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1 \documentclass{article} |
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2 \usepackage{../style} |
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3 \usepackage{../langs} |
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4 \usepackage{disclaimer} |
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5 \usepackage{tikz} |
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6 \usepackage{pgf} |
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7 \usepackage{pgfplots} |
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8 \usepackage{stackengine} |
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9 %% \usepackage{accents} |
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10 \newcommand\barbelow[1]{\stackunder[1.2pt]{#1}{\raisebox{-4mm}{\boldmath$\uparrow$}}} |
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11 |
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12 \begin{filecontents}{re-python2.data} |
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28 \end{filecontents} |
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29 |
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30 \begin{filecontents}{re-java.data} |
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74 \end{filecontents} |
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75 |
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76 |
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77 \begin{document} |
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78 |
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79 % BF IDE |
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80 % https://www.microsoft.com/en-us/p/brainf-ck/9nblgggzhvq5 |
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81 |
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82 \section*{Coursework 8 (Regular Expressions and Brainf***)} |
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83 |
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84 This coursework is worth 10\%. It is about regular expressions, |
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85 pattern matching and an interpreter. The first part is due on 30 |
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86 November at 11pm; the second, more advanced part, is due on 21 |
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87 December at 11pm. In the first part, you are asked to implement a |
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88 regular expression matcher based on derivatives of regular |
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89 expressions. The reason is that regular expression matching in Java |
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90 and Python can sometimes be extremely slow. The advanced part is about |
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91 an interpreter for a very simple programming language.\bigskip |
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92 |
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93 \IMPORTANT{} |
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94 |
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95 \noindent |
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96 Also note that the running time of each part will be restricted to a |
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97 maximum of 360 seconds on my laptop. |
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98 |
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99 \DISCLAIMER{} |
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100 |
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101 |
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102 \subsection*{Part 1 (6 Marks)} |
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103 |
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104 The task is to implement a regular expression matcher that is based on |
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105 derivatives of regular expressions. Most of the functions are defined by |
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106 recursion over regular expressions and can be elegantly implemented |
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107 using Scala's pattern-matching. The implementation should deal with the |
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108 following regular expressions, which have been predefined in the file |
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109 \texttt{re.scala}: |
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110 |
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111 \begin{center} |
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112 \begin{tabular}{lcll} |
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113 $r$ & $::=$ & $\ZERO$ & cannot match anything\\ |
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114 & $|$ & $\ONE$ & can only match the empty string\\ |
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115 & $|$ & $c$ & can match a single character (in this case $c$)\\ |
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116 & $|$ & $r_1 + r_2$ & can match a string either with $r_1$ or with $r_2$\\ |
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117 & $|$ & $r_1\cdot r_2$ & can match the first part of a string with $r_1$ and\\ |
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118 & & & then the second part with $r_2$\\ |
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119 & $|$ & $r^*$ & can match zero or more times $r$\\ |
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120 \end{tabular} |
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121 \end{center} |
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122 |
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123 \noindent |
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124 Why? Knowing how to match regular expressions and strings will let you |
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125 solve a lot of problems that vex other humans. Regular expressions are |
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126 one of the fastest and simplest ways to match patterns in text, and |
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127 are endlessly useful for searching, editing and analysing data in all |
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128 sorts of places (for example analysing network traffic in order to |
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129 detect security breaches). However, you need to be fast, otherwise you |
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130 will stumble over problems such as recently reported at |
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131 |
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132 {\small |
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133 \begin{itemize} |
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134 \item[$\bullet$] \url{http://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016} |
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135 \item[$\bullet$] \url{https://vimeo.com/112065252} |
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136 \item[$\bullet$] \url{http://davidvgalbraith.com/how-i-fixed-atom/} |
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137 \end{itemize}} |
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138 |
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139 \subsubsection*{Tasks (file re.scala)} |
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140 |
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141 The file \texttt{re.scala} has already a definition for regular |
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142 expressions and also defines some handy shorthand notation for |
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143 regular expressions. The notation in this document matches up |
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144 with the code in the file as follows: |
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145 |
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146 \begin{center} |
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147 \begin{tabular}{rcl@{\hspace{10mm}}l} |
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148 & & code: & shorthand:\smallskip \\ |
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149 $\ZERO$ & $\mapsto$ & \texttt{ZERO}\\ |
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150 $\ONE$ & $\mapsto$ & \texttt{ONE}\\ |
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151 $c$ & $\mapsto$ & \texttt{CHAR(c)}\\ |
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152 $r_1 + r_2$ & $\mapsto$ & \texttt{ALT(r1, r2)} & \texttt{r1 | r2}\\ |
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153 $r_1 \cdot r_2$ & $\mapsto$ & \texttt{SEQ(r1, r2)} & \texttt{r1 $\sim$ r2}\\ |
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154 $r^*$ & $\mapsto$ & \texttt{STAR(r)} & \texttt{r.\%} |
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155 \end{tabular} |
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156 \end{center} |
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157 |
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158 |
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159 \begin{itemize} |
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160 \item[(1a)] Implement a function, called \textit{nullable}, by |
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161 recursion over regular expressions. This function tests whether a |
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162 regular expression can match the empty string. This means given a |
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163 regular expression it either returns true or false. The function |
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164 \textit{nullable} |
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165 is defined as follows: |
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166 |
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167 \begin{center} |
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168 \begin{tabular}{lcl} |
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169 $\textit{nullable}(\ZERO)$ & $\dn$ & $\textit{false}$\\ |
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170 $\textit{nullable}(\ONE)$ & $\dn$ & $\textit{true}$\\ |
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171 $\textit{nullable}(c)$ & $\dn$ & $\textit{false}$\\ |
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172 $\textit{nullable}(r_1 + r_2)$ & $\dn$ & $\textit{nullable}(r_1) \vee \textit{nullable}(r_2)$\\ |
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173 $\textit{nullable}(r_1 \cdot r_2)$ & $\dn$ & $\textit{nullable}(r_1) \wedge \textit{nullable}(r_2)$\\ |
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174 $\textit{nullable}(r^*)$ & $\dn$ & $\textit{true}$\\ |
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175 \end{tabular} |
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176 \end{center}~\hfill[1 Mark] |
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177 |
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178 \item[(1b)] Implement a function, called \textit{der}, by recursion over |
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179 regular expressions. It takes a character and a regular expression |
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180 as arguments and calculates the derivative regular expression according |
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181 to the rules: |
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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 $\textit{der}\;c\;(\ZERO)$ & $\dn$ & $\ZERO$\\ |
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186 $\textit{der}\;c\;(\ONE)$ & $\dn$ & $\ZERO$\\ |
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187 $\textit{der}\;c\;(d)$ & $\dn$ & $\textit{if}\; c = d\;\textit{then} \;\ONE \; \textit{else} \;\ZERO$\\ |
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188 $\textit{der}\;c\;(r_1 + r_2)$ & $\dn$ & $(\textit{der}\;c\;r_1) + (\textit{der}\;c\;r_2)$\\ |
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189 $\textit{der}\;c\;(r_1 \cdot r_2)$ & $\dn$ & $\textit{if}\;\textit{nullable}(r_1)$\\ |
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190 & & $\textit{then}\;((\textit{der}\;c\;r_1)\cdot r_2) + (\textit{der}\;c\;r_2)$\\ |
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191 & & $\textit{else}\;(\textit{der}\;c\;r_1)\cdot r_2$\\ |
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192 $\textit{der}\;c\;(r^*)$ & $\dn$ & $(\textit{der}\;c\;r)\cdot (r^*)$\\ |
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193 \end{tabular} |
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194 \end{center} |
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195 |
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196 For example given the regular expression $r = (a \cdot b) \cdot c$, the derivatives |
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197 w.r.t.~the characters $a$, $b$ and $c$ are |
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198 |
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199 \begin{center} |
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200 \begin{tabular}{lcll} |
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201 $\textit{der}\;a\;r$ & $=$ & $(\ONE \cdot b)\cdot c$ & ($= r'$)\\ |
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202 $\textit{der}\;b\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$\\ |
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203 $\textit{der}\;c\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$ |
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204 \end{tabular} |
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205 \end{center} |
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206 |
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207 Let $r'$ stand for the first derivative, then taking the derivatives of $r'$ |
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208 w.r.t.~the characters $a$, $b$ and $c$ gives |
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209 |
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210 \begin{center} |
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211 \begin{tabular}{lcll} |
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212 $\textit{der}\;a\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ \\ |
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213 $\textit{der}\;b\;r'$ & $=$ & $((\ZERO \cdot b) + \ONE)\cdot c$ & ($= r''$)\\ |
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214 $\textit{der}\;c\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ |
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215 \end{tabular} |
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216 \end{center} |
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217 |
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218 One more example: Let $r''$ stand for the second derivative above, |
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219 then taking the derivatives of $r''$ w.r.t.~the characters $a$, $b$ |
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220 and $c$ gives |
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221 |
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222 \begin{center} |
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223 \begin{tabular}{lcll} |
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224 $\textit{der}\;a\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$ \\ |
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225 $\textit{der}\;b\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$\\ |
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226 $\textit{der}\;c\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ONE$ & |
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227 (is $\textit{nullable}$) |
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228 \end{tabular} |
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229 \end{center} |
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230 |
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231 Note, the last derivative can match the empty string, that is it is \textit{nullable}.\\ |
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232 \mbox{}\hfill\mbox{[1 Mark]} |
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233 |
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234 \item[(1c)] Implement the function \textit{simp}, which recursively |
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235 traverses a regular expression from the inside to the outside, and |
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236 on the way simplifies every regular expression on the left (see |
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237 below) to the regular expression on the right, except it does not |
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238 simplify inside ${}^*$-regular expressions. |
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239 |
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240 \begin{center} |
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241 \begin{tabular}{l@{\hspace{4mm}}c@{\hspace{4mm}}ll} |
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242 $r \cdot \ZERO$ & $\mapsto$ & $\ZERO$\\ |
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243 $\ZERO \cdot r$ & $\mapsto$ & $\ZERO$\\ |
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244 $r \cdot \ONE$ & $\mapsto$ & $r$\\ |
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245 $\ONE \cdot r$ & $\mapsto$ & $r$\\ |
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246 $r + \ZERO$ & $\mapsto$ & $r$\\ |
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247 $\ZERO + r$ & $\mapsto$ & $r$\\ |
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248 $r + r$ & $\mapsto$ & $r$\\ |
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249 \end{tabular} |
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250 \end{center} |
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251 |
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252 For example the regular expression |
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253 \[(r_1 + \ZERO) \cdot \ONE + ((\ONE + r_2) + r_3) \cdot (r_4 \cdot \ZERO)\] |
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254 |
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255 simplifies to just $r_1$. \textbf{Hint:} Regular expressions can be |
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256 seen as trees and there are several methods for traversing |
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257 trees. One of them corresponds to the inside-out traversal, which is |
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258 sometimes also called post-order traversal. Furthermore, |
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259 remember numerical expressions from school times: there you had expressions |
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260 like $u + \ldots + (1 \cdot x) - \ldots (z + (y \cdot 0)) \ldots$ |
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261 and simplification rules that looked very similar to rules |
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262 above. You would simplify such numerical expressions by replacing |
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263 for example the $y \cdot 0$ by $0$, or $1\cdot x$ by $x$, and then |
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264 look whether more rules are applicable. If you organise the |
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265 simplification in an inside-out fashion, it is always clear which |
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266 rule should be applied next.\hfill[2 Marks] |
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267 |
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268 \item[(1d)] Implement two functions: The first, called \textit{ders}, |
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269 takes a list of characters and a regular expression as arguments, and |
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270 builds the derivative w.r.t.~the list as follows: |
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271 |
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272 \begin{center} |
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273 \begin{tabular}{lcl} |
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274 $\textit{ders}\;(Nil)\;r$ & $\dn$ & $r$\\ |
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275 $\textit{ders}\;(c::cs)\;r$ & $\dn$ & |
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276 $\textit{ders}\;cs\;(\textit{simp}(\textit{der}\;c\;r))$\\ |
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277 \end{tabular} |
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278 \end{center} |
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279 |
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280 Note that this function is different from \textit{der}, which only |
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281 takes a single character. |
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282 |
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283 The second function, called \textit{matcher}, takes a string and a |
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284 regular expression as arguments. It builds first the derivatives |
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285 according to \textit{ders} and after that tests whether the resulting |
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286 derivative regular expression can match the empty string (using |
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287 \textit{nullable}). For example the \textit{matcher} will produce |
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288 true for the regular expression $(a\cdot b)\cdot c$ and the string |
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289 $abc$, but false if you give it the string $ab$. \hfill[1 Mark] |
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290 |
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291 \item[(1e)] Implement a function, called \textit{size}, by recursion |
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292 over regular expressions. If a regular expression is seen as a tree, |
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293 then \textit{size} should return the number of nodes in such a |
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294 tree. Therefore this function is defined as follows: |
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295 |
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296 \begin{center} |
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297 \begin{tabular}{lcl} |
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298 $\textit{size}(\ZERO)$ & $\dn$ & $1$\\ |
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299 $\textit{size}(\ONE)$ & $\dn$ & $1$\\ |
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300 $\textit{size}(c)$ & $\dn$ & $1$\\ |
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301 $\textit{size}(r_1 + r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\ |
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302 $\textit{size}(r_1 \cdot r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\ |
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303 $\textit{size}(r^*)$ & $\dn$ & $1 + \textit{size}(r)$\\ |
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304 \end{tabular} |
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305 \end{center} |
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306 |
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307 You can use \textit{size} in order to test how much the `evil' regular |
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308 expression $(a^*)^* \cdot b$ grows when taking successive derivatives |
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309 according the letter $a$ without simplification and then compare it to |
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310 taking the derivative, but simplify the result. The sizes |
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311 are given in \texttt{re.scala}. \hfill[1 Mark] |
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312 \end{itemize} |
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313 |
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314 \subsection*{Background} |
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315 |
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316 Although easily implementable in Scala, the idea behind the derivative |
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317 function might not so easy to be seen. To understand its purpose |
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318 better, assume a regular expression $r$ can match strings of the form |
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319 $c\!::\!cs$ (that means strings which start with a character $c$ and have |
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320 some rest, or tail, $cs$). If you take the derivative of $r$ with |
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321 respect to the character $c$, then you obtain a regular expression |
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322 that can match all the strings $cs$. In other words, the regular |
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323 expression $\textit{der}\;c\;r$ can match the same strings $c\!::\!cs$ |
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324 that can be matched by $r$, except that the $c$ is chopped off. |
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325 |
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326 Assume now $r$ can match the string $abc$. If you take the derivative |
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327 according to $a$ then you obtain a regular expression that can match |
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328 $bc$ (it is $abc$ where the $a$ has been chopped off). If you now |
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329 build the derivative $\textit{der}\;b\;(\textit{der}\;a\;r)$ you |
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330 obtain a regular expression that can match the string $c$ (it is $bc$ |
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331 where $b$ is chopped off). If you finally build the derivative of this |
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332 according $c$, that is |
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333 $\textit{der}\;c\;(\textit{der}\;b\;(\textit{der}\;a\;r))$, you obtain |
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334 a regular expression that can match the empty string. You can test |
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335 whether this is indeed the case using the function nullable, which is |
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336 what your matcher is doing. |
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337 |
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338 The purpose of the $\textit{simp}$ function is to keep the regular |
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339 expressions small. Normally the derivative function makes the regular |
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340 expression bigger (see the SEQ case and the example in (1b)) and the |
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341 algorithm would be slower and slower over time. The $\textit{simp}$ |
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342 function counters this increase in size and the result is that the |
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343 algorithm is fast throughout. By the way, this algorithm is by Janusz |
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344 Brzozowski who came up with the idea of derivatives in 1964 in his PhD |
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345 thesis. |
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346 |
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347 \begin{center}\small |
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348 \url{https://en.wikipedia.org/wiki/Janusz_Brzozowski_(computer_scientist)} |
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349 \end{center} |
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350 |
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351 |
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352 If you want to see how badly the regular expression matchers do in |
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353 Java\footnote{Version 8 and below; Version 9 does not seem to be as |
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354 catastrophic, but still worse than the regular expression matcher |
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355 based on derivatives.} and in Python with the `evil' regular |
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356 expression $(a^*)^*\cdot b$, then have a look at the graphs below (you |
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357 can try it out for yourself: have a look at the file |
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358 \texttt{catastrophic.java} and \texttt{catastrophic.py} on |
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359 KEATS). Compare this with the matcher you have implemented. How long |
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360 can the string of $a$'s be in your matcher and still stay within the |
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361 30 seconds time limit? |
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362 |
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363 \begin{center} |
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364 \begin{tabular}{@{}cc@{}} |
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365 \multicolumn{2}{c}{Graph: $(a^*)^*\cdot b$ and strings |
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366 $\underbrace{a\ldots a}_{n}$}\bigskip\\ |
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367 |
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368 \begin{tikzpicture} |
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369 \begin{axis}[ |
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370 xlabel={$n$}, |
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371 x label style={at={(1.05,0.0)}}, |
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372 ylabel={time in secs}, |
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373 y label style={at={(0.06,0.5)}}, |
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374 enlargelimits=false, |
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375 xtick={0,5,...,30}, |
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376 xmax=33, |
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377 ymax=45, |
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378 ytick={0,5,...,40}, |
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379 scaled ticks=false, |
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380 axis lines=left, |
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381 width=6cm, |
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382 height=5.5cm, |
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383 legend entries={Python, Java 8}, |
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384 legend pos=north west] |
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385 \addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data}; |
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386 \addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data}; |
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387 \end{axis} |
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388 \end{tikzpicture} |
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389 & |
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390 \begin{tikzpicture} |
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391 \begin{axis}[ |
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392 xlabel={$n$}, |
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393 x label style={at={(1.05,0.0)}}, |
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394 ylabel={time in secs}, |
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395 y label style={at={(0.06,0.5)}}, |
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396 %enlargelimits=false, |
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397 %xtick={0,5000,...,30000}, |
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398 xmax=65000, |
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399 ymax=45, |
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400 ytick={0,5,...,40}, |
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401 scaled ticks=false, |
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402 axis lines=left, |
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403 width=6cm, |
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404 height=5.5cm, |
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405 legend entries={Java 9}, |
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406 legend pos=north west] |
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407 \addplot[cyan,mark=*, mark options={fill=white}] table {re-java9.data}; |
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408 \end{axis} |
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409 \end{tikzpicture} |
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410 \end{tabular} |
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411 \end{center} |
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412 \newpage |
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413 |
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414 \subsection*{Part 2 (4 Marks)} |
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415 |
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416 Coming from Java or C++, you might think Scala is a quite esoteric |
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417 programming language. But remember, some serious companies have built |
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418 their business on |
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419 Scala.\footnote{\url{https://en.wikipedia.org/wiki/Scala_(programming_language)\#Companies}} |
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420 And there are far, far more esoteric languages out there. One is |
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421 called \emph{brainf***}. You are asked in this part to implement an |
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422 interpreter for this language. |
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423 |
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424 Urban M\"uller developed brainf*** in 1993. A close relative of this |
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425 language was already introduced in 1964 by Corado B\"ohm, an Italian |
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426 computer pioneer, who unfortunately died a few months ago. The main |
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427 feature of brainf*** is its minimalistic set of instructions---just 8 |
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428 instructions in total and all of which are single characters. Despite |
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429 the minimalism, this language has been shown to be Turing |
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430 complete\ldots{}if this doesn't ring any bell with you: it roughly |
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431 means that every algorithm we know can, in principle, be implemented in |
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432 brainf***. It just takes a lot of determination and quite a lot of |
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433 memory resources. Some relatively sophisticated sample programs in |
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434 brainf*** are given in the file \texttt{bf.scala}.\bigskip |
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435 |
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436 \noindent |
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437 As mentioned above, brainf*** has 8 single-character commands, namely |
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438 \texttt{'>'}, \texttt{'<'}, \texttt{'+'}, \texttt{'-'}, \texttt{'.'}, |
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439 \texttt{','}, \texttt{'['} and \texttt{']'}. Every other character is |
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440 considered a comment. Brainf*** operates on memory cells containing |
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441 integers. For this it uses a single memory pointer that points at each |
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442 stage to one memory cell. This pointer can be moved forward by one |
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443 memory cell by using the command \texttt{'>'}, and backward by using |
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444 \texttt{'<'}. The commands \texttt{'+'} and \texttt{'-'} increase, |
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445 respectively decrease, by 1 the content of the memory cell to which |
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446 the memory pointer currently points to. The commands for input/output |
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447 are \texttt{','} and \texttt{'.'}. Output works by reading the content |
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448 of the memory cell to which the memory pointer points to and printing |
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449 it out as an ASCII character. Input works the other way, taking some |
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450 user input and storing it in the cell to which the memory pointer |
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451 points to. The commands \texttt{'['} and \texttt{']'} are looping |
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452 constructs. Everything in between \texttt{'['} and \texttt{']'} is |
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453 repeated until a counter (memory cell) reaches zero. A typical |
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454 program in brainf*** looks as follows: |
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455 |
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456 \begin{center} |
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457 \begin{verbatim} |
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458 ++++++++[>++++[>++>+++>+++>+<<<<-]>+>+>->>+[<]<-]>>.>---.+++++++ |
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459 ..+++.>>.<-.<.+++.------.--------.>>+.>++. |
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460 \end{verbatim} |
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461 \end{center} |
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462 |
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463 \noindent |
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464 This one prints out Hello World\ldots{}obviously. |
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465 |
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466 \subsubsection*{Tasks (file bf.scala)} |
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467 |
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468 \begin{itemize} |
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469 \item[(2a)] Brainf*** memory is represented by a \texttt{Map} from |
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470 integers to integers. The empty memory is represented by |
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471 \texttt{Map()}, that is nothing is stored in the |
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472 memory. \texttt{Map(0 -> 1, 2 -> 3)} clearly stores \texttt{1} at |
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473 memory location \texttt{0}; at \texttt{2} it stores \texttt{3}. The |
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474 convention is that if we query the memory at a location that is |
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475 \emph{not} defined in the \texttt{Map}, we return \texttt{0}. Write |
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476 a function, \texttt{sread}, that takes a memory (a \texttt{Map}) and |
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477 a memory pointer (an \texttt{Int}) as argument, and safely reads the |
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478 corresponding memory location. If the \texttt{Map} is not defined at |
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479 the memory pointer, \texttt{sread} returns \texttt{0}. |
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480 |
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481 Write another function \texttt{write}, which takes a memory, a |
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482 memory pointer and an integer value as argument and updates the |
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483 \texttt{Map} with the value at the given memory location. As usual |
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484 the \texttt{Map} is not updated `in-place' but a new map is created |
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485 with the same data, except the value is stored at the given memory |
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486 pointer.\hfill[1 Mark] |
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487 |
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488 \item[(2b)] Write two functions, \texttt{jumpRight} and |
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489 \texttt{jumpLeft} that are needed to implement the loop constructs |
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490 of brainf***. They take a program (a \texttt{String}) and a program |
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491 counter (an \texttt{Int}) as argument and move right (respectively |
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492 left) in the string in order to find the \textbf{matching} |
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493 opening/closing bracket. For example, given the following program |
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494 with the program counter indicated by an arrow: |
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495 |
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496 \begin{center} |
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497 \texttt{--[\barbelow{.}.+>--],>,++} |
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498 \end{center} |
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499 |
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500 then the matching closing bracket is in 9th position (counting from 0) and |
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501 \texttt{jumpRight} is supposed to return the position just after this |
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502 |
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503 \begin{center} |
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504 \texttt{--[..+>--]\barbelow{,}>,++} |
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505 \end{center} |
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506 |
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507 meaning it jumps to after the loop. Similarly, if you are in 8th position |
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508 then \texttt{jumpLeft} is supposed to jump to just after the opening |
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509 bracket (that is jumping to the beginning of the loop): |
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510 |
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511 \begin{center} |
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512 \texttt{--[..+>-\barbelow{-}],>,++} |
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513 \qquad$\stackrel{\texttt{jumpLeft}}{\longrightarrow}$\qquad |
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514 \texttt{--[\barbelow{.}.+>--],>,++} |
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515 \end{center} |
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516 |
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517 Unfortunately we have to take into account that there might be |
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518 other opening and closing brackets on the `way' to find the |
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519 matching bracket. For example in the brainf*** program |
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520 |
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521 \begin{center} |
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522 \texttt{--[\barbelow{.}.[+>]--],>,++} |
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523 \end{center} |
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524 |
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525 we do not want to return the index for the \texttt{'-'} in the 9th |
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526 position, but the program counter for \texttt{','} in 12th |
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527 position. The easiest to find out whether a bracket is matched is by |
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528 using levels (which are the third argument in \texttt{jumpLeft} and |
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529 \texttt{jumpLeft}). In case of \texttt{jumpRight} you increase the |
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530 level by one whenever you find an opening bracket and decrease by |
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531 one for a closing bracket. Then in \texttt{jumpRight} you are looking |
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532 for the closing bracket on level \texttt{0}. For \texttt{jumpLeft} you |
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533 do the opposite. In this way you can find \textbf{matching} brackets |
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534 in strings such as |
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535 |
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536 \begin{center} |
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537 \texttt{--[\barbelow{.}.[[-]+>[.]]--],>,++} |
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538 \end{center} |
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539 |
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540 for which \texttt{jumpRight} should produce the position: |
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541 |
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542 \begin{center} |
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543 \texttt{--[..[[-]+>[.]]--]\barbelow{,}>,++} |
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544 \end{center} |
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545 |
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546 It is also possible that the position returned by \texttt{jumpRight} or |
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547 \texttt{jumpLeft} is outside the string in cases where there are |
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548 no matching brackets. For example |
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549 |
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550 \begin{center} |
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551 \texttt{--[\barbelow{.}.[[-]+>[.]]--,>,++} |
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552 \qquad$\stackrel{\texttt{jumpRight}}{\longrightarrow}$\qquad |
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553 \texttt{--[..[[-]+>[.]]-->,++\barbelow{\;\phantom{+}}} |
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554 \end{center} |
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555 \hfill[1 Mark] |
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556 |
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557 |
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558 \item[(2c)] Write a recursive function \texttt{run} that executes a |
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559 brainf*** program. It takes a program, a program counter, a memory |
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560 pointer and a memory as arguments. If the program counter is outside |
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561 the program string, the execution stops and \texttt{run} returns the |
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562 memory. If the program counter is inside the string, it reads the |
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563 corresponding character and updates the program counter \texttt{pc}, |
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564 memory pointer \texttt{mp} and memory \texttt{mem} according to the |
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565 rules shown in Figure~\ref{comms}. It then calls recursively |
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566 \texttt{run} with the updated data. |
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567 |
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568 Write another function \texttt{start} that calls \texttt{run} with a |
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569 given brainfu** program and memory, and the program counter and memory pointer |
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570 set to~$0$. Like \texttt{run} it returns the memory after the execution |
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571 of the program finishes. You can test your brainf**k interpreter with the |
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572 Sierpinski triangle or the Hello world programs or have a look at |
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573 |
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574 \begin{center} |
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575 \url{https://esolangs.org/wiki/Brainfuck} |
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576 \end{center}\hfill[2 Marks] |
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577 |
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578 \begin{figure}[p] |
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579 \begin{center} |
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580 \begin{tabular}{|@{}p{0.8cm}|l|} |
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581 \hline |
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582 \hfill\texttt{'>'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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583 $\bullet$ & $\texttt{pc} + 1$\\ |
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584 $\bullet$ & $\texttt{mp} + 1$\\ |
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585 $\bullet$ & \texttt{mem} unchanged |
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586 \end{tabular}\\\hline |
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587 \hfill\texttt{'<'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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588 $\bullet$ & $\texttt{pc} + 1$\\ |
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589 $\bullet$ & $\texttt{mp} - 1$\\ |
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590 $\bullet$ & \texttt{mem} unchanged |
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591 \end{tabular}\\\hline |
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592 \hfill\texttt{'+'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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593 $\bullet$ & $\texttt{pc} + 1$\\ |
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594 $\bullet$ & $\texttt{mp}$ unchanged\\ |
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595 $\bullet$ & \texttt{mem} updated with \texttt{mp -> mem(mp) + 1}\\ |
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596 \end{tabular}\\\hline |
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597 \hfill\texttt{'-'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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598 $\bullet$ & $\texttt{pc} + 1$\\ |
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599 $\bullet$ & $\texttt{mp}$ unchanged\\ |
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600 $\bullet$ & \texttt{mem} updated with \texttt{mp -> mem(mp) - 1}\\ |
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601 \end{tabular}\\\hline |
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602 \hfill\texttt{'.'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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603 $\bullet$ & $\texttt{pc} + 1$\\ |
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604 $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\ |
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605 $\bullet$ & print out \,\texttt{mem(mp)} as a character\\ |
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606 \end{tabular}\\\hline |
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607 \hfill\texttt{','} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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608 $\bullet$ & $\texttt{pc} + 1$\\ |
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609 $\bullet$ & $\texttt{mp}$ unchanged\\ |
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610 $\bullet$ & \texttt{mem} updated with \texttt{mp -> \textrm{input}}\\ |
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611 \multicolumn{2}{@{}l}{the input is given by \texttt{Console.in.read().toByte}} |
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612 \end{tabular}\\\hline |
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613 \hfill\texttt{'['} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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614 \multicolumn{2}{@{}l}{if \texttt{mem(mp) == 0} then}\\ |
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615 $\bullet$ & $\texttt{pc = jumpRight(prog, pc + 1, 0)}$\\ |
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616 $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\medskip\\ |
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617 \multicolumn{2}{@{}l}{otherwise if \texttt{mem(mp) != 0} then}\\ |
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618 $\bullet$ & $\texttt{pc} + 1$\\ |
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619 $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\ |
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620 \end{tabular} |
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621 \\\hline |
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622 \hfill\texttt{']'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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623 \multicolumn{2}{@{}l}{if \texttt{mem(mp) != 0} then}\\ |
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624 $\bullet$ & $\texttt{pc = jumpLeft(prog, pc - 1, 0)}$\\ |
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625 $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\medskip\\ |
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626 \multicolumn{2}{@{}l}{otherwise if \texttt{mem(mp) == 0} then}\\ |
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627 $\bullet$ & $\texttt{pc} + 1$\\ |
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628 $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\ |
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629 \end{tabular}\\\hline |
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630 any other char & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} |
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631 $\bullet$ & $\texttt{pc} + 1$\\ |
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632 $\bullet$ & \texttt{mp} and \texttt{mem} unchanged |
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633 \end{tabular}\\ |
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634 \hline |
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635 \end{tabular} |
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636 \end{center} |
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637 \caption{The rules for how commands in the brainf*** language update the program counter \texttt{pc}, |
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638 memory pointer \texttt{mp} and memory \texttt{mem}.\label{comms}} |
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639 \end{figure} |
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640 \end{itemize}\bigskip |
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641 |
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642 |
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643 |
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644 |
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645 \end{document} |
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646 |
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647 |
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648 %%% Local Variables: |
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649 %%% mode: latex |
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650 %%% TeX-master: t |
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651 %%% End: |