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

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_.