vault backup: 2026-04-16 11:54:17

00 Inbox/29593975 - Regular Expressions.md

@@ -1,33 +0,0 @@
----
-created: 2026-04-08 10:55
-course: "[[29593850 - Automationtheory]]"
-topic: languages
-related: "[[29593940 - Formal Languages#Finite representation of languages]]"
-type: lecture
-status: 🔴
-tags:
-  - university
----
-## 📌 Summary
-
-> [!abstract]
-> 
-
----
-
-## 📝 Content
-
-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 these are also regular expressions:
-	- $(r + s)$ with $L(r + s) = L(r) union L(s)$
-	- $(r s)$ with $L(r s) = L(r)L(s)$
-	- $r^*$ with $L(r^*) = L(r)^*$
-
-
-> [!EXAMPLE]
-> The language $L$ over $Sigma = {a, b}$ containing the substring $a b$ is regular, since it can be expressed using the regular expression
-> $r = (a +b)^* a b (a + b)^*$
-
-## Equivalence of regular expressions
-Two regular expressiosn $r$ and $s$ are _equivalent_ ($r eq.triple s$ or $r hat(eq) s$) if thy generate the same language ($L(r) eq L(s)$).

00 Inbox/29605546 - 29593852 - Strings.md

@@ -1,71 +0,0 @@
----
-created: 2026-04-08 08:52
-course: "[[29593850 - Automationtheory]]"
-topic: strings
-related: "[[29593929 - Alphabets]]"
-type: lecture
-status: 🔴
-tags:
-  - university
----
-## 📌 Summary
-
-> [!abstract]
-> Overview of lecture 1 on `Wednesday, 2026/Apr/08`
-
----
-
-## 📝 Content
-
-A word (or _string_) is a finite sequence $w = a_1 a_2 ... a_n$ if characters from $Sigma$.
-
-> [!CONVENTION]
-> We will use small letters to describe strings that are part of a language.
-
-> [!EXAMPLE]
-> $"aa", "ab", "bba"$ and $"baab"$ are strings over $Sigma = {a, b}.
-
-#### Length of a string
-The _length_ $abs(x)$ of a string $x = a_1 ... a_n$ is its number $abs(x) = n$ of characters.
-#### 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
-$$x circle.small y = x y = a_1 a_2 ... a_n b_1 b_2 ... b_m$$
-> This string is of the length $abs(x y) = n + m$
-
-> [!EXAMPLE]
-> $x = "apple"$
-> $y = "pie"$ 
-> $x circle.small y = "applepie"$
-
-Order of operations / Brackets do _not matter_. (Concatenation is associative but **not** commutative $x y eq.not y x$)
-> $(x circle.small y) circle.small z = x circle.small (y circle.small z)$
-
-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.
-> $x^0 := epsilon$
-> $x^n := x^(n-1) circle.small x$ for $n in NN$
-
-> [!Example]
-> $x^4 = x x x x$
-> $(a b)^3 = a b a b a b$
-
-### 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
-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 $y$.
-

00 Inbox/29605547 - 29605397 - OOP.md

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----
-created: 2026-04-16 11:47
-course: "[[29605547 - 29605397 - OOP]]"
-tags:
-  - university
-topic:
-related:
-type: lecture
-status: 🔴
----
-## 📌 Summary
-
-> [!abstract]
-> 
-
----
-
-## 📝 Content

00 Inbox/29605548 - 29605397 - OOP.md

@@ -1,18 +0,0 @@
----
-created: 2026-04-16 11:48
-course: "[[29605548 - 29605397 - OOP]]"
-tags:
-  - university
-topic:
-related:
-type: lecture
-status: 🔴
----
-## 📌 Summary
-
-> [!abstract]
-> 
-
----
-
-## 📝 Content