vault backup: 2026-04-08 15:13:13

00 Inbox/29593975 - Regular Expressions.md

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