From 9ac7514ab6c32c10f7400cdafbcb5d89dbf3728b Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 10 Nov 2025 07:49:35 +0100 Subject: [PATCH] [TI] Grammar fixes, update to new helpers --- .../02_proofs-of-nonexistance.tex | 6 +++--- semester3/ti/ti-summary.pdf | Bin 967509 -> 967696 bytes semester3/ti/ti-summary.tex | 4 +--- 3 files changed, 4 insertions(+), 6 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 11db675..e606dd2 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 @@ -24,7 +24,7 @@ so heisst das für uns von jetzt an, dass $A$ nicht zwischen $x$ und $y$ untersc Das obenstehende Lemma 3.3 ist ein Spezialfall einer Eigenschaft, die für jedes (deterministische) Rechnermodell gilt. Es besagt eigentlich nichts anderes, als dass wenn das Wort $xz$ akzeptiert wird, so wird auch das Wort $yz$ -Mithilfe von Lemma 3.3 kann man für viele Sprachen deren Nichtregularität beweisen. +Mithilfe von Lemma 3.3 kann man für einen grossteil Sprachen deren Nichtregularität beweisen. \numberingOff \inlineex Sei $L = \{ 0^n1^n \divides n \in \N \}$. @@ -36,7 +36,7 @@ Wir nehmen an, dass $L$ regulär ist und betrachten die Wörter $0^1, 0^2, \ldot Weil wir $|Q| + 1$ Wörter haben, existiert per Pigeonhole-Principle o.B.d.A $i < j \in \{ 1, 2, \ldots, |Q| + 1 \}$ (die Ungleichheit kann in komplexeren Beweisen sehr nützlich werden, da wir dann besser mit Längen argumentieren können), so dass $\hdelta_A(q_0, 0^i) = \hdelta_A(q_0, 0^j)$, also gilt nach Lemma $0^i z \in L \Leftrightarrow 0^j z \in L \smallhspace \forall z \in \wordbool$. -Dies gilt jedoch nicht, weil für jedes $z = 1^i$ zwar jedes $0^i 1^i \in L$ gilt, aber $0^j 1^j \notin L$ +Dies gilt jedoch nicht, weil für jedes $z = 1^i$ zwar jedes $0^i 1^i \in L$ gilt, aber $0^i 1^j \notin Lh$ \numberingOn @@ -52,7 +52,7 @@ denn wenn eine Sprache eine dieser Eigenschaften \textit{nicht} erfüllt, so ist \fhlc{Cyan}{Pumping} -Eine Methode zum Beweis von Aussagen $L \notin \mathcal{L}_{\text{EA}}$ nennt sich \bi{Pumping} und basiert auf folgender Idee: +Eine weitere Methode zum Beweis von Aussagen $L \notin \mathcal{L}_{\text{EA}}$ nennt sich \bi{Pumping} und basiert auf folgender Idee:\\ Wenn für ein Wort $x$ und einen Zustand $p$ gilt, dass $(p, x) \bigvdash{A}{*} (p, \lambda)$, so gilt auch für alle $i \in \N$, dass $(p, x^i) \bigvdash{A}{*} (p, \lambda)$. 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b/semester3/ti/ti-summary.tex @@ -1,8 +1,6 @@ \documentclass{article} -\newcommand{\dir}{~/projects/latex} -\input{\dir/include.tex} -\load{full} +\input{~/projects/latex/dist/full.tex} \setLang{de} \setup{Theoretische Informatik}