diff -r 20f02c5ff53f -r a2c4c6bf319d cws/cw04-new.tex --- a/cws/cw04-new.tex Fri Apr 26 17:29:30 2024 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,651 +0,0 @@ -\documentclass{article} -\usepackage{../style} -\usepackage{../langs} -\usepackage{disclaimer} -\usepackage{tikz} -\usepackage{pgf} -\usepackage{pgfplots} -\usepackage{stackengine} -%% \usepackage{accents} -\newcommand\barbelow[1]{\stackunder[1.2pt]{#1}{\raisebox{-4mm}{\boldmath$\uparrow$}}} - -\begin{filecontents}{re-python2.data} -1 0.033 -5 0.036 -10 0.034 -15 0.036 -18 0.059 -19 0.084 -20 0.141 -21 0.248 -22 0.485 -23 0.878 -24 1.71 -25 3.40 -26 7.08 -27 14.12 -28 26.69 -\end{filecontents} - -\begin{filecontents}{re-java.data} -5 0.00298 -10 0.00418 -15 0.00996 -16 0.01710 -17 0.03492 -18 0.03303 -19 0.05084 -20 0.10177 -21 0.19960 -22 0.41159 -23 0.82234 -24 1.70251 -25 3.36112 -26 6.63998 -27 13.35120 -28 29.81185 -\end{filecontents} - -\begin{filecontents}{re-java9.data} -1000 0.01410 -2000 0.04882 -3000 0.10609 -4000 0.17456 -5000 0.27530 -6000 0.41116 -7000 0.53741 -8000 0.70261 -9000 0.93981 -10000 0.97419 -11000 1.28697 -12000 1.51387 -14000 2.07079 -16000 2.69846 -20000 4.41823 -24000 6.46077 -26000 7.64373 -30000 9.99446 -34000 12.966885 -38000 16.281621 -42000 19.180228 -46000 21.984721 -50000 26.950203 -60000 43.0327746 -\end{filecontents} - - -\begin{document} - -% BF IDE -% https://www.microsoft.com/en-us/p/brainf-ck/9nblgggzhvq5 - -\section*{Coursework 8 (Regular Expressions and Brainf***)} - -This coursework is worth 10\%. It is about regular expressions, -pattern matching and an interpreter. The first part is due on 30 -November at 11pm; the second, more advanced part, is due on 21 -December at 11pm. In the first part, you are asked to implement a -regular expression matcher based on derivatives of regular -expressions. The reason is that regular expression matching in Java -and Python can sometimes be extremely slow. The advanced part is about -an interpreter for a very simple programming language.\bigskip - -\IMPORTANT{} - -\noindent -Also note that the running time of each part will be restricted to a -maximum of 360 seconds on my laptop. - -\DISCLAIMER{} - - -\subsection*{Part 1 (6 Marks)} - -The task is to implement a regular expression matcher that is based on -derivatives of regular expressions. Most of the functions are defined by -recursion over regular expressions and can be elegantly implemented -using Scala's pattern-matching. The implementation should deal with the -following regular expressions, which have been predefined in the file -\texttt{re.scala}: - -\begin{center} -\begin{tabular}{lcll} - $r$ & $::=$ & $\ZERO$ & cannot match anything\\ - & $|$ & $\ONE$ & can only match the empty string\\ - & $|$ & $c$ & can match a single character (in this case $c$)\\ - & $|$ & $r_1 + r_2$ & can match a string either with $r_1$ or with $r_2$\\ - & $|$ & $r_1\cdot r_2$ & can match the first part of a string with $r_1$ and\\ - & & & then the second part with $r_2$\\ - & $|$ & $r^*$ & can match zero or more times $r$\\ -\end{tabular} -\end{center} - -\noindent -Why? Knowing how to match regular expressions and strings will let you -solve a lot of problems that vex other humans. Regular expressions are -one of the fastest and simplest ways to match patterns in text, and -are endlessly useful for searching, editing and analysing data in all -sorts of places (for example analysing network traffic in order to -detect security breaches). However, you need to be fast, otherwise you -will stumble over problems such as recently reported at - -{\small -\begin{itemize} -\item[$\bullet$] \url{http://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016} -\item[$\bullet$] \url{https://vimeo.com/112065252} -\item[$\bullet$] \url{http://davidvgalbraith.com/how-i-fixed-atom/} -\end{itemize}} - -\subsubsection*{Tasks (file re.scala)} - -The file \texttt{re.scala} has already a definition for regular -expressions and also defines some handy shorthand notation for -regular expressions. The notation in this document matches up -with the code in the file as follows: - -\begin{center} - \begin{tabular}{rcl@{\hspace{10mm}}l} - & & code: & shorthand:\smallskip \\ - $\ZERO$ & $\mapsto$ & \texttt{ZERO}\\ - $\ONE$ & $\mapsto$ & \texttt{ONE}\\ - $c$ & $\mapsto$ & \texttt{CHAR(c)}\\ - $r_1 + r_2$ & $\mapsto$ & \texttt{ALT(r1, r2)} & \texttt{r1 | r2}\\ - $r_1 \cdot r_2$ & $\mapsto$ & \texttt{SEQ(r1, r2)} & \texttt{r1 $\sim$ r2}\\ - $r^*$ & $\mapsto$ & \texttt{STAR(r)} & \texttt{r.\%} -\end{tabular} -\end{center} - - -\begin{itemize} -\item[(1a)] Implement a function, called \textit{nullable}, by - recursion over regular expressions. This function tests whether a - regular expression can match the empty string. This means given a - regular expression it either returns true or false. The function - \textit{nullable} - is defined as follows: - -\begin{center} -\begin{tabular}{lcl} -$\textit{nullable}(\ZERO)$ & $\dn$ & $\textit{false}$\\ -$\textit{nullable}(\ONE)$ & $\dn$ & $\textit{true}$\\ -$\textit{nullable}(c)$ & $\dn$ & $\textit{false}$\\ -$\textit{nullable}(r_1 + r_2)$ & $\dn$ & $\textit{nullable}(r_1) \vee \textit{nullable}(r_2)$\\ -$\textit{nullable}(r_1 \cdot r_2)$ & $\dn$ & $\textit{nullable}(r_1) \wedge \textit{nullable}(r_2)$\\ -$\textit{nullable}(r^*)$ & $\dn$ & $\textit{true}$\\ -\end{tabular} -\end{center}~\hfill[1 Mark] - -\item[(1b)] Implement a function, called \textit{der}, by recursion over - regular expressions. It takes a character and a regular expression - as arguments and calculates the derivative regular expression according - to the rules: - -\begin{center} -\begin{tabular}{lcl} -$\textit{der}\;c\;(\ZERO)$ & $\dn$ & $\ZERO$\\ -$\textit{der}\;c\;(\ONE)$ & $\dn$ & $\ZERO$\\ -$\textit{der}\;c\;(d)$ & $\dn$ & $\textit{if}\; c = d\;\textit{then} \;\ONE \; \textit{else} \;\ZERO$\\ -$\textit{der}\;c\;(r_1 + r_2)$ & $\dn$ & $(\textit{der}\;c\;r_1) + (\textit{der}\;c\;r_2)$\\ -$\textit{der}\;c\;(r_1 \cdot r_2)$ & $\dn$ & $\textit{if}\;\textit{nullable}(r_1)$\\ - & & $\textit{then}\;((\textit{der}\;c\;r_1)\cdot r_2) + (\textit{der}\;c\;r_2)$\\ - & & $\textit{else}\;(\textit{der}\;c\;r_1)\cdot r_2$\\ -$\textit{der}\;c\;(r^*)$ & $\dn$ & $(\textit{der}\;c\;r)\cdot (r^*)$\\ -\end{tabular} -\end{center} - -For example given the regular expression $r = (a \cdot b) \cdot c$, the derivatives -w.r.t.~the characters $a$, $b$ and $c$ are - -\begin{center} - \begin{tabular}{lcll} - $\textit{der}\;a\;r$ & $=$ & $(\ONE \cdot b)\cdot c$ & ($= r'$)\\ - $\textit{der}\;b\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$\\ - $\textit{der}\;c\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$ - \end{tabular} -\end{center} - -Let $r'$ stand for the first derivative, then taking the derivatives of $r'$ -w.r.t.~the characters $a$, $b$ and $c$ gives - -\begin{center} - \begin{tabular}{lcll} - $\textit{der}\;a\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ \\ - $\textit{der}\;b\;r'$ & $=$ & $((\ZERO \cdot b) + \ONE)\cdot c$ & ($= r''$)\\ - $\textit{der}\;c\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ - \end{tabular} -\end{center} - -One more example: Let $r''$ stand for the second derivative above, -then taking the derivatives of $r''$ w.r.t.~the characters $a$, $b$ -and $c$ gives - -\begin{center} - \begin{tabular}{lcll} - $\textit{der}\;a\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$ \\ - $\textit{der}\;b\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$\\ - $\textit{der}\;c\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ONE$ & - (is $\textit{nullable}$) - \end{tabular} -\end{center} - -Note, the last derivative can match the empty string, that is it is \textit{nullable}.\\ -\mbox{}\hfill\mbox{[1 Mark]} - -\item[(1c)] Implement the function \textit{simp}, which recursively - traverses a regular expression from the inside to the outside, and - on the way simplifies every regular expression on the left (see - below) to the regular expression on the right, except it does not - simplify inside ${}^*$-regular expressions. - - \begin{center} -\begin{tabular}{l@{\hspace{4mm}}c@{\hspace{4mm}}ll} -$r \cdot \ZERO$ & $\mapsto$ & $\ZERO$\\ -$\ZERO \cdot r$ & $\mapsto$ & $\ZERO$\\ -$r \cdot \ONE$ & $\mapsto$ & $r$\\ -$\ONE \cdot r$ & $\mapsto$ & $r$\\ -$r + \ZERO$ & $\mapsto$ & $r$\\ -$\ZERO + r$ & $\mapsto$ & $r$\\ -$r + r$ & $\mapsto$ & $r$\\ -\end{tabular} - \end{center} - - For example the regular expression - \[(r_1 + \ZERO) \cdot \ONE + ((\ONE + r_2) + r_3) \cdot (r_4 \cdot \ZERO)\] - - simplifies to just $r_1$. \textbf{Hint:} Regular expressions can be - seen as trees and there are several methods for traversing - trees. One of them corresponds to the inside-out traversal, which is - sometimes also called post-order traversal. Furthermore, - remember numerical expressions from school times: there you had expressions - like $u + \ldots + (1 \cdot x) - \ldots (z + (y \cdot 0)) \ldots$ - and simplification rules that looked very similar to rules - above. You would simplify such numerical expressions by replacing - for example the $y \cdot 0$ by $0$, or $1\cdot x$ by $x$, and then - look whether more rules are applicable. If you organise the - simplification in an inside-out fashion, it is always clear which - rule should be applied next.\hfill[2 Marks] - -\item[(1d)] Implement two functions: The first, called \textit{ders}, - takes a list of characters and a regular expression as arguments, and - builds the derivative w.r.t.~the list as follows: - -\begin{center} -\begin{tabular}{lcl} -$\textit{ders}\;(Nil)\;r$ & $\dn$ & $r$\\ - $\textit{ders}\;(c::cs)\;r$ & $\dn$ & - $\textit{ders}\;cs\;(\textit{simp}(\textit{der}\;c\;r))$\\ -\end{tabular} -\end{center} - -Note that this function is different from \textit{der}, which only -takes a single character. - -The second function, called \textit{matcher}, takes a string and a -regular expression as arguments. It builds first the derivatives -according to \textit{ders} and after that tests whether the resulting -derivative regular expression can match the empty string (using -\textit{nullable}). For example the \textit{matcher} will produce -true for the regular expression $(a\cdot b)\cdot c$ and the string -$abc$, but false if you give it the string $ab$. \hfill[1 Mark] - -\item[(1e)] Implement a function, called \textit{size}, by recursion - over regular expressions. If a regular expression is seen as a tree, - then \textit{size} should return the number of nodes in such a - tree. Therefore this function is defined as follows: - -\begin{center} -\begin{tabular}{lcl} -$\textit{size}(\ZERO)$ & $\dn$ & $1$\\ -$\textit{size}(\ONE)$ & $\dn$ & $1$\\ -$\textit{size}(c)$ & $\dn$ & $1$\\ -$\textit{size}(r_1 + r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\ -$\textit{size}(r_1 \cdot r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\ -$\textit{size}(r^*)$ & $\dn$ & $1 + \textit{size}(r)$\\ -\end{tabular} -\end{center} - -You can use \textit{size} in order to test how much the `evil' regular -expression $(a^*)^* \cdot b$ grows when taking successive derivatives -according the letter $a$ without simplification and then compare it to -taking the derivative, but simplify the result. The sizes -are given in \texttt{re.scala}. \hfill[1 Mark] -\end{itemize} - -\subsection*{Background} - -Although easily implementable in Scala, the idea behind the derivative -function might not so easy to be seen. To understand its purpose -better, assume a regular expression $r$ can match strings of the form -$c\!::\!cs$ (that means strings which start with a character $c$ and have -some rest, or tail, $cs$). If you take the derivative of $r$ with -respect to the character $c$, then you obtain a regular expression -that can match all the strings $cs$. In other words, the regular -expression $\textit{der}\;c\;r$ can match the same strings $c\!::\!cs$ -that can be matched by $r$, except that the $c$ is chopped off. - -Assume now $r$ can match the string $abc$. If you take the derivative -according to $a$ then you obtain a regular expression that can match -$bc$ (it is $abc$ where the $a$ has been chopped off). If you now -build the derivative $\textit{der}\;b\;(\textit{der}\;a\;r)$ you -obtain a regular expression that can match the string $c$ (it is $bc$ -where $b$ is chopped off). If you finally build the derivative of this -according $c$, that is -$\textit{der}\;c\;(\textit{der}\;b\;(\textit{der}\;a\;r))$, you obtain -a regular expression that can match the empty string. You can test -whether this is indeed the case using the function nullable, which is -what your matcher is doing. - -The purpose of the $\textit{simp}$ function is to keep the regular -expressions small. Normally the derivative function makes the regular -expression bigger (see the SEQ case and the example in (1b)) and the -algorithm would be slower and slower over time. The $\textit{simp}$ -function counters this increase in size and the result is that the -algorithm is fast throughout. By the way, this algorithm is by Janusz -Brzozowski who came up with the idea of derivatives in 1964 in his PhD -thesis. - -\begin{center}\small -\url{https://en.wikipedia.org/wiki/Janusz_Brzozowski_(computer_scientist)} -\end{center} - - -If you want to see how badly the regular expression matchers do in -Java\footnote{Version 8 and below; Version 9 does not seem to be as - catastrophic, but still worse than the regular expression matcher -based on derivatives.} and in Python with the `evil' regular -expression $(a^*)^*\cdot b$, then have a look at the graphs below (you -can try it out for yourself: have a look at the file -\texttt{catastrophic.java} and \texttt{catastrophic.py} on -KEATS). Compare this with the matcher you have implemented. How long -can the string of $a$'s be in your matcher and still stay within the -30 seconds time limit? - -\begin{center} -\begin{tabular}{@{}cc@{}} -\multicolumn{2}{c}{Graph: $(a^*)^*\cdot b$ and strings - $\underbrace{a\ldots a}_{n}$}\bigskip\\ - -\begin{tikzpicture} -\begin{axis}[ - xlabel={$n$}, - x label style={at={(1.05,0.0)}}, - ylabel={time in secs}, - y label style={at={(0.06,0.5)}}, - enlargelimits=false, - xtick={0,5,...,30}, - xmax=33, - ymax=45, - ytick={0,5,...,40}, - scaled ticks=false, - axis lines=left, - width=6cm, - height=5.5cm, - legend entries={Python, Java 8}, - legend pos=north west] -\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data}; -\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data}; -\end{axis} -\end{tikzpicture} - & -\begin{tikzpicture} -\begin{axis}[ - xlabel={$n$}, - x label style={at={(1.05,0.0)}}, - ylabel={time in secs}, - y label style={at={(0.06,0.5)}}, - %enlargelimits=false, - %xtick={0,5000,...,30000}, - xmax=65000, - ymax=45, - ytick={0,5,...,40}, - scaled ticks=false, - axis lines=left, - width=6cm, - height=5.5cm, - legend entries={Java 9}, - legend pos=north west] -\addplot[cyan,mark=*, mark options={fill=white}] table {re-java9.data}; -\end{axis} -\end{tikzpicture} -\end{tabular} -\end{center} -\newpage - -\subsection*{Part 2 (4 Marks)} - -Coming from Java or C++, you might think Scala is a quite esoteric -programming language. But remember, some serious companies have built -their business on -Scala.\footnote{\url{https://en.wikipedia.org/wiki/Scala_(programming_language)\#Companies}} -And there are far, far more esoteric languages out there. One is -called \emph{brainf***}. You are asked in this part to implement an -interpreter for this language. - -Urban M\"uller developed brainf*** in 1993. A close relative of this -language was already introduced in 1964 by Corado B\"ohm, an Italian -computer pioneer, who unfortunately died a few months ago. The main -feature of brainf*** is its minimalistic set of instructions---just 8 -instructions in total and all of which are single characters. Despite -the minimalism, this language has been shown to be Turing -complete\ldots{}if this doesn't ring any bell with you: it roughly -means that every algorithm we know can, in principle, be implemented in -brainf***. It just takes a lot of determination and quite a lot of -memory resources. Some relatively sophisticated sample programs in -brainf*** are given in the file \texttt{bf.scala}.\bigskip - -\noindent -As mentioned above, brainf*** has 8 single-character commands, namely -\texttt{'>'}, \texttt{'<'}, \texttt{'+'}, \texttt{'-'}, \texttt{'.'}, -\texttt{','}, \texttt{'['} and \texttt{']'}. Every other character is -considered a comment. Brainf*** operates on memory cells containing -integers. For this it uses a single memory pointer that points at each -stage to one memory cell. This pointer can be moved forward by one -memory cell by using the command \texttt{'>'}, and backward by using -\texttt{'<'}. The commands \texttt{'+'} and \texttt{'-'} increase, -respectively decrease, by 1 the content of the memory cell to which -the memory pointer currently points to. The commands for input/output -are \texttt{','} and \texttt{'.'}. Output works by reading the content -of the memory cell to which the memory pointer points to and printing -it out as an ASCII character. Input works the other way, taking some -user input and storing it in the cell to which the memory pointer -points to. The commands \texttt{'['} and \texttt{']'} are looping -constructs. Everything in between \texttt{'['} and \texttt{']'} is -repeated until a counter (memory cell) reaches zero. A typical -program in brainf*** looks as follows: - -\begin{center} -\begin{verbatim} - ++++++++[>++++[>++>+++>+++>+<<<<-]>+>+>->>+[<]<-]>>.>---.+++++++ - ..+++.>>.<-.<.+++.------.--------.>>+.>++. -\end{verbatim} -\end{center} - -\noindent -This one prints out Hello World\ldots{}obviously. - -\subsubsection*{Tasks (file bf.scala)} - -\begin{itemize} -\item[(2a)] Brainf*** memory is represented by a \texttt{Map} from - integers to integers. The empty memory is represented by - \texttt{Map()}, that is nothing is stored in the - memory. \texttt{Map(0 -> 1, 2 -> 3)} clearly stores \texttt{1} at - memory location \texttt{0}; at \texttt{2} it stores \texttt{3}. The - convention is that if we query the memory at a location that is - \emph{not} defined in the \texttt{Map}, we return \texttt{0}. Write - a function, \texttt{sread}, that takes a memory (a \texttt{Map}) and - a memory pointer (an \texttt{Int}) as argument, and safely reads the - corresponding memory location. If the \texttt{Map} is not defined at - the memory pointer, \texttt{sread} returns \texttt{0}. - - Write another function \texttt{write}, which takes a memory, a - memory pointer and an integer value as argument and updates the - \texttt{Map} with the value at the given memory location. As usual - the \texttt{Map} is not updated `in-place' but a new map is created - with the same data, except the value is stored at the given memory - pointer.\hfill[1 Mark] - -\item[(2b)] Write two functions, \texttt{jumpRight} and - \texttt{jumpLeft} that are needed to implement the loop constructs - of brainf***. They take a program (a \texttt{String}) and a program - counter (an \texttt{Int}) as argument and move right (respectively - left) in the string in order to find the \textbf{matching} - opening/closing bracket. For example, given the following program - with the program counter indicated by an arrow: - - \begin{center} - \texttt{--[\barbelow{.}.+>--],>,++} - \end{center} - - then the matching closing bracket is in 9th position (counting from 0) and - \texttt{jumpRight} is supposed to return the position just after this - - \begin{center} - \texttt{--[..+>--]\barbelow{,}>,++} - \end{center} - - meaning it jumps to after the loop. Similarly, if you are in 8th position - then \texttt{jumpLeft} is supposed to jump to just after the opening - bracket (that is jumping to the beginning of the loop): - - \begin{center} - \texttt{--[..+>-\barbelow{-}],>,++} - \qquad$\stackrel{\texttt{jumpLeft}}{\longrightarrow}$\qquad - \texttt{--[\barbelow{.}.+>--],>,++} - \end{center} - - Unfortunately we have to take into account that there might be - other opening and closing brackets on the `way' to find the - matching bracket. For example in the brainf*** program - - \begin{center} - \texttt{--[\barbelow{.}.[+>]--],>,++} - \end{center} - - we do not want to return the index for the \texttt{'-'} in the 9th - position, but the program counter for \texttt{','} in 12th - position. The easiest to find out whether a bracket is matched is by - using levels (which are the third argument in \texttt{jumpLeft} and - \texttt{jumpLeft}). In case of \texttt{jumpRight} you increase the - level by one whenever you find an opening bracket and decrease by - one for a closing bracket. Then in \texttt{jumpRight} you are looking - for the closing bracket on level \texttt{0}. For \texttt{jumpLeft} you - do the opposite. In this way you can find \textbf{matching} brackets - in strings such as - - \begin{center} - \texttt{--[\barbelow{.}.[[-]+>[.]]--],>,++} - \end{center} - - for which \texttt{jumpRight} should produce the position: - - \begin{center} - \texttt{--[..[[-]+>[.]]--]\barbelow{,}>,++} - \end{center} - - It is also possible that the position returned by \texttt{jumpRight} or - \texttt{jumpLeft} is outside the string in cases where there are - no matching brackets. For example - - \begin{center} - \texttt{--[\barbelow{.}.[[-]+>[.]]--,>,++} - \qquad$\stackrel{\texttt{jumpRight}}{\longrightarrow}$\qquad - \texttt{--[..[[-]+>[.]]-->,++\barbelow{\;\phantom{+}}} - \end{center} - \hfill[1 Mark] - - -\item[(2c)] Write a recursive function \texttt{run} that executes a - brainf*** program. It takes a program, a program counter, a memory - pointer and a memory as arguments. If the program counter is outside - the program string, the execution stops and \texttt{run} returns the - memory. If the program counter is inside the string, it reads the - corresponding character and updates the program counter \texttt{pc}, - memory pointer \texttt{mp} and memory \texttt{mem} according to the - rules shown in Figure~\ref{comms}. It then calls recursively - \texttt{run} with the updated data. - - Write another function \texttt{start} that calls \texttt{run} with a - given brainfu** program and memory, and the program counter and memory pointer - set to~$0$. Like \texttt{run} it returns the memory after the execution - of the program finishes. You can test your brainf**k interpreter with the - Sierpinski triangle or the Hello world programs or have a look at - - \begin{center} - \url{https://esolangs.org/wiki/Brainfuck} - \end{center}\hfill[2 Marks] - - \begin{figure}[p] - \begin{center} - \begin{tabular}{|@{}p{0.8cm}|l|} - \hline - \hfill\texttt{'>'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & $\texttt{mp} + 1$\\ - $\bullet$ & \texttt{mem} unchanged - \end{tabular}\\\hline - \hfill\texttt{'<'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & $\texttt{mp} - 1$\\ - $\bullet$ & \texttt{mem} unchanged - \end{tabular}\\\hline - \hfill\texttt{'+'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & $\texttt{mp}$ unchanged\\ - $\bullet$ & \texttt{mem} updated with \texttt{mp -> mem(mp) + 1}\\ - \end{tabular}\\\hline - \hfill\texttt{'-'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & $\texttt{mp}$ unchanged\\ - $\bullet$ & \texttt{mem} updated with \texttt{mp -> mem(mp) - 1}\\ - \end{tabular}\\\hline - \hfill\texttt{'.'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\ - $\bullet$ & print out \,\texttt{mem(mp)} as a character\\ - \end{tabular}\\\hline - \hfill\texttt{','} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & $\texttt{mp}$ unchanged\\ - $\bullet$ & \texttt{mem} updated with \texttt{mp -> \textrm{input}}\\ - \multicolumn{2}{@{}l}{the input is given by \texttt{Console.in.read().toByte}} - \end{tabular}\\\hline - \hfill\texttt{'['} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - \multicolumn{2}{@{}l}{if \texttt{mem(mp) == 0} then}\\ - $\bullet$ & $\texttt{pc = jumpRight(prog, pc + 1, 0)}$\\ - $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\medskip\\ - \multicolumn{2}{@{}l}{otherwise if \texttt{mem(mp) != 0} then}\\ - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\ - \end{tabular} - \\\hline - \hfill\texttt{']'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - \multicolumn{2}{@{}l}{if \texttt{mem(mp) != 0} then}\\ - $\bullet$ & $\texttt{pc = jumpLeft(prog, pc - 1, 0)}$\\ - $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\medskip\\ - \multicolumn{2}{@{}l}{otherwise if \texttt{mem(mp) == 0} then}\\ - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\ - \end{tabular}\\\hline - any other char & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}} - $\bullet$ & $\texttt{pc} + 1$\\ - $\bullet$ & \texttt{mp} and \texttt{mem} unchanged - \end{tabular}\\ - \hline - \end{tabular} - \end{center} - \caption{The rules for how commands in the brainf*** language update the program counter \texttt{pc}, - memory pointer \texttt{mp} and memory \texttt{mem}.\label{comms}} - \end{figure} -\end{itemize}\bigskip - - - - -\end{document} - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: t -%%% End: