From 7e4fa445394c7936f8f25b7503c11b80a378dcd9 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 2 Dec 2025 17:15:04 +0100 Subject: [PATCH] [TI] Compact: Start --- .../ti-compact/parts/04_computability.tex | 30 +++++++++++++++++- semester3/ti-compact/parts/05_complexity.tex | 0 semester3/ti-compact/ti-compact.pdf | Bin 332115 -> 335815 bytes semester3/ti-compact/ti-compact.tex | 4 +++ 4 files changed, 33 insertions(+), 1 deletion(-) create mode 100644 semester3/ti-compact/parts/05_complexity.tex diff --git a/semester3/ti-compact/parts/04_computability.tex b/semester3/ti-compact/parts/04_computability.tex index 9273a2c..462f4ce 100644 --- a/semester3/ti-compact/parts/04_computability.tex +++ b/semester3/ti-compact/parts/04_computability.tex @@ -28,7 +28,7 @@ Proving that a language \textit{is} recursively enumerable is as difficult as pr 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 +\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 \} \\ @@ -45,3 +45,31 @@ In other, more different, words, $w_i$ being in $L_\text{diag}$ implies (from th \setLabelNumber{theorem}{3} \inlinetheorem $L_\text{diag} \notin \cL_{RE}$ + +% ──────────────────────────────────────────────────────────────────── + + +\subsection{Reductions} +This is the start of the topics that are part of the endterm. + +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 $L_{EQ} = \{ \text{Kod}(M)\# \text{Kod}(\overline{M}) \divides L(M) = L(\overline{M}) \}$ ($\in \cL_{RE}$, but $\notin \cL_R$) + \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} + +\setLabelNumber{theorem}{6} +\fancytheorem{Universal TM} A TM $U$, such that $L(U) = L_U$ + +% ──────────────────────────────────────────────────────────────────── + +\subsection{Rice's Theorem} +\setLabelNumber{theorem}{9} +\fancytheorem{Rice's Theorem} diff --git 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