{shaded}
    Write your answer to Problem Five, part iv. here.
    \end{shaded}

    \newpage
    
    v.
    \begin{shaded}
    Write your answer to Problem Five, part v. here.
    \end{shaded}

    \newpage
    
    vi.
    \begin{shaded}
    Write your answer to Problem Five, part vi. here.
    \end{shaded}

    \newpage
    
    vii.
    \begin{shaded}
    Write your answer to Problem Five, part vii. here.
    \end{shaded}

\newpage

\section*{Problem Six: Monoids and Kleene Stars}
    i.
    \begin{shaded}
    Write your answer to Problem Six, part i. here.
    \end{shaded}

    \newpage
    
    ii.
    \begin{shaded}
    Write your answer to Problem Six, part ii. here.
    \end{shaded}

    \newpage
    
    iii.
    \begin{shaded}
    Write your answer to Problem Six, part iii. here.
    \end{shaded}

    \newpage
    
    iv.
    \begin{shaded}
    Write your answer to Problem Six, part iv. here.
    \end{shaded}
    
\newpage

\section*{Optional Fun Problem: Dual-Use Automata}
\begin{shaded}
Write your answer to the Optional Fun Problem here.
\end{shaded}

\end{document}
