---
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 expressions $r$ and $s$ are _equivalent_ ($r eq.triple s$ or $r hat(eq) s$) if they generate the same language ($L(r) eq L(s)$).