diff --git a/W4/fasto.sty b/W4/fasto.sty new file mode 100644 index 0000000..2897484 --- /dev/null +++ b/W4/fasto.sty @@ -0,0 +1,50 @@ +\usepackage{xcolor} +\usepackage{listings} +\usepackage{tcolorbox} +\tcbuselibrary{listings} + +\definecolor{basicColor}{HTML}{5C6166} + +\definecolor{backgroundColor}{HTML}{FCFCFC} +\definecolor{keywordColor}{HTML}{FA8D3E} +\definecolor{stringColor}{HTML}{86B300} +\definecolor{typeColor}{HTML}{399EE6} +\definecolor{numberColor}{HTML}{A37ACC} + +\lstdefinelanguage{Fasto}{ + alsoletter=0123456789, + keywords={[0]fun,let,in,if,else,then}, + keywords={[1]int,char,bool}, + keywords={[2]@invariant,0,1,2,3,4,5,6,7,8,9}, + sensitive=true, + comment=[l]{//}, + comment=[s]{/*}{*/}, + string=[b]", +} + +\lstdefinelanguage{FSharp}{ + alsoletter=0123456789->|, + keywords={[0]let,rec,|,->,match,for,in,with}, + keywords={[1]int,char,bool,INT,EXP,SymTab,CONSTANT,VARIABLE,OPERATE,LET_IN, OVER,BPLUS,BMINUS,BTIMES,RSUM,RPROD,RMAX,RARGMAX}, + keywords={[2]@invariant,0,1,2,3,4,5,6,7,8,9}, + sensitive=true, + comment=[l]{//}, + comment=[s]{/*}{*/}, + string=[b]", +} + +\lstset{ + basicstyle=\color{basicColor}\small\ttfamily, + keywordstyle=[0]\color{keywordColor}, + keywordstyle=[1]\color{typeColor}, + keywordstyle=[2]\color{numberColor}, + stringstyle=\color{stringColor}, + backgroundcolor=\color{backgroundColor}, + frame=single, framerule=1pt, + numbers=left, + stepnumber=1, + showstringspaces=false, + tabsize=1, + breaklines=true, + breakatwhitespace=true +} diff --git a/W4/main.aux b/W4/main.aux new file mode 100644 index 0000000..28821ec --- /dev/null +++ b/W4/main.aux @@ -0,0 +1,12 @@ +\relax +\@writefile{toc}{\contentsline {section}{\numberline {Task 1}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {Task 2}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsubsection}{\numberline {a)}}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline 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mode 100644 index 0000000..7b57157 Binary files /dev/null and b/W4/main.pdf differ diff --git a/W4/main.synctex.gz b/W4/main.synctex.gz new file mode 100644 index 0000000..26d35e8 Binary files /dev/null and b/W4/main.synctex.gz differ diff --git a/W4/main.tex b/W4/main.tex new file mode 100644 index 0000000..4a0f41a --- /dev/null +++ b/W4/main.tex @@ -0,0 +1,566 @@ +\documentclass[a4paper]{article} +\usepackage{listings} +\usepackage{fontspec} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{multicol} +\usepackage{changepage} +\usepackage[dvipsnames]{xcolor} +\usepackage{tikz} + +\usetikzlibrary{arrows.meta,automata} + +\renewcommand{\thesection}{Task \arabic{section}} +\renewcommand{\thesubsection}{\arabic{subsection})} +\renewcommand{\thesubsubsection}{\alph{subsubsection})} + +\setmonofont[Scale=0.9]{Antikor Mono Medium} + +\input{fasto.sty} + +\setlength{\parskip}{5pt} +\setlength{\parindent}{0pt} + +\title{W4 - IPS} +\author{Nikolaj Gade (qhp695)} +\date{May 2022} + +\makeatletter +\renewenvironment{table}% + {\renewcommand{\familydefault}{\ttdefault}\selectfont + \@float{table}} + {\end@float} +\makeatother + + +\begin{document} +\maketitle +\section{} +\begin{table}[h] +\begin{center} +\begin{tabular}{|r||r|r|r|l|} \hline + SubExp & I/O? & Elim? & UsedVars & OptSubExp \\ \hline \hline + 1 & no & - & u & u \\ \hline + 2 & no & - & x & x+x \\ \hline + 3 & yes & - & x & foo(x) \\ \hline + 4 & no & - & & 7 \\ \hline + 5 & yes & no & x & let x=foo(x) in 7 \\ \hline + 6 & no & - & x,u & x+u \\ \hline + 7 & no & - & y,x & y*x \\ \hline + 8 & no & yes & y,x & y*x \\ \hline + 9 & yes & no & x & let y=(let x=foo(x) in 7) in y*x \\ \hline + 10 & yes & yes & x & let y=(let x=foo(x) in 7) in y*x \\ \hline + 11 & yes & no & u & let x=u in let y=(let x=foo(x) in 7) in y*x \\ \hline +\end{tabular} +\end{center} +\end{table} + +\section{} +\subsubsection{} +\begin{table}[h] +\begin{center} +\begin{tabular}{|r||r|r|r|} \hline + i & succ[i] & gen[i] & kill[i] \\ \hline \hline + 1 & 2 & & \\ \hline + 2 & 3,7 & a,b & \\ \hline + 3 & 4 & & \\ \hline + 4 & 5 & a & t \\ \hline + 5 & 6 & b & a \\ \hline + 6 & 7 & t & b \\ \hline + 7 & 8 & & \\ \hline + 8 & 9 & & z \\ \hline + 9 & 10 & a,b & b \\ \hline + 10 & 1,11 & b,z & \\ \hline + 11 & 12 & & \\ \hline + 12 & & a & \\ \hline +\end{tabular} +\end{center} +\end{table} + +\newpage +\subsubsection{} +\begin{table}[h] +\begin{center} +\begin{tabular}{|r||c|c||c|c||c|c||c|c|} \hline + & \multicolumn{2}{c||}{Initial} & \multicolumn{2}{c||}{Iteration 1} & \multicolumn{2}{c||}{Iteration 2} & \multicolumn{2}{c|}{Iteration 3} \\ + i & out[i] & in[i] & out[i] & in[i] & out[i] & in[i] & out[i] & in[i] \\ \hline \hline + 1 & & & a,b & a,b & a,b & a,b & a,b & a,b \\ \hline + 2 & & & a,b & a,b & a,b & a,b & a,b & a,b \\ \hline + 3 & & & a,b & a,b & a,b & a,b & a,b & a,b \\ \hline + 4 & & & b,t & a,b & b,t & a,b & b,t & a,b \\ \hline + 5 & & & a,t & b,t & a,t & b,t & a,t & b,t \\ \hline + 6 & & & a,b & a,t & a,b & a,t & a,b & a,t \\ \hline + 7 & & & a,b & a,b & a,b & a,b & a,b & a,b \\ \hline + 8 & & & a,b,z & a,b & a,b,z & a,b & a,b,z & a,b \\ \hline + 9 & & & a,b,z & a,b,z & a,b,z & a,b,z & a,b,z & a,b,z \\ \hline + 10 & & & a & a,b,z & a,b & a,b,z & a,b & a,b,z \\ \hline + 11 & & & a & a & a & a & a & a \\ \hline +\end{tabular} +\end{center} +\end{table} + +\subsubsection{} +\begin{table}[h] +\begin{center} +\begin{tabular}{|r||c|c||c|} \hline + i & kill[i] & out[i] & Interferes with \\ \hline \hline + 4 & t & b,t & b \\ \hline + 5 & a & a,t & t \\ \hline + 6 & b & a,b & a \\ \hline + 8 & z & a,b,z & a,b \\ \hline + 9 & b & a,b,z & a,z \\ \hline +\end{tabular} +\end{center} +\end{table} + +\subsubsection{} +\resizebox{100pt}{!}{% +\begin{tikzpicture} + \node[font=\bfseries] (a) {a}; + \node[font=\bfseries] (b) [right of=a] {b}; + \node[font=\bfseries] (t) [below of=a] {t}; + \node[font=\bfseries] (z) [below of=b] {z}; + + \draw (t) to (b); + \draw (a) to (t); + \draw (z) to (a); + \draw (z) to (b); + \draw (b) to (a); +\end{tikzpicture} +} + +\newpage + +\subsubsection{} + +\begin{table}[h] +\begin{center} +\begin{tabular}{|c|c||c|} \hline + Node & Neighbors & Color \\ \hline \hline + z & & \color{blue} 1 \\ \hline + b & z & \color{red} 2 \\ \hline + a & b,z & \color{green} 3 \\ \hline + t & a,b & \color{blue} 1 \\ \hline +\end{tabular} +\end{center} +\end{table} + +\resizebox{100pt}{!}{% +\begin{tikzpicture} + \node[font=\bfseries] (a) [color=red] {a}; + \node[font=\bfseries] (b) [right of=a, color=green] {b}; + \node[font=\bfseries] (t) [below of=a, color=blue] {t}; + \node[font=\bfseries] (z) [below of=b, color=blue] {z}; + + \draw (t) to (b); + \draw (a) to (t); + \draw (z) to (a); + \draw (z) to (b); + \draw (b) to (a); +\end{tikzpicture} +} + +\subsubsection{} + +\begin{table}[h] +\begin{center} +\begin{tabular}{|c|c||c|} \hline + Node & Neighbors & Color \\ \hline \hline + z & & \color{blue} 1 \\ \hline + b & z & \color{red} 2 \\ \hline + t & b & \color{blue} 1 \\ \hline + a & b,t,z & spill \\ \hline +\end{tabular} +\end{center} +\end{table} + +\resizebox{100pt}{!}{% +\begin{tikzpicture} + \node[font=\bfseries] (a) [color=gray] {a}; + \node[font=\bfseries] (b) [right of=a, color=red] {b}; + \node[font=\bfseries] (t) [below of=a, color=blue] {t}; + \node[font=\bfseries] (z) [below of=b, color=blue] {z}; + + \draw (t) to (b); + \draw [color=gray] (a) to (t); + \draw [color=gray] (z) to (a); + \draw (z) to (b); + \draw [color=gray] (b) to (a); +\end{tikzpicture} +} + +\begin{lstlisting} + gcd(a,b) { + M[addra] := a +1: LABEL start + a2 := M[addra] +2: IF a2 < b THEN next ELSE swap +3: LABEL swap + a4 := M[addra] +4: t := a4 +5: a5 := b + M[addra] := a5 +6: b := t +7: LABEL next +8: z := 0 + a9 := M[addra] +9: b := b mod a9 +10: IF b = z THEN end ELSE start +11: LABEL end + a := M[addra] +12: RETURN a + } +\end{lstlisting} + +\section{} +Edges without a label should be assumed to be labeled with \epsilon. + +\begin{tikzpicture} + \begin{scope} + \node[] (0) at (0,0) {start}; + \node[draw, rectangle, rounded corners=2mm] (a) at (3,0) {\color{ForestGreen}\textbf{(a*b|a)*(b|}$\boldsymbol\epsilon$\textbf{)}}; + \node[draw, state, double] (b) at (6,0) {}; + \end{scope} + + \begin{scope}[rounded corners=2mm, -Triangle] + \draw (0) -- (a); + \draw (a) -- (b); + \end{scope} +\end{tikzpicture} + +\begin{tikzpicture} + \begin{scope} + \node[] (0) at (0,0) {start}; + \node[draw, rectangle, rounded corners=2mm] (a) at (2,0) {\color{ForestGreen}\textbf{(a*b|a)*}}; + \node[draw, rectangle, rounded corners=2mm] (c) at (4,0) {\color{red}\textbf{b|}$\boldsymbol\epsilon$}; + \node[draw, state, double] (b) at (6,0) {}; + \end{scope} + + \begin{scope}[rounded corners=2mm, -Triangle] + \draw (0) -- (a); + \draw (a) -- (c); + \draw (c) -- (b); + \end{scope} +\end{tikzpicture} + +\begin{tikzpicture} + \begin{scope} + \node[] (0) at (0,0) {start}; + \node[state] (a) at (2,0) {}; + \node[draw, rectangle, rounded corners=2mm] (d) at (2,2) {\color{ForestGreen}\textbf{a*b|a}}; + \node[draw, rectangle, rounded corners=2mm] (c) at (4,0) {\color{red}\textbf{b|}$\boldsymbol\epsilon$}; + \node[draw, state, double] (b) at (6,0) {}; + \end{scope} + + \begin{scope}[rounded corners=2mm, -Triangle] + \draw (0) -- (a); + \draw (a) -- (c); + \draw (c) -- (b); + \draw (a) to [out = 135, in=225] (d); + \draw (d) to [out = 315, in=45] (a); + \end{scope} +\end{tikzpicture} + +\begin{tikzpicture} + \begin{scope} + \node[] (0) at (0,0) {start}; + \node[state] (a) at (2,0) {}; + \node[state] (d) at (0.5,2.5) {}; + \node[draw, rectangle, rounded corners=2mm] (e) at (2,3.5) {\color{ForestGreen}\textbf{a*b}}; + \node[draw, state] (f) at (2,1.5) {}; + \node[draw, state] (g) at (3.5,2.5) {}; + \node[draw, rectangle, rounded corners=2mm] (c) at (4,0) {\color{red}\textbf{b|}$\boldsymbol\epsilon$}; + \node[draw, state, double] (b) at (6,0) {}; + \end{scope} + + \begin{scope}[rounded corners=2mm, -Triangle] + \draw (0) -- (a); + \draw (a) -- (c); + \draw (c) -- (b); + \draw (a) to [out = 135, in=180] (d); + \draw (g) to [out = 0, in=45] (a); + \draw (d) -- (e); + \draw (d) -- (f); + \draw (e) -- (g); + \draw (f) -- (g) node[midway, above] {a}; + \end{scope} +\end{tikzpicture} + +\begin{tikzpicture} + \begin{scope} + \node[] (0) at (0,0) {start}; + \node[state] (a) at (2,0) {}; + \node[state] (d) at (0,2.5) {}; + \node[draw, rectangle, rounded corners=2mm] (e) at (1.3,3.5) {\color{ForestGreen}\textbf{a*}}; + \node[state] (h) at (2.7,3.5) {}; + \node[draw, state] (f) at (2,1.5) {}; + \node[draw, state] (g) at (4,2.5) {}; + \node[draw, rectangle, rounded corners=2mm] (c) at (4,0) {\color{red}\textbf{b|}$\boldsymbol\epsilon$}; + \node[draw, state, double] (b) at (6,0) {}; + \end{scope} + + \begin{scope}[rounded corners=2mm, -Triangle] + \draw (0) -- (a); + \draw (a) -- (c); + \draw (c) -- (b); + \draw (a) to [out = 135, in=180] (d); + \draw (g) to [out = 0, in=45] (a); + \draw (d) -- (e); + \draw (d) -- (f); + \draw (e) -- (h); + \draw (h) -- (g) node[midway, above] {b}; + \draw (f) -- (g) node[midway, above] {a}; + \end{scope} +\end{tikzpicture} + +\begin{tikzpicture} + \begin{scope} + \node[] (0) at (0,0) {start}; + \node[state] (a) at (2,0) {}; + \node[state] (d) at (0,2.5) {}; + \node[state] (e) at (1.3,3.5) {}; + \node[state] (i) at (0.3,4.5) {}; + \node[state] (h) at (2.7,3.5) {}; + \node[draw, state] (f) at (2,1.5) {}; + \node[draw, state] (g) at (4,2.5) {}; + \node[draw, rectangle, rounded corners=2mm] (c) at (4,0) {\color{red}\textbf{b|}$\boldsymbol\epsilon$}; + \node[draw, state, double] (b) at (6,0) {}; + \end{scope} + + \begin{scope}[rounded corners=2mm, -Triangle] + \draw (0) -- (a); + \draw (a) -- (c); + \draw (c) -- (b); + \draw (a) to [out = 135, in=180] (d); + \draw (g) to [out = 0, in=45] (a); + \draw (d) -- (e); + \draw (d) -- (f); + \draw (e) to [out = 180, in=270] (i) node at (1.3,4.5) {a}; + \draw (i) to [out = 0, in=90] (e); + \draw (e) -- (h); + \draw (h) -- (g) node[midway, above right] {b}; + \draw (f) -- (g) node[midway, above left] {a}; + \end{scope} +\end{tikzpicture} + +\begin{tikzpicture} + \begin{scope} + \node[] (0) at (0,0) {start}; + \node[state] (a) at (2,0) {}; + \node[state] (d) at (0,2.5) {}; + \node[state] (e) at (1.3,3.5) {}; + \node[state] (i) at (0.3,4.5) {}; + \node[state] (h) at (2.7,3.5) {}; + \node[state] (f) at (2,1.5) {}; + \node[state] (g) at (4,2.5) {}; + \node[state] (c) at (5,0) {}; + \node[state] (j) at (6.8,1) {}; + \node[state] (k) at (6.8,-1) {}; + \node[state, double] (b) at (8.6,0) {}; + \end{scope} + + \begin{scope}[rounded corners=2mm, -Triangle] + \draw (0) -- (a); + \draw (a) -- (c); + \draw (a) to [out = 135, in=180] (d); + \draw (g) to [out = 0, in=45] (a); + \draw (d) -- (e); + \draw (d) -- (f); + \draw (e) to [out = 180, in=270] (i) node at (1.3,4.5) {a}; + \draw (i) to [out = 0, in=90] (e); + \draw (e) -- (h); + \draw (h) -- (g) node[midway, above right] {b}; + \draw (f) -- (g) node[midway, above left] {a}; + \draw (c) -- (j); + \draw (c) -- (k); + \draw (j) -- (b) node[midway, above right] {b}; + \draw (k) -- (b); + \end{scope} +\end{tikzpicture} + +\section{} +\begin{table}[h] +\begin{center} +\begin{tabular}{|l||r|c|} \hline + state & epsilon-closure & complete \\ \hline \hline + 0 & \{1,2,4\} & \\ \hline +\end{tabular} +\end{center} +\end{table} + + +\begin{equation} +\begin{split} +move(s'_0, \mathtt{a}) & = \epsilon\text{-closure}(\{t | s \in \{1,2,4\}\text{ and } s^{\mathtt{a}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{3,4\}) \\ +& = \{3,4,6\} \\ +& = s'_1 +\end{split} +\end{equation} +\begin{equation} +\begin{split} +move(s'_0, \mathtt{b}) & = \epsilon\text{-closure}(\{t | s \in \{1,2,4\}\text{ and } s^{\mathtt{b}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{5\}) \\ +& = \{4,5,6\} \\ +& = s'_2 +\end{split} +\end{equation} + +\begin{table}[h] +\begin{center} +\begin{tabular}{|l||r|c|} \hline + state & epsilon-closure & complete \\ \hline \hline + 0 & \{1,2,4\} & \checkmark \\ \hline + 1 & \{3,4,6\} & \\ \hline + 2 & \{4,5,6\} & \\ \hline +\end{tabular} +\end{center} +\end{table} + +\begin{equation} +\begin{split} +move(s'_1, \mathtt{a}) & = \epsilon\text{-closure}(\{t | s \in \{3,4,6\}\text{ and } s^{\mathtt{a}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{4\}) \\ +& = \{4\} \\ +& = s'_3 +\end{split} +\end{equation} +\begin{equation} +\begin{split} +move(s'_1, \mathtt{b}) & = \epsilon\text{-closure}(\{t | s \in \{3,4,6\}\text{ and } s^{\mathtt{b}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{3,5\}) \\ +& = \{3,4,5,6\} \\ +& = s'_4 +\end{split} +\end{equation} + +\newpage +\begin{table}[h] +\begin{center} +\begin{tabular}{|l||r|c|} \hline + state & epsilon-closure & complete \\ \hline \hline + 0 & \{1,2,4\} & \checkmark \\ \hline + 1 & \{3,4,6\} & \checkmark \\ \hline + 2 & \{4,5,6\} & \\ \hline + 3 & \{4\} & \\ \hline + 4 & \{3,4,5,6\} & \\ \hline +\end{tabular} +\end{center} +\end{table} + +\begin{equation} +\begin{split} +move(s'_2, \mathtt{a}) & = \epsilon\text{-closure}(\{t | s \in \{4,5,6\}\text{ and } s^{\mathtt{a}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{4\}) \\ +& = \{4\} \\ +& = s'_3 +\end{split} +\end{equation} +\begin{equation} +\begin{split} +move(s'_2, \mathtt{b}) & = \epsilon\text{-closure}(\{t | s \in \{4,5,6\}\text{ and } s^{\mathtt{b}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{5\}) \\ +& = \{4,5,6\} \\ +& = s'_2 +\end{split} +\end{equation} + + +\begin{table}[h] +\begin{center} +\begin{tabular}{|l||r|c|} \hline + state & epsilon-closure & complete \\ \hline \hline + 0 & \{1,2,4\} & \checkmark \\ \hline + 1 & \{3,4,6\} & \checkmark \\ \hline + 2 & \{4,5,6\} & \checkmark \\ \hline + 3 & \{4\} & \\ \hline + 4 & \{3,4,5,6\} & \\ \hline +\end{tabular} +\end{center} +\end{table} + +\begin{equation} +\begin{split} +move(s'_3, \mathtt{a}) & = \epsilon\text{-closure}(\{t | s \in \{4\}\text{ and } s^{\mathtt{a}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{4\}) \\ +& = \{4\} \\ +& = s'_3 +\end{split} +\end{equation} +\begin{equation} +\begin{split} +move(s'_3, \mathtt{b}) & = \epsilon\text{-closure}(\{t | s \in \{4\}\text{ and } s^{\mathtt{b}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{5\}) \\ +& = \{4,5,6\} \\ +& = s'_2 +\end{split} +\end{equation} + + +\begin{table}[h] +\begin{center} +\begin{tabular}{|l||r|c|} \hline + state & epsilon-closure & complete \\ \hline \hline + 0 & \{1,2,4\} & \checkmark \\ \hline + 1 & \{3,4,6\} & \checkmark \\ \hline + 2 & \{4,5,6\} & \checkmark \\ \hline + 3 & \{4\} & \checkmark \\ \hline + 4 & \{3,4,5,6\} & \\ \hline +\end{tabular} +\end{center} +\end{table} + +\begin{equation} +\begin{split} +move(s'_4, \mathtt{a}) & = \epsilon\text{-closure}(\{t | s \in \{3,4,5,6\}\text{ and } s^{\mathtt{a}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{4\}) \\ +& = \{4\} \\ +& = s'_3 +\end{split} +\end{equation} +\begin{equation} +\begin{split} +move(s'_4, \mathtt{b}) & = \epsilon\text{-closure}(\{t | s \in \{3,4,5,6\}\text{ and } s^{\mathtt{b}}t \in T\}) \\ +& = \epsilon\text{-closure}(\{3,5\}) \\ +& = \{3,4,5,6\} \\ +& = s'_4 +\end{split} +\end{equation} + +\newpage +\begin{table}[h] +\begin{center} +\begin{tabular}{|l||r|c|} \hline + state & epsilon-closure & complete \\ \hline \hline + 0 & \{1,2,4\} & \checkmark \\ \hline + 1 & \{3,4,6\} & \checkmark \\ \hline + 2 & \{4,5,6\} & \checkmark \\ \hline + 3 & \{4\} & \checkmark \\ \hline + 4 & \{3,4,5,6\} & \checkmark \\ \hline +\end{tabular} +\end{center} +\end{table} + +\begin{tikzpicture} + \node[state] (0) at (0,0) {$S_0$}; + \node[state, double] (1) at (3,0) {$S_1$}; + \node[state, double] (2) at (0,-3) {$S_2$}; + \node[state] (3) at (3,-3) {$S_3$}; + \node[state, double] (4) at (6,0) {$S_4$}; + + \begin{scope}[rounded corners=2mm, -Triangle] + \draw (0) -- (1) node [midway, above] {\texttt{a}}; + \draw (0) -- (2) node [midway, left] {\texttt{b}}; + \draw (1) -- (3) node [midway, left] {\texttt{a}}; + \draw (1) -- (4) node [midway, above] {\texttt{b}}; + \draw (2) to [out=315, in=225] (3) node at (1.5,-2) {\texttt{a}}; + \draw (2) to [out=270, in=180, loop] (2) node at (-1.2,-4.1) {\texttt{b}}; + \draw (3) to [out=0, in=270, loop] (3) node at (4.2,-4.1) {\texttt{a}}; + \draw (3) to [out=135, in=45] (2) node at (1.5,-4) {\texttt{a}}; + \draw (4) -- (3) node [midway, above] {\texttt{a}}; + \draw (4) to [out=0, in=270, loop] (4) node at (7.2,-1.1) {\texttt{b}}; + \end{scope} +\end{tikzpicture} + +\end{document} \ No newline at end of file diff --git a/W4/report.pdf b/W4/report.pdf new file mode 100644 index 0000000..7b57157 Binary files /dev/null and b/W4/report.pdf differ