From 07e55b00663261f45bffb7bab4e5b736fff36526 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 17 Oct 2025 07:47:08 +0200 Subject: [PATCH] [TI] Finish section on proofs of nonexistance --- .../02_proofs-of-nonexistance.tex | 6 +++++- semester3/ti/ti-summary.pdf | Bin 935653 -> 936330 bytes semester3/ti/ti-summary.tex | 2 +- 3 files changed, 6 insertions(+), 2 deletions(-) diff --git a/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex b/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex index c9a4de8..506cb40 100644 --- a/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex +++ b/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex @@ -124,8 +124,12 @@ Dies ist aber unmöglich, da: Für komplexere Sprachen ist es oft einfach, $L_x$ so zu wählen, dass $x = a^{\alpha + 1}$ ist, wobei $\alpha$ der Exponent (nach Variabelnwechsel) aus der Sprache ist. Also beispielsweise für $L = \{ 0^{n^2 \cdot 2n \divides n \in \N }\}$ ist $\alpha = m^2 \cdot 2m$, also ist $x = 0^{m^2 \cdot 2m + 1}$. -$y_1$ (das erste Wort der Sprache $L_x$) ist dann $y_1 = 0^{(m + 1)^2 \cdot 2(m + 1) - m^2 \cdot 2m + 1}$ +$y_1$ (das erste Wort der Sprache $L_x$) ist dann $y_1 = 0^{(m + 1)^2 \cdot 2(m + 1) - m^2 \cdot 2m + 1}$. +Wir können dann mit der Länge des Wortes $|y_1|$ und dem Theorem 3.1 argumentieren, dass wir einen Widerspruch erreichen und so also die Sprache nichtregulär ist. +Dazu sagen wir, dass für jedes $m \in \N$ ein $c$ existiert, so dass $K(y_1) \leq \ceil{\log_2(1 + 1)} + c = 1 + c$. +Da unser Wort $y_1$ unendlich lang werden kann, gibt es unendlich viele solcher Wörter. +Dies widerspricht jedoch dem Fakt, dass es nur endlich viele Programme mit Kolmogorov-Komplexität $\leq 1 + c$ gibt. \numberingOn diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 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