55 lines
1.5 KiB
Markdown
55 lines
1.5 KiB
Markdown
---
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created: 2026-04-08 10:20
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course: "[[29593850 - Automationtheory]]"
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topic: languages
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related:
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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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> Definition and example for formal languages
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---
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## 📝 Content
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A formal _language_ of the alphabet $Sigma$ is a subset $L$ of $Sigma^*$.
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> [!EXAMPLE]
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> 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 $b$:
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> $L_1 = {b a^n b | n in NN_0}$
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> Then $b b in L_1, b a b in L_1$, etc.
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>
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>More in [[29593895 - atfl-st2026-l01-formal-languages-full.pdf#page=54]]
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## Language Operations
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### Concatenation of languages
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For languages $X subset Sigma^*_X$ over alphabet $Sigma_X$ and $Y subset Sigma^*_Y$ over alphabet $Sigma_Y$, their _concatenation_ is
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> $X circle.small Y = X Y = {x y bar x in X and y in Y}$
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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 $Y$.
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> [!CONVENTION]
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> Concatenation has a higher precedence than set operations ($union, inter$).
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### Exponentiation of languages
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The $n^"th"$ power of the language $L subset.eq Sigma^*$ over alphabet $Sigma$ is
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- $L^0 := { epsilon }$
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- $L^n := L^(n-1) L$ if $n > 0$
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## Finite representation of languages
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**Goal:** Represent a language using _finite_ information
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### Using set notation
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$S = {a^n b^m bar n, m >= 0} = {epsilon, a, b, "aa", "ab", ...}$
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> This is limited in practice.
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### Using regular expressions
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