Introduction

I typed out some definitions from our theoretical computer science lecture for the fun of it. Unfortunately I noticed, that it didn't look very professional and clean . The reason for that should be wrong alignments etc., but I can't really improve it, even though I tried to define theorem/definition-environments and similar stuff. Unfortunately, my definitions still look bad.


Question

How to improve the visual appearance of my document?

I would like to get suggestions (for packages, environments, spacing, sectioning etc.), which will help me to improve my document.
EDIT: I would like to have real definition environments for every defintion I made in order to label it. Furthermore, some equations are bad aligned (for example the last equations of "formal languages"). Furthermore I would like to get suggestions on how to make the document look more modern (for example by adding colors and changing fonts)


Code

Note: The file is translated poorly, because English is not my native language. I only translated the document (from german to english), in order that you understand the content and can help with formatting based on it. The file is basically a small list of definitions from the TCS lecture.

Here is my code:

\documentclass{memoir}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}

\title{Computer-Science (Poor translation)}
\author{Doesbaddel}
\date{November 2018}

\begin{document}

\maketitle

    \begin{abstract}
        Note: The file is translated poorly, because English is not my native language. I only translated the document (from german to english), in order that you understand the content and can help with formatting based on it. The file is basically a small list of definitions from the TCS lecture. 
    \end{abstract}


\section{Fundamentals}
Definition: 
    \begin{enumerate}
        \item {\em Finite Alphabet:} Finite set $\Sigma := \{a_1,\dots , a_\Delta\}$ with various objects\\ $a_1, \dots , a_\Delta$
        \item A {\em word/string} $w=(w_1,\dots , w_n)$ with $w_i\in \Sigma$ for $1\leq i \leq n$ is called a concatenation of letters in $\Sigma$.\\
        \emph{The length of} $w$ \emph{is} ${\mid w \mid} := n$\\
            Strings with the length of $n$: $\Sigma^n = \underbrace{\Sigma \times \Sigma \dots \times \Sigma}_{n}$\\
            The empty string is $\lambda$: ${\mid \lambda \mid} := 0$\\
            $\Sigma^* := \bigcup^{\infty}_{n=0} \Sigma^n \text{ is the set of all strings with finite length on } \Sigma.$\\
            $\Sigma^* := \{\lambda\}\text{ is not the empty set, because }\lambda\in\Sigma^*$\\
            $\Sigma^+ := \Sigma^*\backslash \{\lambda\}$
        \item Let $X=(x_1,\dots ,x_n)$, $Y=(y_1,\dots ,y_n)$ then
            \[X.Y:=(x_1,\dots ,x_n,y_1,\dots ,y_m)\] is called the concatenation of $x$ and $y$. $X$ is called the prefix of $XY$; $Y$ is called the Suffix.\\
            Obvious:
                \begin{align*}
                    \mid x.y\mid        &= \mid x\mid +\mid y\mid\\
                    \mid x.\lambda \mid &= \mid \lambda .x\mid = \mid x\mid
                \end{align*}
            Notation: $x.y = xy = x_1x_2\dots x_ny_1y_2\dots y_n$
        \item Any $L\subseteq \Sigma^*$ is called a "formal language" on $\Sigma$.
    \end{enumerate}
Definition: Let $\Sigma$ be a finite alphabet, $L_0,L_1,L_2 \subseteq \Sigma^*$ (free formal languages in the finite alphabet.)
    \begin{enumerate}
        \item $\overline{L} := \Sigma^*\backslash L \text{ compliment of } L$
        \item   $L_1 \cap L_2   := \{ x \in \Sigma^* \mid x \in L_1 \land x \in L_2 \} \text{ Intersection}$\\
                $L_1 \cup L_2   := \{ x \in \Sigma^* \mid x \in L_1 \lor x \in L_2 \} \text{ Union}$
        \item   $L_1.L_2        := \{ xy \mid x \in L_1 \land y \in L_2 \} \text{ Concatenation of two languages}$
        \item   $L^*            := \{ w_1 w_2 \dots w_t \mid w_i \in L \lor w_i = \lambda \land t \in N \} \} \text{ Kleene-Star of }L$\\
        \emph{Please note:} $\lambda \in L^*$ with $t = 1$, $w_1 = \lambda$\\
        $L^+ = L.L^*$\\
        $\lambda \in L^+ \Longleftrightarrow \lambda \in L$
    \end{enumerate}


\section{Regular Expressions}
Idea: Recursive construction of expressions, which represent the languages.

Definition: Let $\Sigma$ be a finite alphabet. The set of regular expressions on $\Sigma$ and their represented languages are defined in the following way:
    \begin{enumerate}
        \item $\emptyset$,$\lambda$ and any $a \in \Sigma$ are regular expressions\\

        The related languages are:
        \begin{align*}
            \mathcal{L}(\emptyset)  &:= \emptyset\\
            \mathcal{L}(\lambda)    &:= \lambda\\
            \mathcal{L}(a)          &:=\{a\} \qquad \forall a \in \Sigma
        \end{align*}
        \item Let $x,y$ be regular expressions with related languages $\mathcal{L}(x)$, $\mathcal{L}(y)$.
        Hence, the following expressions are regular too:
            \begin{align*}
                & x.y \text{ and } \mathcal{L}(x.y):= \mathcal{L}(x).\mathcal{L}(y)\\
                & x\cup y \text{ and } \mathcal{L}(x\cup y):= \mathcal{L}(x)\cup \mathcal{L}(y)\\
                & x^* \text{ and } \mathcal{L}(x^*):= \mathcal({L}(x))^*
            \end{align*}
        \item Everything that can be produced with \em{1.} and \em{2} in finite steps is regular.
    \end{enumerate}
The related languages are called regular languages.


\section{Automata}
Definition: The deterministic, finite automaton \emph{(DFA)} $M$ is defined as \[M=(K,\Sigma,s,\delta ,F)\] with:
    \begin{itemize}
        \item $K = K \times \Sigma$ finite set of states
        \item $\Sigma$ finite alphabet
        \item $s$ initial state
        \item $\delta$ transition function
        \item $F$ finite set of final states
    \end{itemize}
\end{document}

Regards,

Doesbaddel

closed as primarily opinion-based by samcarter, Sebastiano, Phelype Oleinik, Troy, Stefan Pinnow Nov 9 at 5:21

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    Please be more specific on how you want to improve. – zyy Nov 8 at 18:58
  • I would like to have real definition environments for every defintion I made in order to label it. Furthermore, some equations are bad aligned (for example the last equations of "formal languages"). Furthermore I would like to get suggestions on how to make the document look more modern (for example by adding colors and changing fonts). – Doesbaddel Nov 8 at 19:03
  • For your first question, please try looking at this resource. As for the equation alignment, please try using \begin{gathered} and \end{gathered} instead. As for making the document more modern, that is a broad question and you may want to do the research with your curiosity. – zyy Nov 8 at 20:36
  • Thank you, the link was helpful. Why is the gathered-environment better? What does is differently when the align-environment? – Doesbaddel Nov 8 at 23:46
  • You could try gathered to see if you would like it, since you did not give specific requests, what I could say is just try. – zyy Nov 9 at 1:04