vault backup: 2026-04-08 10:11:50

00 Inbox/29593852 - Strings.md

@@ -18,37 +18,91 @@ tags:
 ## 📝 Content
 
 ### Alphabets
-Alphabets are formal, non-empty, sets of symbols (usually lowercase letters). They are denoted by $Sigma$.
+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 string is a set of letters. If there is an alphabet $Sigma = {a,b}$ then `abba` is a string made from that alphabet.
+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.
+-> $abs(epsilon) = 0$
 ### Concatenation
-String can be concatenated, where one string is appended to another.
-$"apple" dot "pie" = "applepie"$
+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 $abs(x y) = n + m$
 
-$&x = "apple" \ &y = "pie" \ &x dot y = "applepie"$
+> [!EXAMPLE]
+> $x = "apple"$
+> $y = "pie"$ 
+> $x circle.small y = "applepie"$
+
+Order of operations / Brackets do _not matter_. (Concatenation is associative but **not** commutative $x y eq.not y x$)
+> $(x circle.small y) circle.small z = x circle.small (y circle.small z)$
 
-Order of operations / Brackets do _not matter_.
 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 $y$.
 
 ### Kleene Star
-Denoted by $Sigma^*$. The Kleene Star (or _Kleene Closure_) gives an infinite amount of strings made up of the characters of the alphabet.
+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 $n$
 
-$$
-Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}
-$$
-> Example of the Kleene Star of the alphabet ${a, b}$
+#### 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 $Sigma^*$