From 2fd0ab1981197955804d695fc6da2da97dc6668a Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 31 Jan 2026 14:13:03 +0100 Subject: [PATCH] [TI] Clean up summary --- .../02_finite-automata/00_representation.tex | 1 - .../02_finite-automata/03_non-determinism.tex | 27 ------------------ .../main/parts/04_computability/00_intro.tex | 4 --- .../parts/04_computability/01_reduction.tex | 1 - .../parts/04_computability/03_kolmogorov.tex | 2 -- .../ti/main/parts/05_complexity/00_intro.tex | 1 - semester3/ti/main/ti-summary.pdf | Bin 1011818 -> 1009615 bytes 7 files changed, 36 deletions(-) diff --git a/semester3/ti/main/parts/02_finite-automata/00_representation.tex b/semester3/ti/main/parts/02_finite-automata/00_representation.tex index df5a023..ecc9510 100644 --- a/semester3/ti/main/parts/02_finite-automata/00_representation.tex +++ b/semester3/ti/main/parts/02_finite-automata/00_representation.tex @@ -34,7 +34,6 @@ Heute verwendet man meist einen gerichteten Graphen $G(A)$: \item Jeder Knoten hat den Ausgangsgrad $|\Sigma|$ (wir müssen alle Fälle abdecken) \end{itemize} -% TODO: Clean up (make it look less crowded) \begin{definition}[]{Endlicher Automat} Ist eine Quitupel $M = (Q, \Sigma, \delta, q_0, F)$: \begin{enumerate}[label=\textit{(\roman*)}] diff --git a/semester3/ti/main/parts/02_finite-automata/03_non-determinism.tex b/semester3/ti/main/parts/02_finite-automata/03_non-determinism.tex index eb101f2..1d8c169 100644 --- a/semester3/ti/main/parts/02_finite-automata/03_non-determinism.tex +++ b/semester3/ti/main/parts/02_finite-automata/03_non-determinism.tex @@ -102,30 +102,3 @@ Es gibt gewisse NEA, bei welchen man bei der Simulation des Nichtdeterminismus d Man kann beweisen (siehe Seiten 83 und 84 mit Abbildung 3.19 im Buch (= Seiten 98 \& 99 im PDF)), dass man die Potenzmengenkonstruktion nicht allgemein verbessern kann. \inlinelemma Für alle $k \in \N - \{ 0 \}$ muss jeder EA, der $L_k = \{ x1y \divides x \in \wordbool, y \in (\alphabetbool)^{k - 1} \}$ akzeptiert, mindestens $2^k$ Zustände haben. - -% FIXME: Verify with TA that this is correct too -% Else: Note an example from the worked example, TA's approach from the slides or from the book on P100 (PDF) -\fhlc{ForestGreen}{Worked Example} Zeige, das jeder endliche Automat, der die Sprache -\rmvspace -\begin{align*} - L = \{ w \in \{ a , b \}^* \divides w \text{ enthält Teilwort $ab$ gleich oft wie das Teilwort $ba$ enthält} \} -\end{align*} - -\drmvspace -mindestens $n := 5$ Zustände haben muss. - -\begin{table}[h!] - \begin{center} - \begin{tabular}[c]{c|ccccc} - & $ab$ & $(ab)^2$ & $(ab)^3$ & $(ab)^4$ & $(ab)^5$ \\ - \hline - $ab$ & - & $(ba)^2$ & $(ba)^3$ & $(ba)^4$ & $(ba)^5$ \\ - $(ab)^2$ & & - & $(ba)^3$ & $(ba)^4$ & $(ba)^5$ \\ - $(ab)^3$ & & & - & $(ba)^4$ & $(ba)^5$ \\ - $(ab)^4$ & & & & - & $(ba)^5$ \\ - $(ab)^5$ & & & & & - \\ - \end{tabular} - \end{center} -\end{table} -Sei $S = \{ ab, (ab)^2, (ab)^3, (ab)^4, (ab)^5 \}$. -Laut Lemma 3.3 diff --git a/semester3/ti/main/parts/04_computability/00_intro.tex b/semester3/ti/main/parts/04_computability/00_intro.tex index ca561dc..2282211 100644 --- a/semester3/ti/main/parts/04_computability/00_intro.tex +++ b/semester3/ti/main/parts/04_computability/00_intro.tex @@ -3,10 +3,6 @@ Wir definieren $\text{KodTM}$ als die \textit{Menge der binären Kodierungen all Wir haben $\text{KodTM} \subseteq \wordbool$ und die obere Schranke der Kardinalität ist $|\wordbool|$, da es unendlich viele Turingmaschinen gibt. Im Folgenden wird wieder Cantor's Diagonalisierungsmethode verwendet -\begin{recall}[]{Cantor's Diagonalization Argument} - % TODO: Finish - \TODO Finish -\end{recall} \inlinedef $A$ und $B$ sind Mengen. Dann ist $|A| \leq |B|$ falls eine \textit{injektive} Funktion $f$ von $A$ nach $B$ existiert; $|A| = |B|$ falls $|A| \leq |B|$ und $|B| \leq |A|$ (es existiert eine Bijektion); diff --git a/semester3/ti/main/parts/04_computability/01_reduction.tex b/semester3/ti/main/parts/04_computability/01_reduction.tex index 545ac21..28f92ab 100644 --- a/semester3/ti/main/parts/04_computability/01_reduction.tex +++ b/semester3/ti/main/parts/04_computability/01_reduction.tex @@ -1,6 +1,5 @@ \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/main/parts/04_computability/03_kolmogorov.tex b/semester3/ti/main/parts/04_computability/03_kolmogorov.tex index 01ce6fc..5733fad 100644 --- a/semester3/ti/main/parts/04_computability/03_kolmogorov.tex +++ b/semester3/ti/main/parts/04_computability/03_kolmogorov.tex @@ -2,5 +2,3 @@ \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/main/parts/05_complexity/00_intro.tex b/semester3/ti/main/parts/05_complexity/00_intro.tex index 878c131..2d1356c 100644 --- a/semester3/ti/main/parts/05_complexity/00_intro.tex +++ b/semester3/ti/main/parts/05_complexity/00_intro.tex @@ -48,7 +48,6 @@ Es ist auch möglich $\spc_M(n)$ als eine Summe zu definieren, aber laut Lemma \ \inlinedef Wir notieren mit der big-O-notation folgendermassen: Falls $r \in \tco{f(n)}$, so wächst $r$ asymptotisch nicht schneller als $f$. Äquivalent für $s \in \tcl{g(n)}$ und $l \in \tct{h(n)}$ sagen wir asymptotisch mindestens so (gleich) schnell. Falls $\limni \frac{f(n)}{g(n)} = 0$, dann wächst $g$ asymptotisch schneller als $f$ und $f(n) = o(g(n))$ -% TODO: Check if above really is a typo (pretty sure it is) \inlinetheorem Es existiert ein Entscheidungsproblem $(\alphabetbool, L)$, so dass für jede MTM $A$, die $(\alphabetbool, L)$ entscheidet, eine MTM $B$ existiert, die es auch entscheidet und für die gilt: $\tc_B(n) \leq \log_2(\tc_A(n))$ für alle $n \in \N$ diff --git a/semester3/ti/main/ti-summary.pdf b/semester3/ti/main/ti-summary.pdf index f13f458931362cc04b1e3a9627f43cbed0cd3c25..ed768b96d3830c957375af1aab9387fc34144525 100644 GIT binary patch delta 132370 zcmV)YK&-#&;62aJJ+NH~0x&d}A)N>*e_Ts%BgYZG>sJhNq5w~CzaPmVfaAwzeTWw; zAV4-g&|0k}uSnVw<-oq=ucxZtJr9ygCX|7!u@=c@O?7pBRn=d2&yaRMbhP{W;`6JE z7hiGPIpQ?e-PQX}I-;rWjFZF})m^>o-Y%|gZ#EA%+q=i>OKumpHS4p(zU3a~@c>5jg z-obaTI!c7I-M_r#3M4h5TEP4J?#;#P3mQ$Id-X$-vg48%YZ#7+DN2-JouQh*|D!<( z76xV1U{Ih)b&L_G6!(glBk-;lf09_!j+E;>(idMDSziT;38l8{nIKNa>u$H(pC14C 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