From 6ac6ee24cc5e34ef3c5ecd50526571e7d654ed77 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 25 Sep 2025 16:50:11 +0200 Subject: [PATCH] [TI] Fix errors, add useful property --- .../ti/parts/languages-problems/alphabet.tex | 5 ++++- semester3/ti/ti-summary.pdf | Bin 744069 -> 744557 bytes 2 files changed, 4 insertions(+), 1 deletion(-) diff --git a/semester3/ti/parts/languages-problems/alphabet.tex b/semester3/ti/parts/languages-problems/alphabet.tex index 3d23367..4728927 100644 --- a/semester3/ti/parts/languages-problems/alphabet.tex +++ b/semester3/ti/parts/languages-problems/alphabet.tex @@ -68,7 +68,7 @@ Das Problem hierbei ist jedoch, dass dies nicht so effizient ist, besonders nich u < v \Longleftrightarrow |u| < |v| \lor (|u| = |v| \land u = x \cdot s_i \cdot u' \land v = x \cdot s_j \cdot v') \text{ für beliebige $x, u', v' \in \Sigma^*$ und $i < j$} \end{align*} - Oder in Worten, geordnet nach Länge und dann danach, ob man eine gemeinsames Teilwort finden kann + Oder in Worten, geordnet nach Länge und dann danach für den ersten nicht gemeinsamen Buchstaben, nach dessen Ordnung. \end{definition} @@ -79,6 +79,9 @@ Das Problem hierbei ist jedoch, dass dies nicht so effizient ist, besonders nich Die \bi{Konkatenation} von $L_1$ und $L_2$ ist $L_1 \cdot L_2 = L_1 L_2 = \{ vw \divides v \in L_1 \land w \in L_2 \}$ und $L^0 := L_{\lambda}$ und $L^{i + 1} = L^i \cdot L \smallhspace \forall i \in \N$ und $L^* = \bigcup_{i \in \N} L^{i}$ ist der \bi{Kleene'sche Stern} von $L$, wobei $L^+ = \bigcup_{i \in \N - \{0\}} L^i = L \cdot L^*$ \end{definition} +Für jede Sprache $L$ gilt $L^2 \subseteq L \Longrightarrow L = \emptyset \lor L = \{ \lambda \} \lor L$ ist undendlich. +Diese Aussage muss jedoch an der Prüfung bewiesen werden (nicht im Buch vorhanden) + Da Sprachen Mengen sind, gelten auch die Üblichen Operationen, wie Vereinigung ($\cup$) und Schnitt ($\cap$). Die Gleichheit von zwei Sprachen bestimmen wir weiter mit $A \subseteq B \land B \subseteq A \Rightarrow A = B$. 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