---
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 limited in practice.

### Using regular expressions