\newsection
\section{Alphabets, Words, etc}
\stepcounter{subsection}
\subsection{Alphabets, Words, Languages}
\fancydef{Alphabet} Set $\Sigma$.
Important alphabets: $\alphabetbool$, $\alphabets{lat}$ (all latin chars), $\alphabets{Keyboard}$ (all chars on keyboard), $\Sigma_m$ ($m$-ary numbers)

\fancydef{Word} Possibly empty (denoted $\lambda$) sequences of characters from $\Sigma$. $|w|$ is the length,
$\Sigma^*$ is the set of all words and $\Sigma^+ = \Sigma^* - \{ \lambda \}$

\fancydef{Konkatenation} $\text{Kon}(x, y) = xy$, (so like string concat). $(xy)^n$ is $n$-times repeated concat.

\fancydef{Reversal} $a^R$, simply read the word backwards.

\stepLabelNumber{definition}
\fancydef{Prefix, Suffix, Subword} $v$ in $w = vy$; $s$ in $w = xs$; Subword $u$ in $w = xuy$; $x$, $y$ possibly $\lambda$

\fancydef{Appearance} $|x|_a$ is the number of times $a \in \Sigma$ appears in $x$

\fancydef{Canonical ordering} Ordered by length and then by first non-common letter:
\rmvspace
\begin{align*}
    u < v \Longleftrightarrow |u| < |v| \lor (|u| = |v| \land u = x \cdot s_i \cdot u' \land v = x \cdot s_j \cdot v') \text{ for any } x, u', v' \in \word \text{ and } i < j
\end{align*}

\drmvspace
\fancydef{Language} $L \subseteq \word$, and we define $L^C = \word - L$ as the complement, with $L_{\emptyset}$ being the empty language,
whereas $L_\lambda$ is the language with just the empty word in it.

\bi{Concatenation}: $L_1 \cdot L_2 = \{ vw | v \in L_1 \land w \in L_2 \}$ and $L^{i + 1} = L^i \cdot L \ \forall i \in \N$.

\bi{Cleen Star}: $L^* = \bigcup_{i \in \N} L^i$ and $L^+ = L \cdot L^*$

Of note is that there are irregular languages whose Cleen Star is regular, most notably,
the language $L = \{ w \in \{ 0 \}^* \divides |w| \text{ is prime} \}$'s Cleen Star is regular,
due to the fact that the prime factorization is regular


\inlinelemma $L_1L_2 \cup L_1 L_2 = L_1(L_2 \cup L_3)$
\inlinelemma $L_1(K_2 \cap L_3) \subseteq L_1 L_2 \cap L_1 L_3$

For multiple choice questions, really think of how the sets would look to determine if they fulfill a requirement.


\stepcounter{subsection}
\subsection{Kolmogorov-Complexity}
\setLabelNumber{definition}{17}
\fancydef{Kolmogorov-Complexity} $K(x)$ for $x \in \wordbool$ is the minimum of all binary lengths of Pascal programs that output $x$,
where the Program doesn't have to compile, i.e. we can describe processes informally

\stepLabelNumber{lemma}
\inlinelemma For each word $x$ exists constant $d$ s.t. $K(x) \leq |x| + d$, for which we can use a program that simply includes a \texttt{write(x)} command

\fancydef{Of natural number} $K(n) = K(\text{Bin}(x))$ with $|\text{Bin}(x)| = \ceil{\log_2(x + 1)}$

\inlinelemma For each $n \in \N \exists w_n \in (\alphabetbool)^n$ s.t. $K(w_n) \geq |w_n| = n$, i.e. exists a non-compressible word.

\inlinetheorem Kolmogorov-Complexity doesn't depend on programming language. It only differs in constant

\fancydef{Randomness} $x \in \wordbool$ random if $K(x) \geq |x|$, thus for $n \in \N$, $K(n) \geq \ceil{\log_2(n + 1)} - 1$

\stepLabelNumber{theorem}
\fancytheorem{Prime number} $\displaystyle \limni \frac{\text{Prime}(n)}{\frac{n}{\ln(n)}} = 1$ with $\text{Prime}(n)$ the number of prime numbers on $[0, n] \subseteq \N$

\fhlc{Cyan}{Proofs} Proofs in which we need to show a lower bound for Kolmogorov-Complexity (almost) always work as follows:
Assume for contradiction that there are no words with $K(w) > f$ for all $w \in W$.
We count the number $m$ of words in $W$ and the number $n$ of programs of length $\leq f$ ($f$ being the given, lower bound).
We will have $m - n > 0$, which means, there are more different words than there are Programs with Kolmogorov-Complexity $\leq f$,
which is a contradiction to our assumption.

There are $\floor{\frac{n}{k}} + 1$ numbers divisible by $k$ in the set $\{ 0, 1, \ldots, n \}$.

\shade{Orange}{Laws of logarithm}
\drmvspace
\begin{multicols}{3}
    \begin{itemize}[noitemsep]
        \item $\log_a(x) + \log_a(y) = \log_a(x \cdot y)$
        \item $\log_a(x) - \log_a(y) = \log_a(x \div y)$
        \item $y \log_a(x) = \log_a(x^y)$
        \item $\log_a(x) = \frac{\ln(x)}{\ln(a)}$
        \item $\log_a(1) = 0$
    \end{itemize}
\end{multicols}