From 0a7b8bc75fecda5d17ab89c86ffa511896dbc710 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 29 Sep 2025 13:20:09 +0200 Subject: [PATCH] [TI] More on kolmogorov-complexity --- latex | 2 +- .../kolmogorov-complexity.tex | 24 +++++++++++++----- semester3/ti/ti-summary.pdf | Bin 2356 -> 785683 bytes semester3/ti/ti-summary.tex | 2 +- 4 files changed, 19 insertions(+), 9 deletions(-) diff --git a/latex b/latex index 4a98d7d..6c75936 160000 --- a/latex +++ b/latex @@ -1 +1 @@ -Subproject commit 4a98d7d3af358cc0a20bf9797fda643180f8135f +Subproject commit 6c75936b1872a110a86843b2ec0c25abed3f45f1 diff --git a/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex b/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex index 372b7da..98d9e66 100644 --- a/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex +++ b/semester3/ti/parts/languages-problems/kolmogorov-complexity.tex @@ -12,7 +12,7 @@ Hier kommt die Kolmogorov-Komplexit zum Zuge: Sie bietet eine breit Gültige Def \begin{definition}[]{Kolmogorov-Komplexität} - Für jedes Wort $x \in \wordbool$ ist die \bi{Kolmogorov-Komplexität $K(x)$ des Wortes $x$} das Minimum der binären Längen der Pascal-Programme, die $x$ generieren. + Für jedes Wort $x \in \wordbool$ ist die \bi{Kolmogorov-Komplexität $K(x)$ des Wortes $x$} das Minimum der binären Längen der Pascal-Programme, die $x$ generieren. \end{definition} Hierbei ist mit der binären Länge die Anzahl Bits gemeint, die beim Übersetzen des Programms in einen vordefinierten Maschinencode entsteht. @@ -21,25 +21,35 @@ Ein Pascal-Programm in diesem Kurs ist zudem nicht zwingend ein Programm in der \begin{lemma}[]{Kolmogorov-Komplexität} - Für jedes Wort $x \in \wordbool$ existiert eine Konstante $d$ so dass $K(x) \leq |x| + d$ + Für jedes Wort $x \in \wordbool$ existiert eine Konstante $d$ so dass $K(x) \leq |x| + d$ \end{lemma} \inlineproof Für jedes $x \in \wordbool$ kann folgendes Programm $A_x$ verwendet werden: -\begin{minted}{pascal} -A_x: begin +\begin{code}{pascal} +$A_x$: begin write(x); end -\end{minted} +\end{code} Alle Teile, ausser $x$ sind dabei von konstanter Länge, also ist die Länge der Bit-repräsentation des Programms ausschliesslich von der binären Länge des Wortes $x$ abhängig. \proven -Für regelmässige Wörter gibt es natürlich Programme, bei denen das Wort nicht als komplette Variable vorkommt. +Für regelmässige Wörter gibt es natürlich Programme, bei denen das Wort nicht als komplette Variable vorkommt. Deshalb haben diese Wörter auch (meist) eine kleinere Kolmogorov-Komplexität. \fancydef{$K(n)$ für $n \in \N$} Die \bi{Kolmogorov-Komplexität einer natürlichen Zahl $n$} ist $K(n) = K(\text{Bin}(n))$, wobei $\text{Bin}(x) = \ceil{\log_2(x + 1)}$ % TODO: Verify correctness here -\inlinelemma Für jede Zahl $n \in \N - \{ 0 \}$ existiert ein Wort $w_n \in (\alphabets{bool})^n$ +\inlinelemma Für jede Zahl $n \in \N - \{ 0 \}$ existiert ein Wort $w_n \in (\alphabets{bool})^n$ so dass $K(w_n) \geq |w_n| = n$, oder in Worten, es existiert für jedes $n$ ein nicht komprimierbares Wort. + +Eine wichtige Eigenschaft der Kolmogorov-Komplexität ist, dass sie nicht wirklich von der gewählten Programmiersprache abhängt. +Man kann also beliebig auch \texttt{C++}, \texttt{Swift}, \texttt{Python}, \texttt{Java} oder welche auch immer, ohne dass die Kolmogorov-Komplexität um mehr als eine Konstante wächst (auch wenn diese bei \texttt{Java} sehr gross ist): + +\begin{theorem}[]{Unterschiedliche Programmiersprachen} + Für jede Programmiersprachen $A$ und $B$ existiert eine Konstante $c_{A,B}$, die nur von $A$ und $B$ abhängig ist, so dass für alle $x \in \wordbool$ gilt: + \begin{align*} + |K_A(x) - K_B(x)| \leq c_{A, B} + \end{align*} +\end{theorem} diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 4e4e4a3958190d5442e0ee050735272df6369eb6..19b9414d5aa08f658262eb42ba2f6c7c1db1b869 100644 GIT binary patch literal 785683 zcmeFZWpL)ok|u0sW@ct)W@g4Vyk=%*W@ct)X2v$#ZDwk>-Dbw^Gc)_$GdmNp^XK00 z&niL+g_K#U%&dGWl}Z)4l9&WNGXp0K`TX+e8Vnl|6Op~K6$~FA45ON-1AvH8OVQX0 zVB!M9sOoC$@^4op?44|37)3M~9cIGY?M9l0&|FQjP#nQ$F;6%hIVPoV15CfRl zn*v}21Yn$9oB&3)FdiFAh_gymvwI*wkRQO;FaSH#|G4w#?60nK{%QDMBLdeyO9bwJ znF!2(8Rws32Ijv^{;#nEGs{0sh=1K+V*aO%CYHZ!{I9VD%U?$Srwk_+)_%R`2SpRA0#QLxE3D$p_Pq6+K2>ff%iS;k{`=|H> zGt1vS`#)kDb`GxpaZHn{ChMHdgw%IO>k^VlWc?u4OO8JtZ+YnUo!SneF4me80&ieR z+TT|*f@7@_k)J5d*FWFQ=m^rt6m6??a4(N?M5Vfu;w(%`mVIeNOS)E>Og~vX!>+k$ zDoejN$^*r&nF>QBEZJC9nn_;ln_}f6i@_IDv4Ae 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