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58 lines
2.7 KiB
TeX
58 lines
2.7 KiB
TeX
\newsection
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\section{Alphabets, Words, etc}
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\stepcounter{subsection}
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\subsection{Alphabets, Words, Languages}
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\fancydef{Alphabet} Set $\Sigma$.
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Important alphabets: $\alphabetbool$, $\alphabets{lat}$ (all latin chars), $\alphabets{Keyboard}$ (all chars on keyboard), $\Sigma_m$ ($m$-ary numbers)
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\fancydef{Word} Possibly empty (denoted $\lambda$) sequences of characters from $\Sigma$. $|w|$ is the length,
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$\Sigma^*$ is the set of all words and $\Sigma^+ = \Sigma^* - \{ \lambda \}$
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\fancydef{Konkatenation} $\text{Kon}(x, y) = xy$, (so like string concat). $(xy)^n$ is $n$-times repeated concat.
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\fancydef{Reversal} $a^R$, simply read the word backwards.
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\stepLabelNumber{definition}
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\fancydef{Prefix, Suffix, Subword} $v$ in $w = vy$; $s$ in $w = sx$; Subword is $u$ in $w = xuy$
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\fancydef{Appearance} $|x|_a$ is the number of times $a \in \Sigma$ appears in $x$
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\fancydef{Canonical ordering} Ordered by length and then by first non-common letter:
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\rmvspace
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\begin{align*}
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u < v \Longleftrightarrow |u| < |v| \lor (|u| = |v| \land u = x \cdot s_i \cdot u' \land v = x \cdot s_j \cdot v') \text{ for any } x, u', v' \in \word \text{ and } i < j
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\end{align*}
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\drmvspace
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\fancydef{Language} $L \subseteq \word$, and we define $L^C = \word - L$ as the complement, with $L_{\emptyset}$ being the empty language,
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whereas $L_\lambda$ is the language with just the empty word in it.
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\bi{Concatenation}: $L_1 \cdot L_2 = \{ vw | v \in L_1 \land w \in L_2 \}$ and $L^{i + 1} = L^i \cdot L \ \forall i \in \N$.
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\bi{Cleen Star}: $L^* = \bigcup_{i \in \N} L^i$ and $L^+ = L \cdot L^*$
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\inlinelemma $L_1L_2 \cup L_1 L_2 = L_1(L_2 \cup L_3)$
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\inlinelemma $L_1(K_2 \cap L_3) \subseteq L_1 L_2 \cap L_1 L_3$
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\stepcounter{subsection}
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\subsection{Kolmogorov-Complexity}
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\setLabelNumber{definition}{17}
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\fancydef{Kolmogorov-Complexity} $K(x)$ for $x \in \wordbool$ is the minimum of all binary lengths of Pascal programs that output $x$,
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where the Program doesn't have to compile, i.e. we can describe processes informally
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\stepLabelNumber{lemma}
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\inlinelemma For each word $x$ exists constant $d$ s.t. $K(x) \leq |x| + d$, for which we can use a program that simply includes a \texttt{write(x)} command
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\fancydef{Of natural number} $K(n) = K(\text{Bin}(x))$ with $|\text{Bin}(x)| = \ceil{\log_2(x + 1)}$
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\inlinelemma For each $n \in \N \exists w_n \in (\alphabetbool)^n$ s.t. $K(w_n) \geq |w_n| = n$, i.e. exists a non-compressible word.
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\inlinetheorem Kolmogorov-Complexity doesn't depend on programming language. It only differs in constant
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\fancydef{Randomness} $x$ random if $K(x) \geq |x|$, thus for $n$, $K(n) \geq \ceil{\log_2(n + 1)} - 1$
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\stepLabelNumber{theorem}
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\fancytheorem{Prime number} $\displaystyle \limni \frac{\text{Prime}(n)}{\frac{n}{\ln(n)}}$
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