72 lines
2.3 KiB
Markdown
72 lines
2.3 KiB
Markdown
---
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created: 2026-04-08 08:52
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course: "[[29593850 - Automationtheory]]"
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topic: strings
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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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> Overview of lecture 1 on `Wednesday, 2026/Apr/08`
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---
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## 📝 Content
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A word (or _string_) is a finite sequence $w = a_1 a_2 ... a_n$ if characters from $Sigma$.
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> [!CONVENTION]
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> We will use small letters to describe strings that are part of a language.
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> [!EXAMPLE]
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> $"aa", "ab", "bba"$ and $"baab"$ are strings over $Sigma = {a, b}.
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#### Length of a string
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The _length_ $abs(x)$ of a string $x = a_1 ... a_n$ is its number $abs(x) = n$ of characters.
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#### Empty String
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The empty string is denoted by $epsilon$, this is the neutral element.
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-> $abs(epsilon) = 0$
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## String Operations
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### Concatenation
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String can be concatenated, where one string is appended to another.
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For strings $x = a_1 ... a_n$ and $y = b_1 ... b_m$ over alphabets $Sigma_x$ and $Sigma_y$, their _concatenation_ over the alphabet $Sigma = Sigma_x union Sigma_y$ is the string
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$$x circle.small y = x y = a_1 a_2 ... a_n b_1 b_2 ... b_m$$
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> This string is of the length $abs(x y) = n + m$
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> [!EXAMPLE]
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> $x = "apple"$
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> $y = "pie"$
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> $x circle.small y = "applepie"$
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Order of operations / Brackets do _not matter_. (Concatenation is associative but **not** commutative $x y eq.not y x$)
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> $(x circle.small y) circle.small z = x circle.small (y circle.small z)$
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Any string concatenated with the empty string $epsilon$ will result in itself.
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> $x circle.small epsilon = x = epsilon circle.small x$
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### Exponentiation
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The $n^"th"$ power $x^n$ of a string $x$ is the $(n-1)$-fold concatenation of $x$ with itself.
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> $x^0 := epsilon$
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> $x^n := x^(n-1) circle.small x$ for $n in NN$
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> [!Example]
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> $x^4 = x x x x$
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> $(a b)^3 = a b a b a b$
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### Reversing / Mirroring
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For a string $x = a_1 a_2 ... a_(n-1) a_n$ of length $n$, it's _mirrored string_ is given by
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$$ x^("Rev") = a_n a_(n-1)...a_2 a_1$$
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## Substrings
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A string $x$ is a _substring_ of a string $y$ if $y = u x v$, where $u$ and $v$ can be arbitrary strings.
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- If $u = epsilon$ then $x$ is a _prefix_ of $y$.
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- If $v = epsilon$ then $x$ is a suffix of $y$.
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For strings $x$ and $y$ the quantity $abs(y)_x$ is the number of times that $x$ is a substring of $y$.
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