From 6d631f6aa8cf0ae9d3bd4e7313ba7706ffffc25c Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 12 Nov 2025 16:44:30 +0100 Subject: [PATCH] [TI] Finished chapter 5 mostly --- .../parts/04_computability/01_reduction.tex | 1 + .../ti/parts/04_computability/02_rice.tex | 18 ++++++++++++++++++ .../parts/04_computability/03_kolmogorov.tex | 6 ++++++ semester3/ti/parts/05_complexity/00_intro.tex | 0 semester3/ti/ti-summary.pdf | Bin 975671 -> 979540 bytes semester3/ti/ti-summary.tex | 2 ++ 6 files changed, 27 insertions(+) create mode 100644 semester3/ti/parts/04_computability/02_rice.tex create mode 100644 semester3/ti/parts/04_computability/03_kolmogorov.tex create mode 100644 semester3/ti/parts/05_complexity/00_intro.tex diff --git a/semester3/ti/parts/04_computability/01_reduction.tex b/semester3/ti/parts/04_computability/01_reduction.tex index 99c9959..c35880a 100644 --- a/semester3/ti/parts/04_computability/01_reduction.tex +++ b/semester3/ti/parts/04_computability/01_reduction.tex @@ -1,5 +1,6 @@ \newpage \subsection{Die Methode der Reduktion} +% TODO: Add guide for reducing languages \fancydef{Rekursiv reduzierbare Sprache} Eine Sprache $L_1 \subseteq \word_1$ ist auf $L_2 \subseteq \word_2$ rekursiv reduzierbar, geschrieben $L_1 \leq_R L_2$, falls $L_2 \in \cL_R \Rightarrow L_1 \in \cL_R$. diff --git a/semester3/ti/parts/04_computability/02_rice.tex b/semester3/ti/parts/04_computability/02_rice.tex new file mode 100644 index 0000000..7a10a39 --- /dev/null +++ b/semester3/ti/parts/04_computability/02_rice.tex @@ -0,0 +1,18 @@ +\subsection{Der Satz von Rice} +\inlinedef $L$ heisst \bi{semantisch nichttriviales Entscheidungsproblem über Turingmaschinen}, falls folgende Bedingungen gelten: +\begin{enumerate}[label=(\roman*)] + \item Es gibt eine TM $M_1$, so dass $\text{Kod}(M_1) \in L$ (also $L \neq \emptyset$) + \item Es gibt eine TM $M_2$, so dass $\text{Kod}(M_2) \notin L$ (also sind nicht alle Kodierungen in $L$) + \item für zwei TM $A$ und $B$: $L(A) = L(B) \Rightarrow \text{Kod}(A) \in L \Leftrightarrow \text{Kod}(B) \in L$ +\end{enumerate} + +Sei $L_{H, \lambda} = \{ \text{Kod}(M) \divides M \text{ hält auf } \lambda \}$ ein spezifisches Halteproblem. + +\inlinelemma $L_{H, \lambda} \notin \cL_R$ +\inlineproof Auf Seite 146 im Buch (= 159 im PDF) + +\begin{theorem}[]{Satz von Rice} + Jedes semantisch nichttriviale Entscheidungsproblem über Turingmaschinen ist unentscheidbar. +\end{theorem} +\inlineproof Ausführlich im Buch auf Seiten 146 - 149 beschrieben (= 159 - 162 im PDF) +\stepcounter{subsection} diff --git a/semester3/ti/parts/04_computability/03_kolmogorov.tex b/semester3/ti/parts/04_computability/03_kolmogorov.tex new file mode 100644 index 0000000..01ce6fc --- /dev/null +++ b/semester3/ti/parts/04_computability/03_kolmogorov.tex @@ -0,0 +1,6 @@ +\subsection{Die Methode der Kolmogorov-Komplexität} +\inlinetheorem Das Probelem, für jedes $x \in \wordbool$ die Kolmogorov-Komplexität $K(x)$ von $x$ zu berechnen ist algorithmisch unlösbar. + +\inlinelemma Falls $L_H \in \cL_R$, dann existiert ein Algorithmus zur Berechnung der Kolmogorov-Komplexität $K(x)$ für jedes $x\in \wordbool$ + +% TODO: See if we need to do these kinds of proofs and if so, elaborate diff --git a/semester3/ti/parts/05_complexity/00_intro.tex b/semester3/ti/parts/05_complexity/00_intro.tex new file mode 100644 index 0000000..e69de29 diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 56b3921da8acf0cb198a385c17950c0dff5549f5..f1ec0a122d6f17b4279e027082aff093b872e3a3 100644 GIT binary patch delta 68090 zcmXV$Q+Q^Zwr$(CZB0D!B>7|8wvCBx+qP{dXZGIb;_Higs(V%S>Ze!rySinV zckz(7P7)e~i!)IgS_V*1LJ>tN{JKmLV;FuF#Wp6liDwngVjXEFt1?EG`u)UEmq$0I zGOvILqSA&8(sPi!FV6OrMSZzGUj7 zAv0PW2mOsJMMWI*sld9jiZORMw=`AEpA53tV4tOOV!D3p)Elq_aEB{gI6JM4KQIas zG#sE-q)%5b?3js~+`CL*qOp_d_L_#_LLEneeoT;)xRInw^F6)LSBZcUm$zws2>&I~ zbU0WS)PEDKnoy`Jal-{3!M2yBQyr{1NDoev5>aBYp&DPs&+luxSb|1X5>-rgoS=h# 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