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00 Inbox/29593940 - Formal Languages.md

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+---
+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
+	- $(r+s)$