31 lines
899 B
Markdown
31 lines
899 B
Markdown
---
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created: 2026-04-08 10:55
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course: "[[29593850 - Automationtheory]]"
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topic: languages
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related: "[[29593940 - Formal Languages#Finite representation of languages]]"
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type: lecture
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status: 🔴
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tags:
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- university
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---
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## 📌 Summary
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> [!abstract]
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>
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---
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## 📝 Content
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A _regular expression_ $r$ over an alphabet $Sigma$ is defined recursively:
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- $emptyset, epsilon$ and each $a in Sigma$ are regular expression, which represent the Languages $L(emptyset) = emptyset, L(epsilon) = {epsilon}$ and $L(a) = {a}$
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- If $r$ and $s$ are regular expressions then these are also regular expressions:
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- $(r + s)$ with $L(r + s) = L(r) union L(s)$
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- $(r s)$ with $L(r s) = L(r)L(s)$
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- $r^*$ with $L(r^*) = L(r)^*$
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> [!EXAMPLE]
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> The language $L$ over $Sigma = {a, b}$ containing the substring $a b$ is regular, since it can be expressed using the regular expression
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> $r = (a +b)^* a b (a + b)^*$
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