From e38d8c6c946304c2e58df2986b7516702a57f293 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 29 Sep 2025 17:29:31 +0200 Subject: [PATCH] [TI] Kolmogorov-Complexity --- .../kolmogorov-complexity.tex | 35 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 785683 -> 802259 bytes 2 files changed, 35 insertions(+) diff --git a/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex b/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex index 98d9e66..a1962fe 100644 --- a/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex +++ b/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex @@ -1,5 +1,6 @@ \newpage \subsection{Kolmogorov-Komplexität} +\setcounter{definitions}{17} Falls ein Wort $x$ eine kürzere Darstellung hat, wird es \bi{komprimierbar genannt} und wir nennen die Erzeugung dieser Darstellung eine \bi{Komprimierung} von $x$. Eine mögliche Idee, um den Informationsgehalt eines Wortes zu bestimmen, wäre einem komprimierbaren Wort einen kleinen Informationsgehalt zuzuordnen und einem unkomprimierbaren Wort einen grossen Informationsgehalt zuzuordnen. @@ -53,3 +54,37 @@ Man kann also beliebig auch \texttt{C++}, \texttt{Swift}, \texttt{Python}, \text |K_A(x) - K_B(x)| \leq c_{A, B} \end{align*} \end{theorem} + +\fhlc{orange}{Anwendungen der Kolmogorov-Komplexität} + +\shade{orange}{Zufall} Der Zufall ist ein intuitiver, aber nicht sehr formeller Begriff, der mit der Kolmogorov-Komplexität formalisiert werden kann: +\begin{definition}[]{Zufall} + Ein Wort $x \in \wordbool$ (eine Zahl $n$) heisst \bi{zufällig}, falls $K(x) \geq |x|$ ($K(n) = K(\text{Bin}(n)) \geq \ceil{\log_2(n + 1)} - 1$) +\end{definition} + +\shade{orange}{Existenz eines Programms vs Kolmogorov-Komplexität} +\begin{theorem}[]{Programm vs Komplexität} + Sei $L$ eine Sprache über $\alphabets{bool}$ und für jedes $n \in \N - \{0\}$ sei $z_n$ das $n$-te Wort in $L$ bezüglich der kanonischen Ordnung. + Falls ein Programm $A_L$ existiert, das das Entscheidungsproblem $(\alphabets{bool}, L)$ löst, so gilt für alle $n \in \N - \{ 0 \}$ dass + \begin{align*} + K(z_n) \leq \ceil{\log_2(n + 1)} + c & & (c \text{ ist eine von } n \text{ unabhängige Konstante }) + \end{align*} +\end{theorem} + +\shade{orange}{Primality testing} +\begin{theorem}[]{Primzahlensatz} + \vspace{-0.3cm} + \begin{align*} + \limni \frac{\text{Prim}(n)}{\frac{n}{\ln(n)}} = 1 + \end{align*} +\end{theorem} +Die Annäherung von $\text{Prim}(n)$ and $\frac{n}{\ln(n)}$ wird durch folgende Ungleichung gezeigt: +\begin{align*} + \ln(n) - \frac{3}{2} < \frac{n}{\text{Prim}(n)} < \ln(n) - \frac{1}{2} \smallhspace \forall n \geq 67 \in \N +\end{align*} + +\begin{lemma}[]{Anzahl Primzahlen mit Eigenschaften} + Sei $n_1, n_2, \ldots$ eine stetig steigende unendliche Folge natürlicher Zahlen mit $K(n_i) \geq \frac{\ceil{\log_2(n_i)}}{2}$. + Für jedes $i \in \N - \{ 0 \}$ sei $q_i$ die grösste Primzahl, die $n_i$ teilt. + Dann ist die Menge $Q = \{ q_i \divides i \in \N - \{ 0 \} \}$ +\end{lemma} diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 19b9414d5aa08f658262eb42ba2f6c7c1db1b869..21bfb2744bfac54d2a43ad116d6428f47f8739cb 100644 GIT binary patch delta 118754 zcmV)1K+V6C`Y+S_Fo1*sgaU*Egam{Iga(8Mv^XnbRqeQO>7G(){yd_Vj2AX?tZ3Hyq zsikS4S(yAJ&W}*&HDRY2yN{H5%@dSZlDW3$a!P zQyI)BROaLw!Qz^>)chI`Jb~s#9e`)zp@LW_A9&4f1J_E zZcO(@FAn4|c%eV<6UMorq9rWVvjy&KSoiK7r!mo>beVh}Ywu60yK5=H&b!Y}hF{U3 z9kCg2Rbq>lzFngMd$buKao0u3X-8k+MnOK5w|DgX9X$O%$+yF*LFV$K`*wY)Z+98v zdju+Zx!WK-?h(^~i_iJy$L$#Lf2ahYln+0D}VBt$nC0X zKAXLdCN-@0=-9}m_t9vCA-V@EaA|c3R+}N9gdIkbaWaEcS=Z?PKukWOeE-!IZd;0h znz&#r`NQLZApy((u_{3uw?^&$0M8NSdo*ZH=QAlWNf~IvK){Va1o}H^f9(kA3Jice zFhsm`S$B3~Kg?-c!-aU@6>_%X@7K6S6yeXF44s@n6-l<2j@$|xfYHSVD?T2 zThxWeBy7E-MfUkne{A9mOL=okM>~pC8h%0T^*SYHd0dT=3$0VAQ$c?|0T+>?!+U%)`oQ_zP9i;=je?n6DG3t(MtRU-UJrTFy-peE%Yy{?zY_A(1`5s+J;kAUFF9p z-PVWTDZ5nHbf6)e#Q-ii?`&aJwYPK62utx7Ufo%R{-$1j1fUChsQJ7TnLXPeSYaot zvq(USqc%Hze>>=7Kmp*=GG1iG01C+QV-s3LP(UtHrIn1JfIx|$fIy3&fI#!&^vs>W zy$JtWD1a3o5(^MqrVtFA!~%StsUyoU$qP`_Lvewq+DTl16_)@5(Xe$!fIr z&_@B7u{YtTPZQhg*5^(Wqh&c9=J+(R8l5IyDnsWGF;p_DpNU|Mi>V{Ys4*d8E_Nom zwV$N`20!76%SjDFMa^^=QPA0N<{2@n*?XS-#N-ZUBB9EfDlT|;dn|F`^|GQJe zbbc^8e?<$lQ%q05>K6R^d=_1w+yO{7yY&Q4#_>28hO>BsR!2+5@oFSh;GCmO50l$s z@t<7VA3nD=kW@1hC^tE*-^~RRC>}OqcI>|cp3dZG5@jG;o)_^X3StyRJc$wk<{301 zt|49tL|~PPROg%~Q6gp%LSKd;xaS@yUIb+3e`(Pq8fadep1Bja7vcZYB+4OJ$W(T6 z67_io+pu=_)DN9Jqw3yc@{ES;@zNgGe7TWBso!P)v?%;nMIZ&v_!8cq{yObsvJ97N?AoSYPP!X5Y2oXcIUyhHdKQKI{8te^-vce%&$jGt0v8u7gXNVT zf8gR%8Yrw{qf9e*mOvRNwi^Ubb*4Yyb^|-Ll{_`yz=|B#vr#U)-CGnVwtL~_!!Hs| zDH|4hipwco4;Vu(A!iz_Ce$tTwN3qcu^@P)LZ0pOBRPJZ?eov4H3x5|Cw-jkVKn=L z_RdHh*0WSn;_0_x*b%-o*0@N)?*|PIe*!2M-4;AGnYkGDIKT1M%%VMHgZ}V;j};*4 z_)BFYe97Se1!|d3e&CJk*Gh5_`yZYt+VTD-o-9PrXyXK))eN(tooxvg&7!N=U@PZ+snOSrWv&SPMtjc(r~a4{z57D1KY{|=C!>_2ig-8Yb9 zzHi{Ay<8MpWkzhxVL3t>Lo@JuZqGgmmTAs*en)!HnqDat%(v@zN!a*RK zpPH*r-!9(#+v^$DveAm>; 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