diff -r 19b75e899d37 -r 9c03b5e89a2a cws/cw08.tex --- a/cws/cw08.tex Fri Apr 26 17:29:30 2024 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,267 +0,0 @@ -\documentclass{article} -\usepackage{../style} -\usepackage{../langs} -\usepackage{../graphics} - -\begin{document} - -\section*{Replacement Coursework 2 (Automata)} - -This coursework is worth 10\%. It is about deterministic and -non-deterministic finite automata. The coursework is due on 21 March -at 5pm. Make sure the files you submit can be processed by just -calling \texttt{scala <>}.\bigskip - -\noindent -\textbf{Important:} Do not use any mutable data structures in your -submission! They are not needed. This means you cannot use -\texttt{ListBuffer}s, for example. Do not use \texttt{return} in your -code! It has a different meaning in Scala, than in Java. Do not use -\texttt{var}! This declares a mutable variable. Make sure the -functions you submit are defined on the ``top-level'' of Scala, not -inside a class or object. Also note that when marking, the running time -will be restricted to a maximum of 360 seconds on my laptop. - - -\subsection*{Disclaimer} - -It should be understood that the work you submit represents your own -effort! You have not copied from anyone else. An exception is the -Scala code I showed during the lectures or uploaded to KEATS, which -you can freely use.\bigskip - - -\subsection*{Part 1 (Deterministic Finite Automata)} - -\noindent -There are many uses for Deterministic Finite Automata (DFAs), for -example for testing whether a string matches a pattern or not. DFAs -consist of some states (circles) and some transitions (edges) between -states. For example the DFA - -\begin{center} -\begin{tikzpicture}[scale=1.5,>=stealth',very thick,auto, - every state/.style={minimum size=4pt, - inner sep=4pt,draw=blue!50,very thick, - fill=blue!20}] - \node[state, initial] (q0) at ( 0,1) {$Q_0$}; - \node[state] (q1) at ( 1,1) {$Q_1$}; - \node[state, accepting] (q2) at ( 2,1) {$Q_2$}; - \path[->] (q0) edge[bend left] node[above] {$a$} (q1) - (q1) edge[bend left] node[above] {$b$} (q0) - (q2) edge[bend left=50] node[below] {$b$} (q0) - (q1) edge node[above] {$a$} (q2) - (q2) edge [loop right] node {$a$} () - (q0) edge [loop below] node {$b$} (); -\end{tikzpicture} -\end{center} - -\noindent -has three states ($Q_0$, $Q_1$ and $Q_2$), whereby $Q_0$ is the -starting state of the DFA and $Q_2$ is the accepting state. The latter -is indicated by double lines. In general, a DFA can have any number of -accepting states, but only a single starting state. - -Transitions are edges between states labelled with a character. The -idea is that if we are in state $Q_0$, say, and get an $a$, we can go -to state $Q_1$. If we are in state $Q_2$ and get an $a$, we can stay -in state $Q_2$; if we get a $b$ in $Q_2$, then can go to state -$Q_0$. The main point of DFAs is that if we are in a state and get a -character, it is always clear which is the next state---there can only -be at most one. The task of Part 1 is to implement such DFAs in Scala -using partial functions for the transitions. - -A string is accepted by a DFA, if we start in the starting state, -follow all transitions according to the string; if we end up in an -accepting state, then the string is accepted. If not, the string is -not accepted. The technical idea is that DFAs can be used to -accept strings from \emph{regular} languages. - -\subsubsection*{Tasks} - -\begin{itemize} -\item[(1)] Write a polymorphic function, called \texttt{share}, that - decides whether two sets share some elements (i.e.~the intersection - is not empty).\hfill[1 Mark] - -\item[(2)] The transitions of DFAs will be implemented as partial - functions. These functions will have the type (state, - character)-pair to state, that is their input will be a (state, - character)-pair and they return a state. For example the transitions - of the DFA shown above can be defined as the following - partial function: - -\begin{lstlisting}[language=Scala,numbers=none] -val dfa_trans : PartialFunction[(State,Char), State] = - { case (Q0, 'a') => Q1 - case (Q0, 'b') => Q0 - case (Q1, 'a') => Q2 - case (Q1, 'b') => Q0 - case (Q2, 'a') => Q2 - case (Q2, 'b') => Q0 - } -\end{lstlisting} - - The main point of partial functions (as opposed to ``normal'' - functions) is that they do not have to be defined everywhere. For - example the transitions above only mention characters $a$ and $b$, - but leave out any other characters. Partial functions come with a - method \texttt{isDefinedAt} that can be used to check whether an - input produces a result or not. For example - -\begin{lstlisting}[language=Scala,numbers=none] - dfa_trans.isDefinedAt((Q0, 'a')) - dfa_trans.isDefinedAt((Q0, 'c')) -\end{lstlisting} - - \noindent - gives \texttt{true} in the first case and \texttt{false} in the - second. There is also a method \texttt{lift} that transforms a - partial function into a ``normal'' function returning an optional - value: if the partial function is defined on the input, the lifted - function will return \texttt{Some}; if it is not defined, then - \texttt{None}. - - Write a function that takes a transition and a (state, character)-pair as arguments - and produces an optional state (the state specified by the partial transition - function whenever it is defined; if the transition function is undefined, - return \texttt{None}).\hfill\mbox{[1 Mark]} - -\item[(3)] - Write a function that ``lifts'' the function in (2) from characters to strings. That - is, write a function that takes a transition, a state and a list of characters - as arguments and produces the state generated by following the transitions for - each character in the list. For example if you are in state $Q_0$ in the DFA above - and have the list \texttt{List(a,a,a,b,b,a)}, then you need to return the - state $Q_1$ (as option since there might not be such a state in general).\\ - \mbox{}\hfill\mbox{[1 Mark]} - -\item[(4)] DFAs are defined as a triple: (starting state, transitions, - set of accepting states). Write a function \texttt{accepts} that tests whether - a string is accepted by an DFA or not. For this start in the - starting state of the DFA, use the function under (3) to calculate - the state after following all transitions according to the - characters in the string. If the resulting state is an accepting state, - return \texttt{true}; otherwise \texttt{false}.\\\mbox{}\hfill\mbox{[1 Mark]} -\end{itemize} - - -\subsection*{Part 2 (Non-Deterministic Finite Automata)} - -The main point of DFAs is that for every given state and character -there is at most one next state (one if the transition is defined; -none otherwise). However, this restriction to at most one state can be -quite limiting for some applications.\footnote{Though there is a - curious fact that every (less restricted) NFA can be translated into - an ``equivalent'' DFA, whereby accepting means accepting the same - set of strings. However this might increase drastically the number - of states in the DFA.} Non-Deterministic Automata (NFAs) remove -this restriction: there can be more than one starting state, and given -a state and a character there can be more than one next -state. Consider for example the NFA - -\begin{center} -\begin{tikzpicture}[scale=0.7,>=stealth',very thick, - every state/.style={minimum size=0pt, - draw=blue!50,very thick,fill=blue!20},] -\node[state,initial] (R_1) {$R_1$}; -\node[state,initial] (R_2) [above=of R_1] {$R_2$}; -\node[state, accepting] (R_3) [right=of R_1] {$R_3$}; -\path[->] (R_1) edge node [below] {$b$} (R_3); -\path[->] (R_2) edge [bend left] node [above] {$a$} (R_3); -\path[->] (R_1) edge [bend left] node [left] {$c$} (R_2); -\path[->] (R_2) edge [bend left] node [right] {$a$} (R_1); -\end{tikzpicture} -\end{center} - -\noindent -where in state $R_2$ if we get an $a$, we can go to state $R_1$ -\emph{or} $R_3$. If we want to find out whether an NFA accepts a -string, then we need to explore both possibilities. We will do this -``exploration'' in the tasks below in a breadth-first manner. - - -The feature of having more than one next state in NFAs will be -implemented by having a \emph{set} of partial transition functions -(DFAs had only one). For example the NFA shown above will be -represented by the set of partial functions - -\begin{lstlisting}[language=Scala,numbers=none] -val nfa_trans : NTrans = Set( - { case (R1, 'c') => R2 }, - { case (R1, 'b') => R3 }, - { case (R2, 'a') => R1 }, - { case (R2, 'a') => R3 } -) -\end{lstlisting} - -\noindent -The point is that the 3rd element in this set makes sure that in state $R_2$ and -given an $a$, we can go to state $R_1$; and the 4th element, in $R_2$, -given an $a$, we can also go to state $R_3$. When following -transitions from a state, we have to look at all partial functions in -the set and generate the set of \emph{all} possible next states. - -\subsubsection*{Tasks} - -\begin{itemize} -\item[(5)] - Write a function \texttt{nnext} which takes a transition set, a state - and a character as arguments, and calculates all possible next states - (returned as set).\\ - \mbox{}\hfill\mbox{[1 Mark]} - -\item[(6)] Write a function \texttt{nnexts} which takes a transition - set, a \emph{set} of states and a character as arguments, and - calculates \emph{all} possible next states that can be reached from - any state in the set.\mbox{}\hfill\mbox{[1 Mark]} - -\item[(7)] Like in (3), write a function \texttt{nnextss} that lifts - \texttt{nnexts} from (6) from single characters to lists of characters. - \mbox{}\hfill\mbox{[1 Mark]} - -\item[(8)] NFAs are also defined as a triple: (set of staring states, - set of transitions, set of accepting states). Write a function - \texttt{naccepts} that tests whether a string is accepted by an NFA - or not. For this start in all starting states of the NFA, use the - function under (7) to calculate the set of states following all - transitions according to the characters in the string. If the - resulting set of states shares at least a single state with the set - of accepting states, return \texttt{true}; otherwise \texttt{false}. - Use the function under (1) in order to test whether these two sets - of states share any states or not.\mbox{}\hfill\mbox{[1 Mark]} - -\item[(9)] Since we explore in functions (6) and (7) all possible next - states, we decide whether a string is accepted in a breadth-first - manner. (Depth-first would be to choose one state, follow all next - states of this single state; check whether it leads to an accepting - state. If not, we backtrack and choose another state). The - disadvantage of breadth-first search is that at every step a - non-empty set of states are ``active''\ldots{} states that need to - be followed at the same time. Write similar functions as in (7) and - (8), but instead of returning states or a boolean, calculate the - number of states that need to be followed in each step. The function - \texttt{max\_accept} should then return the maximum of all these - numbers. - - As a test case, consider again the NFA shown above. At the beginning - the number of active states will be 2 (since there are two starting - states, namely $R_1$ and $R_2$). If we get an $a$, there will be - still 2 active states, namely $R_1$ and $R_3$ both reachable from - $R_2$. There is no transition for $a$ and $R_1$. So for a string, - say, $ab$ which is accepted by the NFA, the maximum number of active - states is 2 (it is not possible that all three states of this NFA - are active at the same time; is it possible that no state is - active?). \hfill\mbox{[2 Marks]} - - -\end{itemize} - - -\end{document} - - -%%% Local Variables: -%%% mode: latex -%%% TeX-master: t -%%% End: