From d507c8ba554a33dc7a231a45748d5a9fa1c82252 Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Sat, 5 Oct 2024 10:05:02 +0800 Subject: [PATCH] =?UTF-8?q?add=20example=20B=C3=BCchi=20Game?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content.tex => Büchi Game-content.tex | 70 +++++++------ simple.pdf => Büchi Games.pdf | Bin 214401 -> 226113 bytes simple.tex => Büchi Games.tex | 15 +-- beamerthemeSimple.sty | 145 +++++++++++++++++++++++++- 4 files changed, 182 insertions(+), 48 deletions(-) rename content.tex => Büchi Game-content.tex (87%) rename simple.pdf => Büchi Games.pdf (58%) rename simple.tex => Büchi Games.tex (59%) diff --git a/content.tex b/Büchi Game-content.tex similarity index 87% rename from content.tex rename to Büchi Game-content.tex index 2283541..69c7e8b 100644 --- a/content.tex +++ b/Büchi Game-content.tex @@ -1,6 +1,5 @@ \definecolor{lightblue}{rgb}{0.67,0.87,0.9} - %------------------------------------------------ \section{Motivation \& References} %------------------------------------------------ @@ -91,7 +90,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ $R_i:=\{v\in V|$ P0 can force a visit from v to a vertex in $T$ in i steps$\}$\\~ - Define Reachability set of $T$ for P0, $Reach(T,0) := \bigcup_{i=1}^{n-1}R_i$\\~ + Define Reachability set of $T$ for P0, $\Reach(T,0) := \bigcup_{i=1}^{n-1}R_i$\\~ A vertex $v\in R_i$: \\~ @@ -184,7 +183,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \end{adjustbox} \column{.5\textwidth} % Right column and width - \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 $Reach (T_i, 0)$. + \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 $\Reach (T_i, 0)$. \end{columns} \end{frame} %frame 11 @@ -204,8 +203,9 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ The minimum base of $T$ is the minimum subset of $T$ which can generate the same Reachability set as $T$.\\~ Computing the minimum base is NP-hard.\\~ - - 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 $S$. + \begin{problem}[Set cover] + 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 $S$. + \end{problem} \end{frame} %frame 13 @@ -250,10 +250,12 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \section{Büchi Game} %frame 14 \begin{frame}{Büchi Game} - \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.\\~ - Play: The definition of play in Büchi Game is the same as in Reachability game.\\~ - Definition of winning: We assume the play $P$ is infinite here. - if there exists infinite many vertices $v\in T$ in $P$, P0 wins. Otherwise P1 wins. + \begin{definition}[Büchi Game] + 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.\\~ + Play: The definition of play in Büchi Game is the same as in Reachability game.\\~ + Definition of winning: We assume the play $P$ is infinite here. + if there exists infinite many vertices $v\in T$ in $P$, P0 wins. Otherwise P1 wins. + \end{definition} \end{frame} %frame 15 @@ -299,15 +301,15 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \draw[thick,fill=lightblue, fill opacity=0.3] (2,-2) -- (2,2) -- (6,2) -- (6,-2) -- cycle; % reach \node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$}; \node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T$}; - \node[shape=circle,draw opacity=0](txt) at (4,1.7) {$Reach(T,0)$}; + \node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)$}; \end{tikzpicture} \end{adjustbox} \column{.35\textwidth} % Right column and width - If $v\notin Reach(T,0)\cup T$, $v$ can not reach $T$, P0 will lose.\\~ + If $v\notin \Reach(T,0)\cup T$, $v$ can not reach $T$, P0 will lose.\\~ - Some vertices in $T$ can not reach $Reach(T,0)\cup T$, P0 will also lose on these vertices. + Some vertices in $T$ can not reach $\Reach(T,0)\cup T$, P0 will also lose on these vertices. \end{columns} \end{frame} @@ -323,19 +325,19 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$}; \node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_2$}; \node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T_1$}; - \node[shape=circle,draw opacity=0](txt) at (4,-1.7) {$Reach(T_2,0)$}; - \node[shape=circle,draw opacity=0](txt) at (4,1.7) {$Reach(T,0)\backslash Reach(T_2,0)$}; + \node[shape=circle,draw opacity=0](txt) at (4,-1.7) {$\Reach(T_2,0)$}; + \node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)\backslash \Reach(T_2,0)$}; \end{tikzpicture} \end{adjustbox} \column{.37\textwidth} % Right column and width - $T_1=\{v\in T|v$ can't reach $T\cup Reach(T,0)\}$\\~ + $T_1=\{v\in T|v$ can't reach $T\cup \Reach(T,0)\}$\\~ - Some vertices in $T_2$ can only reach $Reach(T,0)\backslash Reach(T_2,0)$\\~ + Some vertices in $T_2$ can only reach $\Reach(T,0)\backslash \Reach(T_2,0)$\\~ - We find $T_3=\{v\in T_2|v$ can't reach $T_2\cup Reach(T_2,0)\}$ + We find $T_3=\{v\in T_2|v$ can't reach $T_2\cup \Reach(T_2,0)\}$ \end{columns} \end{frame} \begin{frame}{Algorithm for Büchi Game 1} @@ -352,7 +354,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \draw[thick,fill=red, fill opacity=0.3] (0,0.125) -- (0,0.25) -- (6,0.25) -- (6,0.125) -- cycle; \node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$}; \node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T_1$}; - \node[shape=circle,draw opacity=0](txt) at (4,1.7) {$Reach(T,0)\backslash Reach(T_2,0)$}; + \node[shape=circle,draw opacity=0](txt) at (4,1.7) {$\Reach(T,0)\backslash \Reach(T_2,0)$}; \node[shape=circle,draw opacity=0](txt) at (2.1,-1.5) {Winning set for P0}; \end{tikzpicture} @@ -361,17 +363,17 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \column{.35\textwidth} % Right column and width We repeat this process until $T_k$ does not shrink.\\~ - The remaining part of $T_k\cup Reach(T_k,0)$ is the winning set for P0. + The remaining part of $T_k\cup \Reach(T_k,0)$ is the winning set for P0. \end{columns} \end{frame} \begin{frame}{Algorithm for Büchi Game 1} \begin{itemize} \item How to find $T_1$\\ - $T_1=\{v\in T|v$ can't reach $T\cup Reach(T,0)\}$\\ - $T_1=\{v\in T|v$ can only reach $V\backslash \{T\cup Reach(T,0)\}\}$\\ - P1 wants to reach $V\backslash \{T\cup Reach(T,0)\}\}$, P0 tries to avoid $V\backslash \{T\cup Reach(T,0)\}\}$.\\ - compute $Reach(V\backslash \{T\cup Reach(T,0)\}\},1)$ + $T_1=\{v\in T|v$ can't reach $T\cup \Reach(T,0)\}$\\ + $T_1=\{v\in T|v$ can only reach $V\backslash \{T\cup \Reach(T,0)\}\}$\\ + P1 wants to reach $V\backslash \{T\cup \Reach(T,0)\}\}$, P0 tries to avoid $V\backslash \{T\cup \Reach(T,0)\}\}$.\\ + compute $\Reach(V\backslash \{T\cup \Reach(T,0)\}\},1)$ \item Time complexity\\ $O(m)$ to find $T_i$, at most $O(n)$ times. Worst-case $O(nm)$. @@ -390,7 +392,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \node[shape=circle,draw opacity=0](txt) at (3.5,-2.3) {$G$}; \node[shape=circle,draw opacity=0](txt) at (1,1.7) {$T$}; \node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$}; - \node[shape=circle,draw opacity=0](txt) at (2,-1.7) {$Reach(C_0\cup C_1,1)$}; + \node[shape=circle,draw opacity=0](txt) at (2,-1.7) {$\Reach(C_0\cup C_1,1)$}; \end{tikzpicture} \end{adjustbox} @@ -402,7 +404,7 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ $C_0$ is a set of vertices in $V_0\backslash T$ having all outgoing edges to vertices in $V\backslash T$.\\ $C_1$ is a set of vertices in $V_1\backslash T$ having an outgoing edge to vertices in $V\backslash T$.\\~ - Compute $Reach(C_0\cup C_1,1)$ + Compute $\Reach(C_0\cup C_1,1)$ \end{columns} \end{frame} \begin{frame}{Algorithm for Büchi Game 2} @@ -421,21 +423,21 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$}; \node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_1$}; \node[shape=circle,draw opacity=0](txt) at (2.5,-1.7) {$D$}; - \node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$Reach(T_1\cup D,0)$}; + \node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$\Reach(T_1\cup D,0)$}; \node[shape=circle,draw opacity=0](txt) at (4,-0.75) {$E$}; \end{tikzpicture} \end{adjustbox} \column{.39\textwidth} % Right column and width - Some vertices in $Reach(C_0\cup C_1,1)$ can "reach" $T_1$.($D$ in the left picture)\\~ + Some vertices in $\Reach(C_0\cup C_1,1)$ can "reach" $T_1$.($D$ in the left picture)\\~ - Compute $Reach(T_1\cup D,0)$.\\~ + Compute $\Reach(T_1\cup D,0)$.\\~ - $E=Reach(C_0\cup C_1,1)\backslash \{T_1\cup D \cup Reach(T_1\cup D,0)\}$\\~ + $E=\Reach(C_0\cup C_1,1)\backslash \{T_1\cup D \cup \Reach(T_1\cup D,0)\}$\\~ - $\{E\cup C_0\cup C_1\}\backslash Reach(T_1\cup D,0)$ is the set of vertices which can't "reach" $T$. + $\{E\cup C_0\cup C_1\}\backslash \Reach(T_1\cup D,0)$ is the set of vertices which can't "reach" $T$. \end{columns} \end{frame} \begin{frame}{Algorithm for Büchi Game 2} @@ -454,18 +456,18 @@ Vertices in $T$ are red, the initial vertex $v_I$ is blue.\\~ \node[shape=circle,draw opacity=0](txt) at (6.5,0) {$C_0\cup C_1$}; \node[shape=circle,draw opacity=0](txt) at (1,-1.7) {$T_1$}; \node[shape=circle,draw opacity=0](txt) at (2.5,-1.7) {$D$}; - \node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$Reach(T_1\cup D,0)$}; + \node[shape=circle,draw opacity=0](txt) at (5,-1.7) {$\Reach(T_1\cup D,0)$}; \node[shape=circle,draw opacity=0](txt) at (4,-0.75) {$E$}; \end{tikzpicture} \end{adjustbox} \column{.4\textwidth} % Right column and width - $S=\{E\cup C_0\cup C_1\}\backslash Reach(T_1\cup D,0)$ is the same as $V\backslash \{T\cup Reach(T)\}$ in Algorithm 1.\\~ + $S=\{E\cup C_0\cup C_1\}\backslash \Reach(T_1\cup D,0)$ is the same as $V\backslash \{T\cup \Reach(T)\}$ in Algorithm 1.\\~ - Then we can compute $Reach(S,1)$ to delete some losing vertices for P0 in $T$.\\~ + Then we can compute $\Reach(S,1)$ to delete some losing vertices for P0 in $T$.\\~ - Repeat the same process on $G\backslash\{T\backslash Reach(S,1)\}$ + Repeat the same process on $G\backslash\{T\backslash \Reach(S,1)\}$ \end{columns} \end{frame} \begin{frame}{Algorithm for Büchi Game 2} diff --git a/simple.pdf b/Büchi Games.pdf similarity index 58% rename from simple.pdf rename to Büchi Games.pdf index edfc7659783f31a792ba539f83f95c92e0955799..b1934e2c78121cbda442a73aae2dbffc1a999f3d 100644 GIT binary patch delta 95960 zcmZs?Q(z@ru&o=bW81cE+qP|c#r6t1wr$(Ct?t-PC%OOL=iK{r-{#w#4>fC4jruA$ z7GdxYLJ~zrBOW*{pjA~tX_y(l_f_ldw=8>X?z~|nHY(Q0$}ds;uDVOwMg_d>U^D=M zl!(}s7(6+nPqLAHzm=2-G@K)wN6ziri*ITP)jt;| z8!;jDc)v%#dDR*vIAmoEv}s%zGfSeWE=OD}*@NOb*Qm}MP;Y=!kB^2PB{gX+R*>qI zb#!Hth9C(As<0xdJWJfr1A^VsGAPOsPQ~JQX=$kRkr9|}hwkK4n2Yu7n#DNW({Lov zz7}J^c5Ai}U1!)*|L*4TnsH8eWx>2u6J?`entB#WuVY^rNmV}-edy)5JjCg#K_9R^zUD5KTzhq8XZ 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