7 \usepackage{pgfplots} |
7 \usepackage{pgfplots} |
8 \usepackage{stackengine} |
8 \usepackage{stackengine} |
9 %% \usepackage{accents} |
9 %% \usepackage{accents} |
10 \newcommand\barbelow[1]{\stackunder[1.2pt]{#1}{\raisebox{-4mm}{\boldmath$\uparrow$}}} |
10 \newcommand\barbelow[1]{\stackunder[1.2pt]{#1}{\raisebox{-4mm}{\boldmath$\uparrow$}}} |
11 |
11 |
12 \begin{filecontents}{re-python2.data} |
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22 23 0.878 |
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24 25 3.40 |
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28 \end{filecontents} |
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29 |
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47 \end{filecontents} |
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48 |
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49 \begin{filecontents}{re-js.data} |
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60 32 32.190 |
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61 \end{filecontents} |
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62 |
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63 \begin{filecontents}{re-java9.data} |
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64 1000 0.01410 |
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73 10000 0.97419 |
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74 11000 1.28697 |
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76 14000 2.07079 |
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77 16000 2.69846 |
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78 20000 4.41823 |
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81 30000 9.99446 |
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83 38000 16.281621 |
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85 46000 21.984721 |
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86 50000 26.950203 |
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87 60000 43.0327746 |
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88 \end{filecontents} |
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89 |
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90 |
12 |
91 \begin{document} |
13 \begin{document} |
92 |
14 |
93 % BF IDE |
15 % BF IDE |
94 % https://www.microsoft.com/en-us/p/brainf-ck/9nblgggzhvq5 |
16 % https://www.microsoft.com/en-us/p/brainf-ck/9nblgggzhvq5 |
95 |
17 |
96 \section*{Coursework 9 (Scala)} |
18 \section*{Coursework 9 (Scala)} |
97 |
19 |
98 This coursework is worth 10\%. It is about a regular expression |
20 This coursework is worth 10\%. It is about a small programming |
99 matcher and the shunting yard algorithm by Dijkstra. The first part is |
21 language called brainf***. The first part is due on 13 December at |
100 due on 6 December at 11pm; the second, more advanced part, is due on |
22 11pm; the second, more advanced part, is due on 20 December at |
101 21 December at 11pm. In the first part, you are asked to implement a |
23 11pm.\bigskip |
102 regular expression matcher based on derivatives of regular |
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103 expressions. The reason is that regular expression matching in |
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104 languages like Java, JavaScipt and Python can sometimes be extremely |
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105 slow. The advanced part is about the shunting yard algorithm that |
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106 transforms the usual infix notation of arithmetic expressions into the |
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107 postfix notation, which is for example used in compilers.\bigskip |
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108 |
24 |
109 \IMPORTANT{} |
25 \IMPORTANT{} |
110 |
26 |
111 \noindent |
27 \noindent |
112 Also note that the running time of each part will be restricted to a |
28 Also note that the running time of each part will be restricted to a |
113 maximum of 30 seconds on my laptop. |
29 maximum of 30 seconds on my laptop. |
114 |
30 |
115 \DISCLAIMER{} |
31 \DISCLAIMER{} |
116 |
32 |
117 |
33 |
118 \subsection*{Part 1 (6 Marks, Regular Expression Matcher)} |
34 |
119 |
35 \subsection*{Part 1 (6 Marks)} |
120 The task is to implement a regular expression matcher that is based on |
36 |
121 derivatives of regular expressions. Most of the functions are defined by |
37 Coming from Java or C++, you might think Scala is a rather esoteric |
122 recursion over regular expressions and can be elegantly implemented |
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123 using Scala's pattern-matching. The implementation should deal with the |
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124 following regular expressions, which have been predefined in the file |
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125 \texttt{re.scala}: |
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126 |
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127 \begin{center} |
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128 \begin{tabular}{lcll} |
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129 $r$ & $::=$ & $\ZERO$ & cannot match anything\\ |
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130 & $|$ & $\ONE$ & can only match the empty string\\ |
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131 & $|$ & $c$ & can match a single character (in this case $c$)\\ |
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132 & $|$ & $r_1 + r_2$ & can match a string either with $r_1$ or with $r_2$\\ |
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133 & $|$ & $r_1\cdot r_2$ & can match the first part of a string with $r_1$ and\\ |
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134 & & & then the second part with $r_2$\\ |
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135 & $|$ & $r^*$ & can match a string with zero or more copies of $r$\\ |
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136 \end{tabular} |
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137 \end{center} |
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138 |
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139 \noindent |
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140 Why? Knowing how to match regular expressions and strings will let you |
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141 solve a lot of problems that vex other humans. Regular expressions are |
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142 one of the fastest and simplest ways to match patterns in text, and |
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143 are endlessly useful for searching, editing and analysing data in all |
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144 sorts of places (for example analysing network traffic in order to |
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145 detect security breaches). However, you need to be fast, otherwise you |
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146 will stumble over problems such as recently reported at |
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147 |
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148 {\small |
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149 \begin{itemize} |
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150 \item[$\bullet$] \url{http://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016} |
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151 \item[$\bullet$] \url{https://vimeo.com/112065252} |
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152 \item[$\bullet$] \url{http://davidvgalbraith.com/how-i-fixed-atom/} |
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153 \end{itemize}} |
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154 |
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155 \subsubsection*{Tasks (file re.scala)} |
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156 |
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157 The file \texttt{re.scala} has already a definition for regular |
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158 expressions and also defines some handy shorthand notation for |
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159 regular expressions. The notation in this document matches up |
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160 with the code in the file as follows: |
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161 |
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162 \begin{center} |
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163 \begin{tabular}{rcl@{\hspace{10mm}}l} |
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164 & & code: & shorthand:\smallskip \\ |
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165 $\ZERO$ & $\mapsto$ & \texttt{ZERO}\\ |
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166 $\ONE$ & $\mapsto$ & \texttt{ONE}\\ |
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167 $c$ & $\mapsto$ & \texttt{CHAR(c)}\\ |
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168 $r_1 + r_2$ & $\mapsto$ & \texttt{ALT(r1, r2)} & \texttt{r1 | r2}\\ |
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169 $r_1 \cdot r_2$ & $\mapsto$ & \texttt{SEQ(r1, r2)} & \texttt{r1 $\sim$ r2}\\ |
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170 $r^*$ & $\mapsto$ & \texttt{STAR(r)} & \texttt{r.\%} |
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171 \end{tabular} |
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172 \end{center} |
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173 |
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174 |
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175 \begin{itemize} |
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176 \item[(1)] Implement a function, called \textit{nullable}, by |
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177 recursion over regular expressions. This function tests whether a |
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178 regular expression can match the empty string. This means given a |
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179 regular expression it either returns true or false. The function |
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180 \textit{nullable} |
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181 is defined as follows: |
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182 |
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183 \begin{center} |
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184 \begin{tabular}{lcl} |
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185 $\textit{nullable}(\ZERO)$ & $\dn$ & $\textit{false}$\\ |
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186 $\textit{nullable}(\ONE)$ & $\dn$ & $\textit{true}$\\ |
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187 $\textit{nullable}(c)$ & $\dn$ & $\textit{false}$\\ |
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188 $\textit{nullable}(r_1 + r_2)$ & $\dn$ & $\textit{nullable}(r_1) \vee \textit{nullable}(r_2)$\\ |
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189 $\textit{nullable}(r_1 \cdot r_2)$ & $\dn$ & $\textit{nullable}(r_1) \wedge \textit{nullable}(r_2)$\\ |
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190 $\textit{nullable}(r^*)$ & $\dn$ & $\textit{true}$\\ |
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191 \end{tabular} |
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192 \end{center}~\hfill[1 Mark] |
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193 |
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194 \item[(2)] Implement a function, called \textit{der}, by recursion over |
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195 regular expressions. It takes a character and a regular expression |
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196 as arguments and calculates the derivative regular expression according |
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197 to the rules: |
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198 |
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199 \begin{center} |
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200 \begin{tabular}{lcl} |
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201 $\textit{der}\;c\;(\ZERO)$ & $\dn$ & $\ZERO$\\ |
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202 $\textit{der}\;c\;(\ONE)$ & $\dn$ & $\ZERO$\\ |
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203 $\textit{der}\;c\;(d)$ & $\dn$ & $\textit{if}\; c = d\;\textit{then} \;\ONE \; \textit{else} \;\ZERO$\\ |
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204 $\textit{der}\;c\;(r_1 + r_2)$ & $\dn$ & $(\textit{der}\;c\;r_1) + (\textit{der}\;c\;r_2)$\\ |
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205 $\textit{der}\;c\;(r_1 \cdot r_2)$ & $\dn$ & $\textit{if}\;\textit{nullable}(r_1)$\\ |
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206 & & $\textit{then}\;((\textit{der}\;c\;r_1)\cdot r_2) + (\textit{der}\;c\;r_2)$\\ |
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207 & & $\textit{else}\;(\textit{der}\;c\;r_1)\cdot r_2$\\ |
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208 $\textit{der}\;c\;(r^*)$ & $\dn$ & $(\textit{der}\;c\;r)\cdot (r^*)$\\ |
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209 \end{tabular} |
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210 \end{center} |
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211 |
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212 For example given the regular expression $r = (a \cdot b) \cdot c$, the derivatives |
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213 w.r.t.~the characters $a$, $b$ and $c$ are |
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214 |
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215 \begin{center} |
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216 \begin{tabular}{lcll} |
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217 $\textit{der}\;a\;r$ & $=$ & $(\ONE \cdot b)\cdot c$ & \quad($= r'$)\\ |
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218 $\textit{der}\;b\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$\\ |
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219 $\textit{der}\;c\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$ |
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220 \end{tabular} |
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221 \end{center} |
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222 |
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223 Let $r'$ stand for the first derivative, then taking the derivatives of $r'$ |
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224 w.r.t.~the characters $a$, $b$ and $c$ gives |
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225 |
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226 \begin{center} |
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227 \begin{tabular}{lcll} |
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228 $\textit{der}\;a\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ \\ |
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229 $\textit{der}\;b\;r'$ & $=$ & $((\ZERO \cdot b) + \ONE)\cdot c$ & \quad($= r''$)\\ |
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230 $\textit{der}\;c\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ |
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231 \end{tabular} |
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232 \end{center} |
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233 |
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234 One more example: Let $r''$ stand for the second derivative above, |
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235 then taking the derivatives of $r''$ w.r.t.~the characters $a$, $b$ |
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236 and $c$ gives |
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237 |
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238 \begin{center} |
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239 \begin{tabular}{lcll} |
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240 $\textit{der}\;a\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$ \\ |
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241 $\textit{der}\;b\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$\\ |
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242 $\textit{der}\;c\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ONE$ & |
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243 (is $\textit{nullable}$) |
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244 \end{tabular} |
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245 \end{center} |
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246 |
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247 Note, the last derivative can match the empty string, that is it is \textit{nullable}.\\ |
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248 \mbox{}\hfill\mbox{[1 Mark]} |
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249 |
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250 \item[(3)] Implement the function \textit{simp}, which recursively |
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251 traverses a regular expression from the inside to the outside, and |
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252 on the way simplifies every regular expression on the left (see |
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253 below) to the regular expression on the right, except it does not |
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254 simplify inside ${}^*$-regular expressions. |
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255 |
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256 \begin{center} |
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257 \begin{tabular}{l@{\hspace{4mm}}c@{\hspace{4mm}}ll} |
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258 $r \cdot \ZERO$ & $\mapsto$ & $\ZERO$\\ |
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259 $\ZERO \cdot r$ & $\mapsto$ & $\ZERO$\\ |
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260 $r \cdot \ONE$ & $\mapsto$ & $r$\\ |
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261 $\ONE \cdot r$ & $\mapsto$ & $r$\\ |
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262 $r + \ZERO$ & $\mapsto$ & $r$\\ |
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263 $\ZERO + r$ & $\mapsto$ & $r$\\ |
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264 $r + r$ & $\mapsto$ & $r$\\ |
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265 \end{tabular} |
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266 \end{center} |
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267 |
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268 For example the regular expression |
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269 \[(r_1 + \ZERO) \cdot \ONE + ((\ONE + r_2) + r_3) \cdot (r_4 \cdot \ZERO)\] |
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270 |
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271 simplifies to just $r_1$. \textbf{Hint:} Regular expressions can be |
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272 seen as trees and there are several methods for traversing |
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273 trees. One of them corresponds to the inside-out traversal, which is |
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274 sometimes also called post-order traversal'' you traverse inside the |
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275 tree and on the way up, you apply simplification rules. |
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276 Furthermore, |
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277 remember numerical expressions from school times: there you had expressions |
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278 like $u + \ldots + (1 \cdot x) - \ldots (z + (y \cdot 0)) \ldots$ |
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279 and simplification rules that looked very similar to rules |
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280 above. You would simplify such numerical expressions by replacing |
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281 for example the $y \cdot 0$ by $0$, or $1\cdot x$ by $x$, and then |
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282 look whether more rules are applicable. If you organise the |
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283 simplification in an inside-out fashion, it is always clear which |
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284 rule should be applied next.\hfill[1 Mark] |
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285 |
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286 \item[(4)] Implement two functions: The first, called \textit{ders}, |
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287 takes a list of characters and a regular expression as arguments, and |
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288 builds the derivative w.r.t.~the list as follows: |
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289 |
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290 \begin{center} |
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291 \begin{tabular}{lcl} |
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292 $\textit{ders}\;(Nil)\;r$ & $\dn$ & $r$\\ |
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293 $\textit{ders}\;(c::cs)\;r$ & $\dn$ & |
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294 $\textit{ders}\;cs\;(\textit{simp}(\textit{der}\;c\;r))$\\ |
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295 \end{tabular} |
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296 \end{center} |
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297 |
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298 Note that this function is different from \textit{der}, which only |
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299 takes a single character. |
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300 |
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301 The second function, called \textit{matcher}, takes a string and a |
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302 regular expression as arguments. It builds first the derivatives |
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303 according to \textit{ders} and after that tests whether the resulting |
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304 derivative regular expression can match the empty string (using |
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305 \textit{nullable}). For example the \textit{matcher} will produce |
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306 true for the regular expression $(a\cdot b)\cdot c$ and the string |
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307 $abc$, but false if you give it the string $ab$. \hfill[1 Mark] |
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308 |
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309 \item[(5)] Implement a function, called \textit{size}, by recursion |
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310 over regular expressions. If a regular expression is seen as a tree, |
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311 then \textit{size} should return the number of nodes in such a |
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312 tree. Therefore this function is defined as follows: |
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313 |
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314 \begin{center} |
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315 \begin{tabular}{lcl} |
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316 $\textit{size}(\ZERO)$ & $\dn$ & $1$\\ |
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317 $\textit{size}(\ONE)$ & $\dn$ & $1$\\ |
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318 $\textit{size}(c)$ & $\dn$ & $1$\\ |
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319 $\textit{size}(r_1 + r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\ |
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320 $\textit{size}(r_1 \cdot r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\ |
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321 $\textit{size}(r^*)$ & $\dn$ & $1 + \textit{size}(r)$\\ |
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322 \end{tabular} |
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323 \end{center} |
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324 |
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325 You can use \textit{size} in order to test how much the `evil' regular |
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326 expression $(a^*)^* \cdot b$ grows when taking successive derivatives |
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327 according the letter $a$ without simplification and then compare it to |
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328 taking the derivative, but simplify the result. The sizes |
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329 are given in \texttt{re.scala}. \hfill[1 Mark] |
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330 |
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331 \item[(6)] You do not have to implement anything specific under this |
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332 task. The purpose is that you will be marked for some ``power'' |
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333 test cases. For example can your matcher decide withing 30 seconds |
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334 whether the regular expression $(a^*)^*\cdot b$ matches strings of the |
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335 form $aaa\ldots{}aaaa$, for say 1 Million $a$'s. And does simplification |
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336 simplify the regular expression |
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337 |
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338 \[ |
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339 \texttt{SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE)} |
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340 \] |
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341 |
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342 \noindent correctly to just \texttt{ONE}, where \texttt{SEQ} is nested |
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343 50 or more times.\\ |
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344 \mbox{}\hfill[1 Mark] |
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345 \end{itemize} |
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346 |
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347 \subsection*{Background} |
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348 |
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349 Although easily implementable in Scala, the idea behind the derivative |
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350 function might not so easy to be seen. To understand its purpose |
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351 better, assume a regular expression $r$ can match strings of the form |
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352 $c\!::\!cs$ (that means strings which start with a character $c$ and have |
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353 some rest, or tail, $cs$). If you take the derivative of $r$ with |
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354 respect to the character $c$, then you obtain a regular expression |
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355 that can match all the strings $cs$. In other words, the regular |
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356 expression $\textit{der}\;c\;r$ can match the same strings $c\!::\!cs$ |
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357 that can be matched by $r$, except that the $c$ is chopped off. |
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358 |
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359 Assume now $r$ can match the string $abc$. If you take the derivative |
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360 according to $a$ then you obtain a regular expression that can match |
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361 $bc$ (it is $abc$ where the $a$ has been chopped off). If you now |
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362 build the derivative $\textit{der}\;b\;(\textit{der}\;a\;r)$ you |
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363 obtain a regular expression that can match the string $c$ (it is $bc$ |
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364 where $b$ is chopped off). If you finally build the derivative of this |
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365 according $c$, that is |
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366 $\textit{der}\;c\;(\textit{der}\;b\;(\textit{der}\;a\;r))$, you obtain |
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367 a regular expression that can match the empty string. You can test |
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368 whether this is indeed the case using the function nullable, which is |
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369 what your matcher is doing. |
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370 |
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371 The purpose of the $\textit{simp}$ function is to keep the regular |
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372 expressions small. Normally the derivative function makes the regular |
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373 expression bigger (see the SEQ case and the example in (2)) and the |
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374 algorithm would be slower and slower over time. The $\textit{simp}$ |
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375 function counters this increase in size and the result is that the |
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376 algorithm is fast throughout. By the way, this algorithm is by Janusz |
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377 Brzozowski who came up with the idea of derivatives in 1964 in his PhD |
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378 thesis. |
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379 |
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380 \begin{center}\small |
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381 \url{https://en.wikipedia.org/wiki/Janusz_Brzozowski_(computer_scientist)} |
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382 \end{center} |
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383 |
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384 |
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385 If you want to see how badly the regular expression matchers do in |
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386 Java\footnote{Version 8 and below; Version 9 and above does not seem to be as |
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387 catastrophic, but still much worse than the regular expression |
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388 matcher based on derivatives.}, JavaScript and in Python with the |
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389 `evil' regular expression $(a^*)^*\cdot b$, then have a look at the |
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390 graphs below (you can try it out for yourself: have a look at the file |
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391 \texttt{catastrophic9.java}, \texttt{catastrophic.js} and |
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392 \texttt{catastrophic.py} on KEATS). Compare this with the matcher you |
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393 have implemented. How long can the string of $a$'s be in your matcher |
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394 and still stay within the 30 seconds time limit? |
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395 |
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396 \begin{center} |
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397 \begin{tabular}{@{}cc@{}} |
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398 \multicolumn{2}{c}{Graph: $(a^*)^*\cdot b$ and strings |
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399 $\underbrace{a\ldots a}_{n}$}\bigskip\\ |
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400 |
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401 \begin{tikzpicture} |
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402 \begin{axis}[ |
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403 xlabel={$n$}, |
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404 x label style={at={(1.05,0.0)}}, |
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405 ylabel={time in secs}, |
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406 y label style={at={(0.06,0.5)}}, |
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407 enlargelimits=false, |
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408 xtick={0,5,...,30}, |
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409 xmax=33, |
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410 ymax=45, |
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411 ytick={0,5,...,40}, |
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412 scaled ticks=false, |
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413 axis lines=left, |
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414 width=6cm, |
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415 height=5.5cm, |
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416 legend entries={Python, Java 8, JavaScript}, |
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417 legend pos=north west] |
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418 \addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data}; |
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419 \addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data}; |
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420 \addplot[red,mark=*, mark options={fill=white}] table {re-js.data}; |
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421 \end{axis} |
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422 \end{tikzpicture} |
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423 & |
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424 \begin{tikzpicture} |
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425 \begin{axis}[ |
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426 xlabel={$n$}, |
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427 x label style={at={(1.05,0.0)}}, |
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428 ylabel={time in secs}, |
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429 y label style={at={(0.06,0.5)}}, |
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430 %enlargelimits=false, |
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431 %xtick={0,5000,...,30000}, |
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432 xmax=65000, |
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433 ymax=45, |
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434 ytick={0,5,...,40}, |
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435 scaled ticks=false, |
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436 axis lines=left, |
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437 width=6cm, |
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438 height=5.5cm, |
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439 legend entries={Java 9}, |
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440 legend pos=north west] |
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441 \addplot[cyan,mark=*, mark options={fill=white}] table {re-java9.data}; |
|
442 \end{axis} |
|
443 \end{tikzpicture} |
|
444 \end{tabular} |
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445 \end{center} |
|
446 \newpage |
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447 |
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448 \subsection*{Part 2 (4 Marks)} |
|
449 |
|
450 Coming from Java or C++, you might think Scala is a quite esoteric |
|
451 programming language. But remember, some serious companies have built |
38 programming language. But remember, some serious companies have built |
452 their business on |
39 their business on |
453 Scala.\footnote{\url{https://en.wikipedia.org/wiki/Scala_(programming_language)\#Companies}} |
40 Scala.\footnote{\url{https://en.wikipedia.org/wiki/Scala_(programming_language)\#Companies}} |
454 And there are far, far more esoteric languages out there. One is |
41 And there are far, far more esoteric languages out there. One is |
455 called \emph{brainf***}. You are asked in this part to implement an |
42 called \emph{brainf***}. You are asked in this part to implement an |
456 interpreter for this language. |
43 interpreter and compiler for this language. |
457 |
44 |
458 Urban M\"uller developed brainf*** in 1993. A close relative of this |
45 Urban M\"uller developed brainf*** in 1993. A close relative of this |
459 language was already introduced in 1964 by Corado B\"ohm, an Italian |
46 language was already introduced in 1964 by Corado B\"ohm, an Italian |
460 computer pioneer, who unfortunately died a few months ago. The main |
47 computer pioneer. The main feature of brainf*** is its minimalistic |
461 feature of brainf*** is its minimalistic set of instructions---just 8 |
48 set of instructions---just 8 instructions in total and all of which |
462 instructions in total and all of which are single characters. Despite |
49 are single characters. Despite the minimalism, this language has been |
463 the minimalism, this language has been shown to be Turing |
50 shown to be Turing complete\ldots{}if this doesn't ring any bell with |
464 complete\ldots{}if this doesn't ring any bell with you: it roughly |
51 you: it roughly means that every algorithm we know can, in principle, |
465 means that every algorithm we know can, in principle, be implemented in |
52 be implemented in brainf***. It just takes a lot of determination and |
466 brainf***. It just takes a lot of determination and quite a lot of |
53 quite a lot of memory resources. Some relatively sophisticated sample |
467 memory resources. Some relatively sophisticated sample programs in |
54 programs in brainf*** are given in the file \texttt{bf.scala}, including |
468 brainf*** are given in the file \texttt{bf.scala}.\bigskip |
55 a brainf*** program for the Sierpinski triangle and Mandelbot set.\bigskip |
469 |
56 |
470 \noindent |
57 \noindent |
471 As mentioned above, brainf*** has 8 single-character commands, namely |
58 As mentioned above, brainf*** has 8 single-character commands, namely |
472 \texttt{'>'}, \texttt{'<'}, \texttt{'+'}, \texttt{'-'}, \texttt{'.'}, |
59 \texttt{'>'}, \texttt{'<'}, \texttt{'+'}, \texttt{'-'}, \texttt{'.'}, |
473 \texttt{','}, \texttt{'['} and \texttt{']'}. Every other character is |
60 \texttt{','}, \texttt{'['} and \texttt{']'}. Every other character is |