\newsection
\section{Computability}
\stepcounter{subsection}
\subsection{Diagonalization}
The \bi{set of binary encodings of all TMs} is denoted $\text{KodTM}$ and $\text{KodTM} \subseteq \wordbool$ and the upper bound of the cardinality is $|\wordbool|$,
as there are infinitely many TMs.

Below is a list of countable objects. They all have corresponding Lemmas in the script, but omitted here:
\drmvspace
\begin{multicols}{4}
    \begin{itemize}
        \item $\word$ for any $\Sigma$
        \item $\text{KodTM}$
        \item $\N \times \N$
        \item $\Q^+$
    \end{itemize}
\end{multicols}

\rmvspace
\drmvspace
The following objects are uncountable: $[0, 1]$, $\R$, $\cP(\wordbool)$

\inlinecorollary $|\text{KodTM}| < |\cP(\wordbool)|$ and thus there exist infinitely many not recursively enumerable languages over $\alphabetbool$

\setLabelNumber{theorem}{3}
\inlinetheorem $L_\text{diag} \notin \cL_{RE}$

\fhlc{Cyan}{Proof of $L$ (not) recursively enumerable}

Proving that a language \textit{is} recursively enumerable is as difficult as providing a Turing Machine that accepts it.

Proving that a language is \textit{not} recursively enumerable is likely easier. For it, let $d_{ij} = 1 \Longleftrightarrow M_i$ accepts $w_j$.

\inlineex Assume towards contradiction that $L_\text{diag} \in \cL_{RE}$. Let
\rmvspace
\begin{align*}
    L_{\text{diag}} & = \{ w \in \wordbool \divides w = w_i \text{ for an } i \in \N - \{ 0 \} \text{ and $M_i$ does not accept } w_i \} \\
                    & = \{ w \in \wordbool \divides w = w_i \text{ for an } i \in \N - \{ 0 \} \text{ and } d_{ii} = 0\}
\end{align*}
Thus assume that, $L_\text{diag} = L(M)$ for a Turing Machine $M$.
Since $M$ is a Turing Machine in the canonical ordering of all Turing Machines, so there exists an $i \in \N - \{ 0 \}$, such that $M = M_i$.

This however leads to a contradiction, as $w_i \in L_\text{diag} \Longleftrightarrow d_{ii} = 0 \Longleftrightarrow w_i \notin L(M_i)$.

In other words, $w_i$ is in $L_\text{diag}$ if and only if $w_i$ is not in $L(M_i)$, which contradicts our statement above, in which we assumed that $L_\text{diag} \in \cL_{RE}$.

In other, more different, words, $w_i$ being in $L_\text{diag}$ implies (from the definition) that $d_{ii} = 0$, which from its definition implies that $w_i \notin L(M_i)$.

\drmvspace
\proven


Another result (not formally proven in the script, but there is a proof by intimidation) that can come in useful, especially when trying to show $L \notin \cL_{RE}$ is:
\rmvspace
\begin{align*}
    L, L^C \in \cL_{RE} \Longleftrightarrow L \in \cL_R
\end{align*}

\drmvspace
Additionally, as a reminder, $\cL_{RE} = \{ L(M) \divides M \text{ is a TM} \}$, so to prove that a language $L \notin \cL_{RE}$,
we only need to show that there exists no TM $M$, for which $L(M) \in \cL_{RE}$.



% ────────────────────────────────────────────────────────────────────


\newpage
\subsection{Reductions}
\label{sec:reductions}
This is the start of the topics that are explicitly part of the endterm.

For the reductions, it is important to get the order right.\\
To show that a language $L_1$ is not part of e.g. $\cL_R$, show that there exists a reduction into a language $L_2 \notin \cL_R$, i.e. e.g. show $L_2 \leq_R L_1$.\\
To show that a language $L_1$ is part of e.g. $\cL_R$, show that there exists a reduction into a language $L_2 \in \cL_R$, i.e. e.g. show $L_1 \leq_R L_2$.

For a language to be in $\cL_R$, in contrast to $L \in \cL_{RE}$, the TM has to halt also for \texttt{no} instances, i.e. it has to be an algorithm.
In other words: A TM $A$ can enumerate all valid strings of a \textit{recursively enumerable language} ($L \in \cL_{RE}$),
where for \textit{recursive languages}, it has to be able to difinitively answer for both \texttt{yes} and \texttt{no} and thus halt in finite time for both.

First off, a list of important languages for this and the next section:
\begin{itemize}
    \item $L_U = \{ \text{Kod}(M)\# w \divides w \in \wordbool \text{ and TM $M$ accepts } w \}$ ($\in \cL_{RE}$, but $\notin \cL_R$)
    \item $L_H = \{ \text{Kod}(M)\# x \divides x \in \wordbool \text{ and TM $M$ halts on } x \}$ ($\in \cL_{RE}$, but $\notin \cL_R$)
    \item $L_{\text{diag}} = \{ w \in \wordbool \divides w = w_i \text{ for an } i \in \N - \{ 0 \} \text{ and $M_i$ does not accept } w_i \}$ ($\notin \cL_{RE}$ and thus $\notin \cL_R$)
    \item $(L_{\text{diag}})^C$ ($\in \cL_{RE}$, but $\notin \cL_R$)
    \item $L_{EQ} = \{ \text{Kod}(M)\# \text{Kod}(\overline{M}) \divides L(M) = L(\overline{M}) \}$ ($\notin \cL_{RE}$, and thus $\notin \cL_R$)
    \item $(L_{EQ})^C = \{ \text{Kod}(M)\# \text{Kod}(\overline{M}) \divides L(M) \neq L(\overline{M}) \}$ ($\notin \cL_{RE}$, and thus $\notin \cL_R$)
    \item $\lempty = \{ \text{Kod}(M) \divides L(M) = \emptyset \}$ ($\in \cL_{RE}$, but $\notin \cL_R$)
    \item $(\lempty)^C = \{ x \in \wordbool \divides x \notin \text{Kod}(\overline{M}) \forall \text{ TM } \overline{M} \text{ or } x = \text{Kod}(M) \text{ and } L(M) \neq \emptyset \}$
          ($\in \cL_{RE}$, but $\notin \cL_R$)
    \item $L_{H, \lambda} = \{ \text{Kod}(M) \divides M \text{ halts on } \lambda \}$ ($\in \cL_{RE}$, but $\notin \cL_R$)
\end{itemize}
An important consequence of the fact that both $L_{EQ}$ and its complement are $\notin \cL_{RE}$ is that it is not guaranteed
for a language's complement to \textit{necessarily} be in $\cL_{RE}$, if the language is not.


\setLabelNumber{definition}{3}
\fancydef{Recursively reducible languages} $L_1 \leq_R L_2$ ($L_1$ reducible into $L_2$), if $L_2 \in \cL_R \Rightarrow L_1 \in \cL_R$

\fancydef{$EE$-Reductions} $L_1 \leq_{EE} L_2$ if there exists a TM $M$ that implements image $f_M : \word_1 \rightarrow \word_2$,
for which we have $x \in L_1 \Leftrightarrow f_M(x) \in L_2$ for all $x \in \wordbool_1$

\setLabelNumber{lemma}{3}
\inlinelemma If $L_1 \leq_{EE} L_2$ then also $L_1 \leq_R L_2$

\inlinelemma For each language $L \subseteq \word$ we have: $L \leq_R L^C$ and $L^C \leq_R L$

\setLabelNumber{theorem}{6}
\fancytheorem{Universal TM} A TM $U$, such that $L(U) = L_U$


\fhlc{Cyan}{Showing reductions} First, a general guide to reductions and below what else we need to keep in mind for specific reductions:
\begin{enumerate}
    \item We construct a TM $A$ that:
        \begin{enumerate}
            \item Checks if the input has the right form and if it does not, returns some output that is $\notin L_2$
            \item Applies the transformation to all remaining input
        \end{enumerate}
    \item We show $x \in L_1 \Leftrightarrow A(x) \in L_2$ by showing the implications:
        \begin{enumerate}
            \item For $\Rightarrow$, we show it directly, by assuming that $x \in L_1$ (obviously) and we can ignore the invalid input (as that $\notin L_1$ anyway)
            \item For $\Leftarrow$, we have two options (mention what happens to invalid input here):
                \begin{itemize}
                    \item We show $A(x) \in L_2 \Rightarrow x \in L_1$ directly (usually harder)
                    \item We show $x \notin L_1 \Rightarrow A(x) \notin L_2$ (contraposition)
                \end{itemize}
        \end{enumerate}
    \item Show that the TM always halts (for $P$, $EE$ and $R$ reductions at least)
\end{enumerate}

\shade{Cyan}{$EE$-reductions} They follow the above scheme exactly

\shade{Cyan}{$R$-reductions} It is usually a good idea to draw the setup here. We have a TM $C$ that basically executes an $EE$-reduction and we have a TM $A$ that can check $L_2$.
Then, we have a TM $B$ that wraps the whole thing: It first executes TM $C$, which will either output an transformation of $L_1$ for $L_2$ (i.e. execute an $EE$-reduction) or
output some encoding for \textit{invalid word}.
If it outputs the encoding for \textit{invalid word}, $B$ will output $x \notin L_1$.

If $C$ does not output an encoding for \textit{invalid word}, then $B$ will execute $A$ on the output of $C$ and then use the output of $A$ (either accepting or rejecting)
to output the same (i.e. if $A$ accepts, then $B$ will output $x \in L_1$ and if $A$ rejects, $B$ outputs $x \notin L_1$)

\inlineintuition In $R$-reductions, we construct a full verifier for $L_1$ using the verifier for $L_2$, i.e. we can use TM $B$ directly to check if a given word is in $L_1$
given that the transformed word is also in $L_2$.


\shade{Cyan}{$P$-reductions} (Used in Chapter \ref{sec:complexity}). We need to also show that $A$ terminates in polynomial time.

\shade{orange}{Tips \& Tricks:}
\begin{itemize}
    \item The TM $A$ has to terminate always
    \item Check the input for the correct form first
    \item For the correctness, show $x \in L_1 \Leftrightarrow A(x) \in L_2$
    \item The following tricks can be useful:
          \begin{itemize}
              \item Transitions into $\qacc$ and $\qrej$ can be redirected to $\qacc / \qrej$ or into an infinite loop
              \item Construct TM $M'$ that ignores input and does the same, regardless of input
          \end{itemize}
    \item Generate encoding of a $TM$ with special properties (e.g. accepts all input, never halts, \dots)
\end{itemize}


% ────────────────────────────────────────────────────────────────────


\subsection{Rice's Theorem}
\setLabelNumber{definition}{7}
\inlinedef $L$ is called a \bi{semantically non-trivial decision problem}, if these conditions apply:
\begin{enumerate}[label=\textit{(\roman*)}]
    \item There exists a TM $M_1$, such that $\text{Kod}(M_1) \in L$ (i.e. $L \neq \emptyset$)
    \item There exists a TM $M_2$, such that $\text{Kod}(M_2) \notin L$ (not all encodings are in $L$)
    \item For two TM $A$ and $B$: $L(A) = L(B) \Rightarrow \text{Kod}(A) \in L \Rightarrow \text{Kod}(B) \in L$
\end{enumerate}


\setLabelNumber{theorem}{9}
\fancytheorem{Rice's Theorem} Every semantically non-trivial decision problem over TMs is undecidable

\fhlc{Cyan}{Using Rice's Theorem} We only need to show that a language is semantically non-trivial, which we do by checking the above conditions.
For the third condition, intuitively, we only need to check if in the definition of $L$ only $L(M)$ appears and nowhere $M$ directly (except of course, to say that $M$ has to be a TM),
or the condition can be restated such that only $L(M)$ is described by it.

For a more formal proof of that condition, simply show that the implication holds


\stepcounter{subsection}
\subsection{The method of the Kolmogorov-Complexity}
\inlinetheorem The problem of computing the Kolmogorov-Complexity $K(x)$ for each $x$ is algorithmically unsolvable.

\inlinelemma If $L_H \in \cL_R$, then there exists an algorithm to compute the Kolmogorov-Complexity $K(x)$ for each $x \in \wordbool$


As of HS2025, chapters 5.5 and 5.7 are not relevant for the Endterm or Session exam, so they are omitted here
