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add example Büchi Game
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@ -1,6 +1,5 @@
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\definecolor{lightblue}{rgb}{0.67,0.87,0.9}
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%------------------------------------------------
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\section{Motivation \& References}
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%------------------------------------------------
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@ -91,7 +90,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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$R_i:=\{v\in V|$ P0 can force a visit from v to a vertex in $T$ in i steps$\}$\\~
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Define Reachability set of $T$ for P0, $Reach(T,0) := \bigcup_{i=1}^{n-1}R_i$\\~
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Define Reachability set of $T$ for P0, $\Reach(T,0) := \bigcup_{i=1}^{n-1}R_i$\\~
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A vertex $v\in R_i$: \\~
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@ -184,7 +183,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\end{adjustbox}
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\column{.5\textwidth} % Right column and width
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\textbf{Definition} A type of vertex $x$ is a tuple $(y_1, \ldots, y_k)$, where each $x_i \in \{0,1\}$, such that $y_i=1$ iff $x$ is in $Reach (T_i, 0)$.
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\textbf{Definition} A type of vertex $x$ is a tuple $(y_1, \ldots, y_k)$, where each $x_i \in \{0,1\}$, such that $y_i=1$ iff $x$ is in $\Reach (T_i, 0)$.
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\end{columns}
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\end{frame}
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%frame 11
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@ -204,8 +203,9 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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The minimum base of $T$ is the minimum subset of $T$ which can generate the same Reachability set as $T$.\\~
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Computing the minimum base is NP-hard.\\~
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Set cover problem: Given a set $S$ of n elements, a collections $S_1,S_2,...,S_m$ of subsets of $S$, and a number K, does there exists a collection of at most k of these sets whose union is equal to all of $S$.
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\begin{problem}[Set cover]
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Given a set $S$ of n elements, a collections $S_1,S_2,...,S_m$ of subsets of $S$, and a number K, does there exists a collection of at most k of these sets whose union is equal to all of $S$.
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\end{problem}
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\end{frame}
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%frame 13
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@ -250,10 +250,12 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\section{Büchi Game}
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%frame 14
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\begin{frame}{Büchi Game}
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\textbf{Definition} A \textbf{Büchi game} is a game $\mathcal{G}=(G,s,T)$ where $G$ is the Reachability game graph, $V_i$ is an initial vertex, $T\subseteq V$ is the target set as in Reachability game.\\~
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Play: The definition of play in Büchi Game is the same as in Reachability game.\\~
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Definition of winning: We assume the play $P$ is infinite here.
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if there exists infinite many vertices $v\in T$ in $P$, P0 wins. Otherwise P1 wins.
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\begin{definition}[Büchi Game]
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A \textbf{Büchi game} is a game $\mathcal{G}=(G,s,T)$ where $G$ is the Reachability game graph, $V_i$ is an initial vertex, $T\subseteq V$ is the target set as in Reachability game.\\~
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Play: The definition of play in Büchi Game is the same as in Reachability game.\\~
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Definition of winning: We assume the play $P$ is infinite here.
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if there exists infinite many vertices $v\in T$ in $P$, P0 wins. Otherwise P1 wins.
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\end{definition}
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\end{frame}
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%frame 15
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@ -299,15 +301,15 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\draw[thick,fill=lightblue, fill opacity=0.3] (2,-2) -- (2,2) -- (6,2) -- (6,-2) -- cycle; % reach
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\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
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\node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T$};
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\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$Reach(T,0)$};
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\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)$};
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\end{tikzpicture}
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\end{adjustbox}
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\column{.35\textwidth} % Right column and width
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If $v\notin Reach(T,0)\cup T$, $v$ can not reach $T$, P0 will lose.\\~
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If $v\notin \Reach(T,0)\cup T$, $v$ can not reach $T$, P0 will lose.\\~
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Some vertices in $T$ can not reach $Reach(T,0)\cup T$, P0 will also lose on these vertices.
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Some vertices in $T$ can not reach $\Reach(T,0)\cup T$, P0 will also lose on these vertices.
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\end{columns}
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\end{frame}
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@ -323,19 +325,19 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
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\node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_2$};
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\node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T_1$};
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\node[shape=circle,draw opacity=0](txt) at (4,-1.7) {$Reach(T_2,0)$};
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\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$Reach(T,0)\backslash Reach(T_2,0)$};
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\node[shape=circle,draw opacity=0](txt) at (4,-1.7) {$\Reach(T_2,0)$};
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\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)\backslash \Reach(T_2,0)$};
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\end{tikzpicture}
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\end{adjustbox}
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\column{.37\textwidth} % Right column and width
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$T_1=\{v\in T|v$ can't reach $T\cup Reach(T,0)\}$\\~
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$T_1=\{v\in T|v$ can't reach $T\cup \Reach(T,0)\}$\\~
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Some vertices in $T_2$ can only reach $Reach(T,0)\backslash Reach(T_2,0)$\\~
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Some vertices in $T_2$ can only reach $\Reach(T,0)\backslash \Reach(T_2,0)$\\~
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We find $T_3=\{v\in T_2|v$ can't reach $T_2\cup Reach(T_2,0)\}$
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We find $T_3=\{v\in T_2|v$ can't reach $T_2\cup \Reach(T_2,0)\}$
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\end{columns}
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\end{frame}
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\begin{frame}{Algorithm for Büchi Game 1}
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@ -352,7 +354,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\draw[thick,fill=red, fill opacity=0.3] (0,0.125) -- (0,0.25) -- (6,0.25) -- (6,0.125) -- cycle;
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\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
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\node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T_1$};
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\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$Reach(T,0)\backslash Reach(T_2,0)$};
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\node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)\backslash \Reach(T_2,0)$};
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\node[shape=circle,draw opacity=0](txt) at (2.1,-1.5) {Winning set for P0};
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\end{tikzpicture}
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@ -361,17 +363,17 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\column{.35\textwidth} % Right column and width
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We repeat this process until $T_k$ does not shrink.\\~
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The remaining part of $T_k\cup Reach(T_k,0)$ is the winning set for P0.
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The remaining part of $T_k\cup \Reach(T_k,0)$ is the winning set for P0.
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\end{columns}
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\end{frame}
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\begin{frame}{Algorithm for Büchi Game 1}
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\begin{itemize}
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\item How to find $T_1$\\
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$T_1=\{v\in T|v$ can't reach $T\cup Reach(T,0)\}$\\
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$T_1=\{v\in T|v$ can only reach $V\backslash \{T\cup Reach(T,0)\}\}$\\
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P1 wants to reach $V\backslash \{T\cup Reach(T,0)\}\}$, P0 tries to avoid $V\backslash \{T\cup Reach(T,0)\}\}$.\\
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compute $Reach(V\backslash \{T\cup Reach(T,0)\}\},1)$
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$T_1=\{v\in T|v$ can't reach $T\cup \Reach(T,0)\}$\\
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$T_1=\{v\in T|v$ can only reach $V\backslash \{T\cup \Reach(T,0)\}\}$\\
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P1 wants to reach $V\backslash \{T\cup \Reach(T,0)\}\}$, P0 tries to avoid $V\backslash \{T\cup \Reach(T,0)\}\}$.\\
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compute $\Reach(V\backslash \{T\cup \Reach(T,0)\}\},1)$
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\item Time complexity\\
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$O(m)$ to find $T_i$, at most $O(n)$ times. Worst-case $O(nm)$.
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@ -390,7 +392,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$};
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\node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T$};
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\node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$};
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\node[shape=circle,draw opacity=0](txt) at (2,-1.7) {$Reach(C_0\cup C_1,1)$};
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\node[shape=circle,draw opacity=0](txt) at (2,-1.7) {$\Reach(C_0\cup C_1,1)$};
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\end{tikzpicture}
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\end{adjustbox}
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@ -402,7 +404,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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$C_0$ is a set of vertices in $V_0\backslash T$ having all outgoing edges to vertices in $V\backslash T$.\\
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$C_1$ is a set of vertices in $V_1\backslash T$ having an outgoing edge to vertices in $V\backslash T$.\\~
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Compute $Reach(C_0\cup C_1,1)$
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Compute $\Reach(C_0\cup C_1,1)$
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\end{columns}
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\end{frame}
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\begin{frame}{Algorithm for Büchi Game 2}
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@ -421,21 +423,21 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$};
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\node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_1$};
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\node[shape=circle,draw opacity=0](txt) at (2.5,-1.7) {$D$};
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\node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$Reach(T_1\cup D,0)$};
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\node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$\Reach(T_1\cup D,0)$};
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\node[shape=circle,draw opacity=0](txt) at (4,-0.75) {$E$};
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\end{tikzpicture}
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\end{adjustbox}
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\column{.39\textwidth} % Right column and width
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Some vertices in $Reach(C_0\cup C_1,1)$ can "reach" $T_1$.($D$ in the left picture)\\~
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Some vertices in $\Reach(C_0\cup C_1,1)$ can "reach" $T_1$.($D$ in the left picture)\\~
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Compute $Reach(T_1\cup D,0)$.\\~
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Compute $\Reach(T_1\cup D,0)$.\\~
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$E=Reach(C_0\cup C_1,1)\backslash \{T_1\cup D \cup Reach(T_1\cup D,0)\}$\\~
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$E=\Reach(C_0\cup C_1,1)\backslash \{T_1\cup D \cup \Reach(T_1\cup D,0)\}$\\~
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$\{E\cup C_0\cup C_1\}\backslash Reach(T_1\cup D,0)$ is the set of vertices which can't "reach" $T$.
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$\{E\cup C_0\cup C_1\}\backslash \Reach(T_1\cup D,0)$ is the set of vertices which can't "reach" $T$.
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\end{columns}
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\end{frame}
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\begin{frame}{Algorithm for Büchi Game 2}
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@ -454,18 +456,18 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~
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\node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$};
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\node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_1$};
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\node[shape=circle,draw opacity=0](txt) at (2.5,-1.7) {$D$};
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\node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$Reach(T_1\cup D,0)$};
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\node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$\Reach(T_1\cup D,0)$};
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\node[shape=circle,draw opacity=0](txt) at (4,-0.75) {$E$};
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\end{tikzpicture}
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\end{adjustbox}
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\column{.4\textwidth} % Right column and width
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$S=\{E\cup C_0\cup C_1\}\backslash Reach(T_1\cup D,0)$ is the same as $V\backslash \{T\cup Reach(T)\}$ in Algorithm 1.\\~
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$S=\{E\cup C_0\cup C_1\}\backslash \Reach(T_1\cup D,0)$ is the same as $V\backslash \{T\cup \Reach(T)\}$ in Algorithm 1.\\~
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Then we can compute $Reach(S,1)$ to delete some losing vertices for P0 in $T$.\\~
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Then we can compute $\Reach(S,1)$ to delete some losing vertices for P0 in $T$.\\~
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Repeat the same process on $G\backslash\{T\backslash Reach(S,1)\}$
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Repeat the same process on $G\backslash\{T\backslash \Reach(S,1)\}$
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\end{columns}
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\end{frame}
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\begin{frame}{Algorithm for Büchi Game 2}
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Binary file not shown.
@ -1,16 +1,4 @@
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\documentclass{beamer}
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\usepackage[english]{babel}
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\usepackage{fancyhdr} % header footer
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\usepackage{graphicx} % figure
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\usepackage{booktabs}
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\usepackage{xcolor}
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\usepackage{bookmark}
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\usepackage{hyperref}
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\usepackage{graphicx} % Allows including images
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\usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule in tables
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\usepackage{tikz}
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\usepackage[ruled,linesnumbered]{algorithm2e}
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\usepackage{adjustbox}
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\author{Yu Cong}
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\title{Reachability and Büchi games}
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@ -23,6 +11,7 @@
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\usetheme{Simple}
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% \useoutertheme{tree}
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\DeclareMathOperator{\Reach}{Reach}
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\begin{document}
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\begin{frame}[plain]
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@ -34,5 +23,5 @@
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% Throughout your presentation, if you choose to use \section{} and \subsection{} commands, these will automatically be printed on this slide as an overview of your presentation
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\tableofcontents
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\end{frame}
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\input{content.tex}
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\input{Büchi Game-content.tex}
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\end{document}
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@ -2,6 +2,44 @@
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%
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% This file may be distributed and/or modified
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% under the LaTeX Project Public License
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\RequirePackage[english]{babel}
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\RequirePackage{fancyhdr} % header footer
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\RequirePackage{xcolor}
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\RequirePackage{bookmark}
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\RequirePackage{hyperref}[colorlinks=true,urlcolor=Blue,citecolor=Green,linkcolor=BrickRed,unicode]
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\RequirePackage{graphicx} % Allows including images
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\RequirePackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule in tables
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\RequirePackage{tikz}
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\usetikzlibrary{backgrounds}
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\usetikzlibrary{arrows,shapes}
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\usetikzlibrary{tikzmark} % for \tikzmarknode
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\usetikzlibrary{calc} % for computing the midpoint between two nodes, e.g. at ($(p1.north)!0.5!(p2.north)$)
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\RequirePackage[ruled,linesnumbered]{algorithm2e}
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\RequirePackage{adjustbox}
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\RequirePackage{subcaption}
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\RequirePackage{amsmath}
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\RequirePackage{amsthm}
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% a color box
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\RequirePackage[breakable, theorems, skins]{tcolorbox}
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\tcbset{enhanced}
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\DeclareRobustCommand{\mybox}[2][gray!20]{%
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\begin{tcolorbox}[ %% Adjust the following parameters at will.
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breakable,
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left=0pt,
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right=0pt,
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top=0pt,
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bottom=0pt,
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colback=#1,
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colframe=#1,
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width=\dimexpr\textwidth\relax,
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enlarge left by=0mm,
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boxsep=5pt,
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arc=0pt,outer arc=0pt,
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]
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#2
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\end{tcolorbox}
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}
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\definecolor{beamer@simple@color}{RGB}{12 72 66} % bluegreen
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\DeclareOptionBeamer{named}{\definecolor{beamer@simple@color}{named}{#1}}
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\DeclareOptionBeamer{hsb}{\definecolor{beamer@simple@color}{hsb}{#1}}
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\ProcessOptionsBeamer
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\definecolor{oliver}{rgb}{0.33, 0.42, 0.18}
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% footline
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% delete navigation below
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@ -101,7 +140,7 @@
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\setbeamercolor{alerted text}{fg=beamer@simple@color}
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\setbeamerfont{block title alerted}{series=\mdseries}
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\setbeamerfont{alerted text}{series=\bfseries\boldmath}
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\hypersetup{colorlinks,linkcolor=,urlcolor=beamer@simple@color!80!white}
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\hypersetup{colorlinks,linkcolor=,urlcolor=oliver}
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\usefonttheme[onlymath]{serif}
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\setbeamerfont{frametitle}{series=\bfseries\boldmath}
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\setbeamerfont{block title}{series=\bfseries\boldmath}
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@ -122,6 +161,110 @@
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\setbeamercolor{block title}{bg=mygrey!25!white}
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\setbeamercolor{block body}{fg=black,bg=mygrey!13!white}
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% more theorem env
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\newtheorem{observation}{Observation}
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\newtheorem{question}{Question}
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% ----------------------------------------------------------------------
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% Simple math stuff
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% ----------------------------------------------------------------------
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\renewcommand{\subset}{\subseteq}
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% ---- SYMBOLS ----
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\let\e\varepsilon % a ``real'' epsilon — better yet, just use Unicode ε.
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%
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% I give up. These are in the wrong font, but my kludged versions
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% LOOK like kludges, especially \Z, \Q, and \C.
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%
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\def\Real{\mathbb{R}}
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\def\Proj{\mathbb{P}}
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\def\Hyper{\mathbb{H}}
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\def\Integer{\mathbb{Z}}
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\def\Natural{\mathbb{N}}
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\def\Complex{\mathbb{C}}
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\def\Rational{\mathbb{Q}}
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\let\N\Natural
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\let\Q\Rational
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\let\R\Real
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\let\Z\Integer
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\def\Rd{\Real^d}
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\def\RP{\Real\Proj}
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\def\CP{\Complex\Proj}
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% ---- OPERATORS (requires amsmath) ----
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\def\aff{\operatorname{aff}}
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\def\area{\operatorname{area}}
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\def\argmax{\operatornamewithlimits{arg\,max}}
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\def\argmin{\operatornamewithlimits{arg\,min}}
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\def\Aut{\operatorname{Aut}} % Automorphism group
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\def\card{\operatorname{card}} % cardinality, deprecated for \abs
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\def\conv{\operatorname{conv}}
|
||||
\def\E{\operatorname{E}} % Expectation: $\E[X]$ (like \Pr)
|
||||
\def\EE{\operatornamewithlimits{E}}
|
||||
\def\Hom{\operatorname{Hom}} % Homomorphism group
|
||||
\def\id{\operatorname{id}} % identity
|
||||
\def\im{\operatorname{im}} % image
|
||||
\def\lcm{\operatorname{lcm}}
|
||||
\def\lfs{\operatorname{lfs}} % local feature size
|
||||
\def\poly{\operatorname{poly}}
|
||||
\def\polylog{\operatorname{polylog}}
|
||||
\def\rank{\operatorname{rank}}
|
||||
\def\rel{\operatorname{rel\,}} % relative (interior, boundary, etc.)
|
||||
\def\sgn{\operatorname{sgn}}
|
||||
\def\vol{\operatorname{vol}} % volume
|
||||
|
||||
\def\fp#1{^{\underline{#1}}} % falling powers: $n\fp{d}$
|
||||
\def\rp#1{^{\overline{#1}}} % rising powers: $n\rp{d}$
|
||||
|
||||
\def\setsymdiff{\operatorname{\triangle}}
|
||||
% % --- Darts and fences ---
|
||||
% % less nice replacements for stmaryrd characters
|
||||
% \@ifundefined{shortrightarrow}{\let\shortrightarrow\rightarrow}{}
|
||||
% \@ifundefined{shortleftarrow}{\let\shortleftarrow\leftarrow}{}
|
||||
% \@ifundefined{shortuparrow}{\let\shortuparrow\uparrow}{}
|
||||
% \@ifundefined{shortdownarrow}{\let\shortdownarrow\downarrow}{}
|
||||
|
||||
\def\arcto{\mathord\shortrightarrow}
|
||||
\def\arcfrom{\mathord\shortleftarrow}
|
||||
\def\arc#1#2{#1\arcto#2}
|
||||
\def\cra#1#2{#1\mathord\shortleftarrow#2}
|
||||
\def\fence#1#2{#1\mathord\shortuparrow#2}
|
||||
\def\ecnef#1#2{#1\mathord\shortdownarrow#2}
|
||||
|
||||
% --- Cheap displaystyle operators ---
|
||||
\def\Frac#1#2{{\displaystyle\frac{#1}{#2}}}
|
||||
\def\Sum{\sum\limits}
|
||||
\def\Prod{\prod\limits}
|
||||
\def\Union{\bigcup\limits}
|
||||
\def\Inter{\bigcap\limits}
|
||||
\def\Lor{\bigvee\limits}
|
||||
\def\Land{\bigwedge\limits}
|
||||
\def\Lim{\lim\limits}
|
||||
\def\Max{\max\limits}
|
||||
\def\Min{\min\limits}
|
||||
|
||||
% ---- RELATORS ----
|
||||
\def\deq{\stackrel{\scriptscriptstyle\triangle}{=}} % Use := instead.
|
||||
\def\into{\DOTSB\hookrightarrow} % = one-to-one
|
||||
\def\onto{\DOTSB\twoheadrightarrow}
|
||||
\def\inonto{\DOTSB\lhook\joinrel\twoheadrightarrow}
|
||||
\def\from{\leftarrow}
|
||||
\def\tofrom{\leftrightarrow}
|
||||
\def\mapsfrom{\mathrel{\reflectbox{$\mapsto$}}}
|
||||
\def\longmapsfrom{\mathrel{\reflectbox{$\longmapsto$}}}
|
||||
|
||||
% ---- DELIMITER PAIRS ----
|
||||
% --- always self-scaling delmiter pairs ---
|
||||
\def\set#1{\left\{ #1 \right\}}
|
||||
\def\floor#1{\left\lfloor #1 \right\rfloor}
|
||||
\def\ceil#1{\left\lceil #1 \right\rceil}
|
||||
\def\seq#1{\left\langle #1 \right\rangle}
|
||||
\def\abs#1{\left| #1 \right|}
|
||||
\def\norm#1{\left\| #1 \right\|}
|
||||
\def\paren#1{\left( #1 \right)} % need better macro name!
|
||||
\def\brack#1{\left[ #1 \right]} % need better macro name!
|
||||
\def\indic#1{\left[ #1 \right]} % indicator variable; Iverson notation
|
||||
|
||||
|
||||
Loading…
x
Reference in New Issue
Block a user