vault backup: 2026-04-08 10:35:59

.obsidian/community-plugins.json

@@ -9,5 +9,6 @@
   "obsidian-file-color",
   "darlal-switcher-plus",
   "dataview",
-  "obsidian-vimrc-support"
+  "obsidian-vimrc-support",
+  "emoji-shortcodes"
 ]

.obsidian/plugins/emoji-shortcodes/data.json

@@ -0,0 +1,9 @@
+{
+  "immediateReplace": true,
+  "suggester": true,
+  "historyPriority": true,
+  "historyLimit": 100,
+  "history": [
+    ":gear:"
+  ]
+}

.obsidian/plugins/emoji-shortcodes/manifest.json

@@ -0,0 +1,11 @@
+{
+	"id": "emoji-shortcodes",
+	"name": "Emoji Shortcodes",
+	"version": "2.2.0",
+	"minAppVersion": "1.0.0",
+	"description": "This Plugin enables the use of Markdown Emoji Shortcodes :smile:",
+	"author": "phibr0",
+	"authorUrl": "https://github.com/phibr0",
+	"isDesktopOnly": false,
+	"fundingUrl": "https://ko-fi.com/phibr0"
+}

.obsidian/plugins/emoji-shortcodes/styles.css

@@ -0,0 +1,31 @@
+a[href="https://ko-fi.com/phibr0"] > img
+{
+  height: 3em;
+}
+
+a[href="https://ko-fi.com/phibr0"]
+{
+  transform: translate(0, 5%);
+}
+
+.ES-suggester-container {
+  display: flex;
+  place-content: space-between;
+}
+
+.ES-shortcode {
+  margin-right: 8px;
+}
+
+.ES-suggestion-item {
+  border-top: solid var(--background-secondary) 1px;
+  padding-left: 10px;
+}
+
+.ES-sub-setting {
+  padding-left: 2em;
+}
+.ES-sub-setting + .ES-sub-setting {
+  padding-left: 0;
+  margin-left: 2em;
+}

.obsidian/workspace.json

@@ -13,12 +13,12 @@
             "state": {
               "type": "markdown",
               "state": {
-                "file": "00 Inbox/29593929 - Alphabets.md",
+                "file": "00 Inbox/29593940 - Formal Languages.md",
                 "mode": "source",
                 "source": false
               },
               "icon": "lucide-file",
-              "title": "29593929 - Alphabets"
+              "title": "29593940 - Formal Languages"
             }
           }
         ]
@@ -184,10 +184,13 @@
   },
   "active": "140d404d9b2faf63",
   "lastOpenFiles": [
+    "00 Inbox/29593952 - Regular Languages.md",
+    "00 Inbox/29593940 - Formal Languages.md",
+    "00 Inbox/29593935 - Kleene.md",
+    "30 Library/29593895 - atfl-st2026-l01-formal-languages-full.pdf",
     "00 Inbox/29593852 - Strings.md",
     "00 Inbox/29593929 - Alphabets.md",
     "30 Library/29592593 - ET_II_Folien_gesamt_020426.pdf",
-    "30 Library/29593895 - atfl-st2026-l01-formal-languages-full.pdf",
     "30 Library/29592593 - 29592593 - ET_II_Folien_gesamt_020426.pdf",
     "30 Library/atfl-st2026-l01-formal-languages-full.pdf",
     "10 Courses/02 - SoSe 2026/Automatentheorie und formale Sprachen/29593850 - Automationtheory.md",
@@ -213,9 +216,6 @@
     "10 Courses/02 - SoSe 2026/ET II/29592709 - Klausur.md",
     "10 Courses/02 - SoSe 2026/ET II/29592729 - Kirchhoffsche Gesetze.md",
     "00 Inbox/29593224 - 29593223 - 29593223 - Elektrotechnik II.md",
-    "00 Inbox/29593223 - 29593223 - Elektrotechnik II.md",
-    "00 Inbox/29593223 - Elektrotechnik II.md",
-    "10 Courses/02 - SoSe 2026/Mathe II/29592601 - Integration.md",
     "20 Atlas",
     "30 Library/ET_II_Folien_gesamt_020426.pdf",
     "30 Library/d6d88eed747e389e597d2a19bb868ee6?cid=4b45b3a6f32c35a6d9ca5b1b07b760de",

00 Inbox/29593852 - Strings.md

@@ -1,8 +1,8 @@
 ---
 created: 2026-04-08 08:52
 course: "[[29593850 - Automationtheory]]"
-topic: "#languages #string #character #kleene #regularExpressions"
-related: "[[29593850 - Automationtheory]]"
+topic: strings
+related: "[[29593929 - Alphabets]]"
 type: lecture
 status: 🔴
 tags:
@@ -17,16 +17,6 @@ tags:
 
 ## 📝 Content
 
-### Alphabets
-Alphabets are formal, non-empty, finite, sets of characters (or _letters_ or _symbols_). They are denoted by $Sigma$.
-
-$Sigma = {a, b}$
-> Alphabet $Sigma$ contains the characters $a$ and $b$.
-
-$Sigma = {a, ..., z, A, ..., Z, 0, ..., 9}$
-> usual alphabet for writing text
-
-### Strings
 A word (or _string_) is a finite sequence $w = a_1 a_2 ... a_n$ if characters from $Sigma$.
 
 > [!CONVENTION]
@@ -40,6 +30,8 @@ The _length_ $abs(x)$ of a string $x = a_1 ... a_n$ is its number $abs(x) = n$ o
 #### Empty String
 The empty string is denoted by $epsilon$, this is the neutral element.
 -> $abs(epsilon) = 0$
+
+## String Operations
 ### Concatenation
 String can be concatenated, where one string is appended to another. 
 For strings $x = a_1 ... a_n$ and $y  = b_1 ... b_m$ over alphabets $Sigma_x$ and $Sigma_y$, their _concatenation_ over the alphabet $Sigma = Sigma_x union Sigma_y$ is the string
@@ -56,6 +48,7 @@ Order of operations / Brackets do _not matter_. (Concatenation is associative bu
 
 Any string concatenated with the empty string $epsilon$ will result in itself.
 > $x circle.small epsilon = x = epsilon circle.small x$ 
+
 ### Exponentiation
 
 The $n^"th"$ power $x^n$ of a string $x$ is the $(n-1)$-fold concatenation of $x$ with itself.
@@ -69,55 +62,10 @@ The $n^"th"$ power $x^n$ of a string $x$ is the $(n-1)$-fold concatenation of $x
 ### Reversing / Mirroring
 For a string $x = a_1 a_2 ... a_(n-1) a_n$ of length $n$, it's _mirrored string_ is given by 
 $$ x^("Rev") = a_n a_(n-1)...a_2 a_1$$
-### Substrings
+## Substrings
 A string $x$ is a _substring_ of a string $y$ if $y = u x v$, where $u$ and $v$ can be arbitrary strings.
 - If $u = epsilon$ then $x$ is a _prefix_ of $y$.
 - If $v = epsilon$ then $x$ is a suffix of $y$.
 
 For strings $x$ and $y$ the quantity $abs(y)_x$ is the number of times that $x$ is a substring of $y$.
 
-### Kleene Star
-Denoted by $Sigma^*$. The Kleene Star (or _Kleene operator_ or _Kleene Closure_) gives an infinite amount of strings made up of the characters of the alphabet $Sigma ^ *$. 
-$Sigma^*$ is the set of all string that can be generated by arbitrary concatenation of its characters.
-> $Sigma^* := union.big_(n>=0) A_n$
-> where $A_n$ is the set of all string combinations of length $n$
-
-#### Remarks
-- The same character can be used multiple times.
-- The empty string $epsilon$ is also part f $Sigma^*$.
-
-> [!Example]
-> $Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}$
-
-> [!FACT]
-> - The set $Sigma^*$ is infinite, since we defined $Sigma$ to be non-empty.
-> - It is _countable_ and has the same cardinality as the set $NN$ of natural numbers
-
-#### Kleene Plus
-The _Kleene Plus_ of an alphabet $Sigma$ is given by $Sigma^+ = Sigma^* backslash {epsilon}$ 
-
-#### Lemma group structure
-The structure _Lemma_ is induced by the Kleene star - it is a monoid, that is a semigroup with a neutral element.
-
-> [!PROOF]
-> - Associativity has been shown
-> - Existence  of a neutral element has been shown.
-> - Closure under $circle.small$: Let $x in Sigma^*$ and $y in Sigma^*$ be two string over the alphabet $Sigma$. Then $x circle.small y = x y in Sigma^*$
-### Formal Languages
-A formal _language_ of the alphabet $Sigma$ is a subset $L$ of $Sigma^*$
-
-### Finite representation of languages
-**Goal:** Represent a language using _finite_ information
-#### Using set notation
-$S = {a^n b^m bar n, m >= 0} = {epsilon, a, b, "aa", "ab", ...}$
-> This is very inefficient.
-
-#### Using regular expressions
-A _regular expression_ $r$ over an alphabet $Sigma$ is defined recursively:
-- $emptyset, epsilon$ and each $a in Sigma$ are regular expression, which represent the Languages $L(emptyset) = emptyset, L(epsilon) = {epsilon}$ and $L(a) = {a}$
-- If $r$ and $s$ are regular expressions then
-	- $(r+s)$
-<mark style="background: #FF5582A6;">	- !! Complete from leture notes</mark>
-
-### Regular languages
-A language $L$ that can be described by a regular expression $r$ (i. e. $L(r) = L$) is called _regular_.

00 Inbox/29593929 - Alphabets.md

@@ -1,18 +1,26 @@
 ---
 created: 2026-04-08 10:09
 course: "[[29593850 - Automationtheory]]"
-topic: "#alphabets"
+topic: alphabets, characters
 related:
 type: lecture
-status: 🔴
+status: 🟢
 tags:
   - university
 ---
-	## 📌 Summary
+## 📌 Summary
 
 > [!abstract]
-> 
+> Definition and examples of alphabets.
 
 ---
 
 ## 📝 Content
+
+Alphabets are formal, non-empty, finite, sets of characters (or _letters_ or _symbols_). They are denoted by $Sigma$.
+
+$Sigma = {a, b}$
+> Alphabet $Sigma$ contains the characters $a$ and $b$.
+
+$Sigma = {a, ..., z, A, ..., Z, 0, ..., 9}$
+> usual alphabet for writing text

00 Inbox/29593935 - Kleene.md

@@ -0,0 +1,46 @@
+---
+created: 2026-04-08 10:15
+course: "[[29593850 - Automationtheory]]"
+topic: kleen
+related: "[[29593929 - Alphabets]]"
+type: lecture
+status: 🟢
+tags:
+  - university
+---
+## 📌 Summary
+
+> [!abstract]
+> 
+
+---
+
+## 📝 Content
+
+### Kleene Star
+Denoted by $Sigma^*$. The Kleene Star (or _Kleene operator_ or _Kleene Closure_) gives an infinite amount of strings made up of the characters of the alphabet $Sigma ^ *$. 
+$Sigma^*$ is the set of all string that can be generated by arbitrary concatenation of its characters.
+> $Sigma^* := union.big_(n>=0) A_n$
+> where $A_n$ is the set of all string combinations of length $n$
+
+#### Remarks
+- The same character can be used multiple times.
+- The empty string $epsilon$ is also part f $Sigma^*$.
+
+> [!Example]
+> $Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}$
+
+> [!FACT]
+> - The set $Sigma^*$ is infinite, since we defined $Sigma$ to be non-empty.
+> - It is _countable_ and has the same cardinality as the set $NN$ of natural numbers
+
+### Kleene Plus
+The _Kleene Plus_ of an alphabet $Sigma$ is given by $Sigma^+ = Sigma^* backslash {epsilon}$ 
+
+### Lemma group structure
+The structure _Lemma_ is induced by the Kleene star - it is a monoid, that is a semigroup with a neutral element.
+
+> [!PROOF]
+> - Associativity has been shown
+> - Existence  of a neutral element has been shown.
+> - Closure under $circle.small$: Let $x in Sigma^*$ and $y in Sigma^*$ be two string over the alphabet $Sigma$. Then $x circle.small y = x y in Sigma^*$

00 Inbox/29593940 - Formal Languages.md

@@ -0,0 +1,58 @@
+---
+created: 2026-04-08 10:20
+course: "[[29593850 - Automationtheory]]"
+topic: languages
+related:
+type: lecture
+status: 🔴
+tags:
+  - university
+---
+## 📌 Summary
+
+> [!abstract]
+> Definition and example for formal languages
+ 	
+
+---
+
+## 📝 Content
+
+A formal _language_ of the alphabet $Sigma$ is a subset $L$ of $Sigma^*$.
+
+> [!EXAMPLE]
+> For alphabet $Sigma = {a, b}$, let $L_1$ be the set of all string starting with $b$, followed by an arbitrary number of $a$'s, and ending with $b$:
+> $L_1 = {b a^n b | n in NN_0}$
+> Then $b b in L_1, b a b in L_1$, etc.
+>
+>More in  [[29593895 - atfl-st2026-l01-formal-languages-full.pdf#page=54]]
+
+## Language Operations
+
+### Concatenation of languages
+For languages $X subset Sigma^*_X$ over alphabet $Sigma_X$ and $Y subset Sigma^*_Y$ over alphabet $Sigma_Y$, their _concatenation_ is
+> $X circle.small Y = X Y = {x y bar x in X and y in Y}$
+
+The concatenation of $X$ and $Y$ thus contains all string combinations where the prefix is a string from $X$ and the suffix is a string from $Y$.
+
+> [!CONVENTION]
+> Concatenation has a higher precedence than set operations ($union, inter$).
+
+### Exponentiation of languages
+The $n^"th"$  power of the language $L subset.eq Sigma^*$ over alphabet $Sigma$ is
+- $L^0 := { epsilon }$
+- $L^n := L^(n-1) L$ if $n > 0$
+
+
+
+## Finite representation of languages
+**Goal:** Represent a language using _finite_ information
+### Using set notation
+$S = {a^n b^m bar n, m >= 0} = {epsilon, a, b, "aa", "ab", ...}$
+> This is very inefficient.
+
+### Using regular expressions
+A _regular expression_ $r$ over an alphabet $Sigma$ is defined recursively:
+- $emptyset, epsilon$ and each $a in Sigma$ are regular expression, which represent the Languages $L(emptyset) = emptyset, L(epsilon) = {epsilon}$ and $L(a) = {a}$
+- If $r$ and $s$ are regular expressions then
+	- $(r+s)$

00 Inbox/29593952 - Regular Languages.md

@@ -0,0 +1,20 @@
+---
+created: 2026-04-08 10:32
+course: "[[29593850 - Automationtheory]]"
+topic: languages
+related: "[[29593940 - Formal Languages]]"
+type: lecture
+status: 🔴
+tags:
+  - university
+---
+## 📌 Summary
+
+> [!abstract]
+> 
+
+---
+
+## 📝 Content
+
+A language $L$ that can be described by a regular expression $r$ (i. e. $L(r) = L$) is called _regular_.