47 lines
1.7 KiB
Markdown
47 lines
1.7 KiB
Markdown
---
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created: 2026-04-08 10:15
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course: "[[29593850 - Automationtheory]]"
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topic: kleen
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related: "[[29593929 - Alphabets]]"
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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 Examples of the Kleene Star, the Kleene Plus and the Lemma Group Structure
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---
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## 📝 Content
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### Kleene Star
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Denoted by $Sigma^*$. The Kleene Star (or _Kleene operator_ or _Kleene Closure_) gives an infinite amount of [[29593852 - Strings|strings]] made up of the characters of the [[29593929 - Alphabets|alphabets]] $Sigma ^ *$.
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$Sigma^*$ is the set of all string that can be generated by arbitrary concatenation of its characters.
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> $Sigma^* := union.big_(n>=0) A_n$
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> where $A_n$ is the set of all string combinations of length $n$
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#### Remarks
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- The same character can be used multiple times.
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- The empty string $epsilon$ is also part f $Sigma^*$.
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> [!Example]
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> $Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}$
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> [!FACT]
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> - The set $Sigma^*$ is infinite, since we defined $Sigma$ to be non-empty.
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> - It is _countable_ and has the same cardinality as the set $NN$ of natural numbers
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### Kleene Plus
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The _Kleene Plus_ of an alphabet $Sigma$ is given by $Sigma^+ = Sigma^* backslash {epsilon}$
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### Lemma group structure
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The _Lemma group structure_ is induced by the [[29593935 - Kleene Star & Kleene Plus#Kleene Star|Kleene Star]] - it is a monoid, that is a semigroup with a neutral element.
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> [!PROOF]
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> - Associativity has been shown
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> - Existence of a neutral element has been shown.
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> - Closure under $circle.small$: Let $x in Sigma^*$ and $y in Sigma^*$ be two string over the alphabet $Sigma$. Then $x circle.small y = x y in Sigma^*$ |