From 572cbad1c2251c826b351c7479f0b076aba8f1b6 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 7 Nov 2025 12:18:37 +0100 Subject: [PATCH] [TI] Compact: Add some notes --- .../ti-compact/parts/01_words-alphabets.tex | 6 ++++++ .../ti-compact/parts/02_finite-automata.tex | 2 +- semester3/ti-compact/ti-compact.pdf | Bin 330810 -> 331209 bytes semester3/ti-compact/ti-compact.tex | 2 ++ 4 files changed, 9 insertions(+), 1 deletion(-) diff --git a/semester3/ti-compact/parts/01_words-alphabets.tex b/semester3/ti-compact/parts/01_words-alphabets.tex index 23b0444..e184f0f 100644 --- a/semester3/ti-compact/parts/01_words-alphabets.tex +++ b/semester3/ti-compact/parts/01_words-alphabets.tex @@ -31,10 +31,16 @@ whereas $L_\lambda$ is the language with just the empty word in it. \bi{Cleen Star}: $L^* = \bigcup_{i \in \N} L^i$ and $L^+ = L \cdot L^*$ +Of note is that there are irregular languages whose Cleen Star is regular, most notably, +the language $L = \{ w \in \{ 0 \}^* \divides |w| \text{ is prime} \}$'s Cleen Star is regular, +due to the fact that the prime factorization is regular + \inlinelemma $L_1L_2 \cup L_1 L_2 = L_1(L_2 \cup L_3)$ \inlinelemma $L_1(K_2 \cap L_3) \subseteq L_1 L_2 \cap L_1 L_3$ +For multiple choice questions, really think of how the sets would look to determine if they fulfill a requirement. + \stepcounter{subsection} \subsection{Kolmogorov-Complexity} diff --git a/semester3/ti-compact/parts/02_finite-automata.tex b/semester3/ti-compact/parts/02_finite-automata.tex index 237f9c9..a62b96f 100644 --- a/semester3/ti-compact/parts/02_finite-automata.tex +++ b/semester3/ti-compact/parts/02_finite-automata.tex @@ -54,7 +54,7 @@ For all of them start by assuming that $L$ is regular. \setLabelNumber{lemma}{3} \begin{lemma}[]{Regular words} Let $A$ be a FA over $\Sigma$ and let $x \neq y \in \Sigma^*$, such that $\hdelta_A (q_0, x) = \hdelta(q_0, y)$. - Then for each $z \in \Sigma^*$ there exists a $r \in Q$, such that $xz, yz \in \class[r]$, and we thus have + Then for each $z \in \Sigma^*$ there exists an $r \in Q$, such that $xz, yz \in \class[r]$, and we thus have \rmvspace \begin{align*} xz \in L(A) \Longleftrightarrow yz \in L(A) diff --git a/semester3/ti-compact/ti-compact.pdf b/semester3/ti-compact/ti-compact.pdf index d4ff5badda93acc2d58580fc2c8197f09a47f279..c1912f15e37b40aabd02ff575efd81484ad1de06 100644 GIT binary patch delta 40267 zcmV(_K-9mwm=wvI6tK$*0Y08kuF~#R<}lNutfp;7rII9l zQFis{-u&%6{k;8Q_hR$c-R8#)fiQ4FX14^9LM2tZ`R)gt?BU&u1S6)3VMq7W1nrq8p} z;r`E6Y3v?G7ZZn91Xf1ts+1OpOC<;_3K$UDbtUYMX z7HDHoj))XFS#-85P_caBVS&Xktjz!ikIfPcam{~D?6_hemObd;`dfG*gKAdD9?0qI4NEF@=u5X3ONT~Hd3D^3G!8DNMNR9*B3 z4kTnPUwBzym#hZ(g4Nj8$2NP3Qp#qqpomdexzTt8ymK%Wpx~yNp%6+mYyyq0ed>pP zG4)-anbx@;nqH?qyyOD5a%|GWJmAmk6)ArpjKOMK@ED*4ml4VlA*6t`Mz|0XcrDDP zEY-|s0513ppyLxHAJZ8u%-)|btBr;BFST)%P=0=>wA;&!KLSY_-*U4-llpYRIQn3 z2E01VXPmr;TY{l%0rk``3M!?e(OG7LtEyY>hOLkAa&HC|mMY+2T&Nc3UOW^H6sOBV9aw3G3M^ 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