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+\section{Combinatorics} +\label{sec:combinatorics} +\subsection{Introduction} +Combinatorics was developed from the willingness of humans to gamble and the fact that everybody wanted to win as much money as possible. + +\subsection{Simple counting operations} +The easiest way to find the best chance of winning is to write down all possible outcomes. This can be very tedious though when the list gets longer. + +We can note this all down as a list or as a tree diagram. So-called Venn Diagrams might also help represent the relationship between two sets or events. Essentially a Venn Diagram is a graphical representation of set operations such as $A \cup B$. + + +\subsection{Basic rules of counting} +\subsubsection{Multiplication rule} +If one has $n$ possibilities for a first choice and $m$ possibilities for a second choice, then there are a total of $n \cdot m$ possible combinations. + +When we think about a task, and we have an \textbf{and} in between e.g. properties, we need to multiply all the options. + +\subsubsection{Addition rule} +If two events are mutually exclusive, the first has $n$ possibilities and the second one has $m$ possibilities, then both events together have $n+m$ possibilities. + +When we think about a task, and we have an \textbf{or} in between e.g. properties, then we need to add all the options. + + +\newpage +\subsection{Factorial} +\begin{definition}[]{Factorial} + The factorial stands for the product of the first $n$ natural numbers where $n \ge 1$. Notation: $!$ + \[ + n! = n \cdot (n - 1) \cdot (n - 2) \cdot \ldots \cdot 3 \cdot 2 \cdot 1 + \] + Additionally, $0! = 1$. We read $n!$ as ``\textit{n factorial}'' +\end{definition} + +\subsubsection{Operations} +We can rewrite $n!$ as $n \cdot (n - 1)!$ or $n \cdot (n - 1) \cdot (n - 2)!$ and so on. + +It is also possible to write $7 \cdot 6 \cdot 5$ with factorial notation: $\displaystyle \frac{7!}{4!}$, or in other words, for any excerpt of a factorial sequence: \[n \cdot (n - 1) \cdot \ldots \cdot m = \frac{n!}{(m - 1)!}\] + + +\subsection{Permutations} +\begin{definition}[]{Permutations} + A permutation of a group is any possible arrangement of the group's elements in a particular order\\ + + \textbf{Permutation rule without repetition:} The number of $n$ \textbf{\textit{distinguishable}} elements is defined as: $n!$ +\end{definition} + + +\subsubsection{Permutation with repetition} +For $n$ elements $n_1,n_2,\ldots,n_k$ of which some are identical, the number of permutations can be calculated as follows: +\[ + p = \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} +\] +where $n_k$ is the number of times a certain element occurs. +As a matter of fact, this rule also applies to permutations without repetition, as each element occurs only once, which means the denominator is $1$, hence $\displaystyle \frac{n!}{(1!)^n} = n!$ + +\inlineex \smallhspace CANADA has $6$ letters, of which $3$ letters are the same. So the word consists of $3$ A's, which can be arranged in $3!$ different ways, a C, N and D, which can be arranged in $1!$ ways each. Therefore, we have: +\[ + \frac{6!}{3!\cdot 1! \cdot 1! \cdot 1!} = \frac{6!}{3!} = 6 \cdot 5 \cdot 4 = 120 +\] + +Since $1!$ equals $1$, we can always ignore all elements that occur only once, as they won't influence the final result. + + +\newpage +\subsection{Variations} +\begin{definition}[]{Variations} + A \textbf{\textit{variation}} is a selection of $k$ elements from a universal set that consists of $n$ \textit{distinguishable} elements.\\ + + \textbf{Variation rule without repetition:} The $_n\mbox{P}_k$ function is used to \textit{\textbf{place}} $n$ elements on $k$ places. In a more mathematical definition: + The number of different variations consisting of $k$ different elements selected from $n$ distinguishable elements can be calculated as follows: + + \[ + \frac{n!}{(n - k)!} = _n\mbox{P}_k + \] +\end{definition} + +\subsubsection{Variations with repetition} +If an element can be selected more than once and the order matters, the number of different variations consisting of $k$ elements selected from $n$ distinguishable elements can be calculated using $n^k$ + + + +\subsection{Combinations} +\begin{definition}[]{Combination} + A combination is a selection of $k$ elements from $n$ elements in total without any regard to order or arrangement. + + \textbf{Combination rule without repetition:} \[ + _n\mbox{C}_k = {n\choose k} = \frac{_n\mbox{P}_k}{k!} = \frac{n!}{(n - k)! \cdot k!} + \] +\end{definition} + +\subsubsection{Combination with repetition} +In general the question to ask for combinations is, in how many ways can I distribute $k$ objects among $n$ elements? +\[ + _{n + k - 1}\mbox{C}_k = {n + k - 1\choose k} = \frac{(n + k - 1)!}{k!(n - 1)!} +\] + + + +\subsection{Binomial Expansion} +\label{sec:binomial-expansion} +Binomial expansion is usually quite hard, but it can be much easier than it first seems. The first term of the expression of $(a + b)^n$ is always $1 a^n b^0$. Using the formula for combination without repetition, we can find the coefficients of each element: + +\begin{center} + \includegraphics[width=0.6\linewidth]{./assets/binomialExpansion.png} +\end{center} + +This theory is based on the Pascal's Triangle and the numbers of row $n$ correspond to the coefficients of each element of the expanded term. + +We can calculate the coefficient of each part of the expanded term $k$ with combinatorics as follows: $\displaystyle {n\choose k}$ + +\begin{formula}[]{Binomial Expansion} + \textbf{\textit{\underbar{In general:}}} + \[ + (a + b)^n = 1a^nb^0 + {n\choose 1} a^{n-1}b^{1} + {n\choose 2} a^{n-2}b^{2} + \ldots + {n\choose n - 1} a^{1}b^{n - 1} + {n\choose n} a^{0}b^{n} + \] +\end{formula} + + +\subsection{Overview} +\includegraphics[width=1\linewidth]{./assets/overview.png} diff --git a/semester3/ti/parts/languages-problems/algorithmic-problems.tex b/semester3/ti/parts/languages-problems/algorithmic-problems.tex new file mode 100644 index 0000000..aee2137 --- /dev/null +++ b/semester3/ti/parts/languages-problems/algorithmic-problems.tex @@ -0,0 +1,22 @@ +\newpage +\subsection{Algorithmische Probleme} +Ein Algorithmus $A : \Sigma_1^* \rightarrow \Sigma_2^*$ ist eine Teilmenge aller Programme, wobei ein Program ein Algorithmus ist, sofern es für jede zulässige Eingabe eine Ausgabe liefert, es darf also nicht eine endlosschleife enthalten. + +\begin{definition}[]{Entscheidungsproblem} + Das \bi{Entscheidungsproblem} $(\Sigma, L)$ ist für jedes $x \in \Sigma^*$ zu entscheiden, ob $x \in L$ oder $x \notin L$. + Ein Algorithmus $A$ löst $(\Sigma, L)$ (erkennt $L$) falls für alle $x \in \Sigma^*$: $A(x) = \begin{cases} + 1, &\text{ falls } x \in L\\ + 0, &\text{ falls } x \notin L + \end{cases}$. +\end{definition} + +\begin{definition}[]{Funktion} + Algorithmus $A$ berechnet (realisiert) eine \bi{Funktion (Transformation)} $f: \Sigma^* \rightarrow \Gamma^*$ falls $A(x) = f(x) \smallhspace \forall x \in \Sigma^*$ für Alphabete $\Sigma$ und $\Gamma$ +\end{definition} + + +\begin{definition}[]{Berechnung} + Sei $R \subseteq \Sigma^* \times \Gamma^*$ eine Relation in den Alphabeten $\Sigma$ und $\Gamma$. + Ein Algorithmus $A$ \bi{berechnet} $R$ (\bi{löst das Relationsproblem} $R$) falls für jedes $x \in \Sigma^*$, für das ein $y \in \Gamma^*$ mit $(x, y) \in R$ existiert gilt: + $(x, A(x)) \in R$ +\end{definition} diff --git a/semester3/ti/parts/languages-problems/alphabet.tex b/semester3/ti/parts/languages-problems/alphabet.tex new file mode 100644 index 0000000..3d23367 --- /dev/null +++ b/semester3/ti/parts/languages-problems/alphabet.tex @@ -0,0 +1,111 @@ +\subsection{Alphabete, Wörter, Sprachen} + +\begin{definition}[]{Alphabet} + Eine endliche, nicht leere Menge $\Sigma$. Elemente sind Buchstaben (Zeichen \& Symbole). + + Beispiele: $\Sigma_{\text{bool}}$, $\Sigma_{\text{lat}}$ latin characters, $\Sigma_{\text{Tastatur}}$, $\Sigma_m$ $m$-adische Zahlen ($m$-ary numbers, zero index) +\end{definition} + +\begin{definition}[]{Wort} + Über $\Sigma$ eine (möglicherweise leere) Folge von Buchstaben aus $\Sigma$. Leeres Wort $\lambda$ (ab und zu $\varepsilon$) hat keine Buchstaben. + + $|w|$ ist die Länge des Wortes (Anzahl Buchstaben im Wort), während $\Sigma^*$ die Menge aller Wörter über $\Sigma$ ist und $\Sigma^+ = \Sigma^* - \{\lambda\}$ + + In diesem Kurs werden Wörter ohne Komma geschrieben, also $x_1x_2\ldots x_n$ statt $x_1, x_2, \ldots, x_n$. + Für das Leersymbol gilt $|\text{\textvisiblespace}|$, also ist es nicht dasselbe wie $\lambda$ +\end{definition} + +Für viele der Berechnungen in Verbindung mit der Länge der Wörter kann Kombinatorik nützlich werden. +In Kapitel \ref{sec:combinatorics} findet sich eine Zusammenfassung über jenes Thema (in English) + +Ein mögliches Alphabet beispielsweise um einen Graphen darzustellen ist folgendes: + +Angenommen, wir speichern den Graphen als Adjezenzmatrix ab, dann können wir beispielsweise mit dem Alphabet $\Sigma = \{0, 1, \#\}$ diese Matrix darstellen, in dem wir jede neue Linie mit einem $\#$ abgrenzen. +Das Problem hierbei ist jedoch, dass dies nicht so effizient ist, besonders nicht, wenn der Graph sparse ist, da wir dann viele \# im Vergleich zu nützlicher Information haben. + +\begin{definition}[]{Konkatenation} + $\Sigma^* \times \Sigma^* \rightarrow \Sigma^*$, so dass Kon$(x, y) = x \cdot y = xy \smallhspace \forall x, y \in \Sigma^*$. + + Intuitiv ist dies genau das was man denkt: Wörter zusammenhängen (wie in Programmiersprachen). + Die Operation ist assoziativ und hat das Neutralelement $\lambda$, was heisst, dass $(\Sigma^*, \text{Kon})$ ein Monoid ist. + + Offensichtlich ist die Konkatenation nur für ein-elementige Alphabete kommutativ. + + Die Notation $(abc)^n$ wird für die $n$-fache Konkatenation von $abc$ verwendet +\end{definition} + + +\begin{definition}[]{Umkehrung} + Sei $a = a_1 a_2 \ldots a_n$, wobei $a_i \in \Sigma$ für $i \in \{1, 2, \ldots, n\}$, dann ist die Umkehrung von $a$, $a^{\text{R}} = a_n a_{n - 1} \ldots a_1$ +\end{definition} + + +\begin{definition}[]{Iteration} + Die $i$-te Iteration $x^i$ von $x \in \Sigma^*$ für alle $i \in \N$ ist definiert als $x^0 = \lambda$, $x^1 = x$ und $x^i = xx^{i - 1}$ +\end{definition} + + +\begin{definition}[]{Teilwort, Präfix, Suffix} + Seien $v, w \in \Sigma^*$ + \begin{itemize} + \item $v$ heisst \bi{Präfix} von $w \Longleftrightarrow \exists y \in \Sigma^* : w = vy$ + \item $v$ heisst \bi{Suffix} von $w \Longleftrightarrow \exists x \in \Sigma^* : w = xv$ + \item $v$ heisst \bi{Teilwort} von $w \Longleftrightarrow \exists x, y \in \Sigma^* : w = xvy$ + \item $v \neq \lambda$ heisst \bi{echtes} Teilwort (gilt auch für Präfix, Suffix) von $w$ genau dann, wenn $v \neq w$ und $v$ ein Teilwort (oder eben Präfix oder Suffix) von $w$ ist + \end{itemize} +\end{definition} + +\begin{definition}[]{Kardinalität, Vorkommen und Potenzmenge} + Für Wort $x \in \Sigma^*$ und Buchstabe $a \in \Sigma$ ist $|x|_a$ definiert als die Anzahl Male, die $a$ in $x$ vorkommt. + + Für jede Menge $A$ ist $|A|$ die Kardinalität und $\mathcal{P}(A) = \{S | S \subseteq A\}$ die Potenzmenge von $A$ +\end{definition} + + +\begin{definition}[]{Kanonische Ordnung} + Wir definieren eine Ordnung $s_1 < \ldots < s_m$ auf $\Sigma$. Die \bi{kanonische Ordnung} auf $\Sigma^*$ für $u, v \in \Sigma^*$ ist definiert als: + \begin{align*} + 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{ für beliebige $x, u', v' \in \Sigma^*$ und $i < j$} + \end{align*} + + Oder in Worten, geordnet nach Länge und dann danach, ob man eine gemeinsames Teilwort finden kann +\end{definition} + + +\begin{definition}[]{Sprache} + $L \subseteq \Sigma^*$ ist eine Sprache, deren Komplement $L^C = \Sigma^* - L$ ist. + Dabei ist $L_{\emptyset}$ die \bi{leere Sprache} und $L_{\lambda}$ die einelementige Sprache die nur aus dem leeren Wort besteht. + + Die \bi{Konkatenation} von $L_1$ und $L_2$ ist $L_1 \cdot L_2 = L_1 L_2 = \{ vw \divides v \in L_1 \land w \in L_2 \}$ und $L^0 := L_{\lambda}$ und $L^{i + 1} = L^i \cdot L \smallhspace \forall i \in \N$ und $L^* = \bigcup_{i \in \N} L^{i}$ ist der \bi{Kleene'sche Stern} von $L$, wobei $L^+ = \bigcup_{i \in \N - \{0\}} L^i = L \cdot L^*$ +\end{definition} + +Da Sprachen Mengen sind, gelten auch die Üblichen Operationen, wie Vereinigung ($\cup$) und Schnitt ($\cap$). +Die Gleichheit von zwei Sprachen bestimmen wir weiter mit $A \subseteq B \land B \subseteq A \Rightarrow A = B$. +Um $A \subseteq B$ zu zeigen reicht es hier zu zeigen dass für jedes $x \in A$, $x \in B$ hält. +Wir betrachten nun, wie die üblichen Operationen mit der neu hinzugefügten Konkatenation interagieren. + +\begin{lemma}[]{Distributivität von Kon und $\cup$} + Für Sprachen $L_1, L_2$ und $L_3$ über $\Sigma$ gilt: $L_1 L_2 \cup L_1 L_3 = L_1 (L_2 \cup L_3)$ +\end{lemma} + +Der Beweis hierfür läuft über die oben erwähnte ``Regel'' zur Gleichheit. +Um das Ganze einfacher zu machen, teilen wir auf: Wir zeigen also erst $L_1 L_2 \subseteq L_1(L_2 \cup L_3)$ und dann equivalent für $L_1 L_3$. + + +\begin{lemma}[]{Distributivität von Kon und $\cap$} + Für Sprachen $L_1, L_2$ und $L_3$ über $\Sigma$ gilt: $L_1 (L_2 \cap L_3) \subseteq L_1 L_2 \cap L_1 L_3$ +\end{lemma} + +\shortlemma Es existieren $U_1, U_2, U_3 \in (\Sigma_{\text{bool}})^*$, so dass $U_1 (U_2 \cap U_3) \subsetneq U_1 U_2 \cap U_1 U_3$ + + + +\begin{definition}[]{Homomorphismus} + $\Sigma_1, \Sigma_2$ beliebige Alphabete. 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+\usetcolorboxes + +\newpage + +\printtoc{Orange} + +\newpage + + +% Combinatorics +\input{parts/combinatorics.tex} + + +\newsection +\section{Alphabete, Wörter, Sprachen und Darstellung von Problemen} +\input{parts/languages-problems/alphabet.tex} +\input{parts/languages-problems/algorithmic-problems.tex} + + + + +\end{document}