From 179de5da7f000da7d790e88a40c16a1a04a036dd Mon Sep 17 00:00:00 2001 From: Sankalp Gambhir <sankalp.gambhir@epfl.ch> Date: Fri, 28 Feb 2025 11:45:20 +0000 Subject: [PATCH] Add exercise set 1 --- info/exercises/ex-01-sol.pdf | Bin 0 -> 206966 bytes info/exercises/ex-01.pdf | Bin 0 -> 147576 bytes info/exercises/past-exercises/README.md | 5 + .../ex1/CS320_EX01_Solutions.pdf | Bin .../exercises/{ => past-exercises}/ex1/ex1.md | 0 .../{ => past-exercises}/ex1/exes01-2018.pdf | Bin .../{ => past-exercises}/ex2/CS320_ES02.pdf | Bin .../ex2/CS320_ES02_Solutions.pdf | Bin .../{ => past-exercises}/ex3/CS320_ES03.pdf | Bin .../ex3/CS320_ES03_Solutions_v2.pdf | Bin .../{ => past-exercises}/ex4/CS320_ES04.pdf | Bin .../ex4/CS320_ES04_Solutions_v2.pdf | Bin .../{ => past-exercises}/ex5/CS320_ES05.pdf | Bin .../ex5/CS320_ES05_sol.pdf | Bin .../{ => past-exercises}/ex6/CS320_ES06.pdf | Bin .../ex6/CS320_ES06_sol.pdf | Bin .../{ => past-exercises}/ex7/CS320_ES07.pdf | Bin .../ex7/CS320_ES07_Solutions.pdf | Bin info/exercises/src/.gitignore | 7 + info/exercises/src/ex-01/ex/dfa.tex | 774 ++++++++++++++++++ info/exercises/src/ex-01/ex/languages.tex | 140 ++++ info/exercises/src/ex-01/ex/lexer.tex | 154 ++++ info/exercises/src/ex-01/main.tex | 30 + info/exercises/src/macro.tex | 114 +++ 24 files changed, 1224 insertions(+) create mode 100644 info/exercises/ex-01-sol.pdf create mode 100644 info/exercises/ex-01.pdf create mode 100644 info/exercises/past-exercises/README.md rename info/exercises/{ => past-exercises}/ex1/CS320_EX01_Solutions.pdf (100%) rename info/exercises/{ => past-exercises}/ex1/ex1.md (100%) rename info/exercises/{ => past-exercises}/ex1/exes01-2018.pdf (100%) rename info/exercises/{ => past-exercises}/ex2/CS320_ES02.pdf (100%) rename info/exercises/{ => past-exercises}/ex2/CS320_ES02_Solutions.pdf (100%) rename info/exercises/{ => past-exercises}/ex3/CS320_ES03.pdf (100%) rename info/exercises/{ => past-exercises}/ex3/CS320_ES03_Solutions_v2.pdf (100%) rename info/exercises/{ => past-exercises}/ex4/CS320_ES04.pdf (100%) rename info/exercises/{ => past-exercises}/ex4/CS320_ES04_Solutions_v2.pdf (100%) rename info/exercises/{ => past-exercises}/ex5/CS320_ES05.pdf (100%) rename info/exercises/{ => past-exercises}/ex5/CS320_ES05_sol.pdf (100%) rename info/exercises/{ => past-exercises}/ex6/CS320_ES06.pdf (100%) rename info/exercises/{ => past-exercises}/ex6/CS320_ES06_sol.pdf (100%) rename info/exercises/{ => past-exercises}/ex7/CS320_ES07.pdf (100%) rename info/exercises/{ => past-exercises}/ex7/CS320_ES07_Solutions.pdf (100%) create mode 100644 info/exercises/src/.gitignore create mode 100644 info/exercises/src/ex-01/ex/dfa.tex create mode 100644 info/exercises/src/ex-01/ex/languages.tex create mode 100644 info/exercises/src/ex-01/ex/lexer.tex create mode 100644 info/exercises/src/ex-01/main.tex create mode 100644 info/exercises/src/macro.tex diff --git 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b/info/exercises/past-exercises/README.md @@ -0,0 +1,5 @@ +# Past Exercises + +This is the collection of the exercises used in previous iterations of the +course. They are here if you wish to try additional exercises, but note that not +everything may necessarily match the current content of the course. diff --git a/info/exercises/ex1/CS320_EX01_Solutions.pdf b/info/exercises/past-exercises/ex1/CS320_EX01_Solutions.pdf similarity index 100% rename from info/exercises/ex1/CS320_EX01_Solutions.pdf rename to info/exercises/past-exercises/ex1/CS320_EX01_Solutions.pdf diff --git a/info/exercises/ex1/ex1.md b/info/exercises/past-exercises/ex1/ex1.md similarity index 100% rename from info/exercises/ex1/ex1.md rename to info/exercises/past-exercises/ex1/ex1.md diff --git a/info/exercises/ex1/exes01-2018.pdf b/info/exercises/past-exercises/ex1/exes01-2018.pdf similarity index 100% rename from info/exercises/ex1/exes01-2018.pdf rename to info/exercises/past-exercises/ex1/exes01-2018.pdf diff --git a/info/exercises/ex2/CS320_ES02.pdf b/info/exercises/past-exercises/ex2/CS320_ES02.pdf similarity index 100% rename from info/exercises/ex2/CS320_ES02.pdf rename to info/exercises/past-exercises/ex2/CS320_ES02.pdf diff --git a/info/exercises/ex2/CS320_ES02_Solutions.pdf b/info/exercises/past-exercises/ex2/CS320_ES02_Solutions.pdf similarity index 100% rename from info/exercises/ex2/CS320_ES02_Solutions.pdf rename to info/exercises/past-exercises/ex2/CS320_ES02_Solutions.pdf diff --git a/info/exercises/ex3/CS320_ES03.pdf b/info/exercises/past-exercises/ex3/CS320_ES03.pdf similarity index 100% rename from info/exercises/ex3/CS320_ES03.pdf rename to info/exercises/past-exercises/ex3/CS320_ES03.pdf diff --git a/info/exercises/ex3/CS320_ES03_Solutions_v2.pdf b/info/exercises/past-exercises/ex3/CS320_ES03_Solutions_v2.pdf similarity index 100% rename from info/exercises/ex3/CS320_ES03_Solutions_v2.pdf rename to info/exercises/past-exercises/ex3/CS320_ES03_Solutions_v2.pdf diff --git a/info/exercises/ex4/CS320_ES04.pdf b/info/exercises/past-exercises/ex4/CS320_ES04.pdf similarity index 100% rename from info/exercises/ex4/CS320_ES04.pdf rename to info/exercises/past-exercises/ex4/CS320_ES04.pdf diff --git a/info/exercises/ex4/CS320_ES04_Solutions_v2.pdf b/info/exercises/past-exercises/ex4/CS320_ES04_Solutions_v2.pdf similarity index 100% rename from info/exercises/ex4/CS320_ES04_Solutions_v2.pdf rename to info/exercises/past-exercises/ex4/CS320_ES04_Solutions_v2.pdf diff --git a/info/exercises/ex5/CS320_ES05.pdf b/info/exercises/past-exercises/ex5/CS320_ES05.pdf similarity index 100% rename from info/exercises/ex5/CS320_ES05.pdf rename to info/exercises/past-exercises/ex5/CS320_ES05.pdf diff --git a/info/exercises/ex5/CS320_ES05_sol.pdf b/info/exercises/past-exercises/ex5/CS320_ES05_sol.pdf similarity index 100% rename from info/exercises/ex5/CS320_ES05_sol.pdf rename to info/exercises/past-exercises/ex5/CS320_ES05_sol.pdf diff --git a/info/exercises/ex6/CS320_ES06.pdf b/info/exercises/past-exercises/ex6/CS320_ES06.pdf similarity index 100% rename from info/exercises/ex6/CS320_ES06.pdf rename to info/exercises/past-exercises/ex6/CS320_ES06.pdf diff --git a/info/exercises/ex6/CS320_ES06_sol.pdf b/info/exercises/past-exercises/ex6/CS320_ES06_sol.pdf similarity index 100% rename from info/exercises/ex6/CS320_ES06_sol.pdf rename to info/exercises/past-exercises/ex6/CS320_ES06_sol.pdf diff --git a/info/exercises/ex7/CS320_ES07.pdf b/info/exercises/past-exercises/ex7/CS320_ES07.pdf similarity index 100% rename from info/exercises/ex7/CS320_ES07.pdf rename to info/exercises/past-exercises/ex7/CS320_ES07.pdf diff --git a/info/exercises/ex7/CS320_ES07_Solutions.pdf b/info/exercises/past-exercises/ex7/CS320_ES07_Solutions.pdf similarity index 100% rename from info/exercises/ex7/CS320_ES07_Solutions.pdf rename to info/exercises/past-exercises/ex7/CS320_ES07_Solutions.pdf diff --git a/info/exercises/src/.gitignore b/info/exercises/src/.gitignore new file mode 100644 index 0000000..0aa0fc5 --- /dev/null +++ b/info/exercises/src/.gitignore @@ -0,0 +1,7 @@ +*.aux +*.fdb_latexmk +*.fls +*.log +*.out +*.synctex.gz +*.pdf diff --git a/info/exercises/src/ex-01/ex/dfa.tex b/info/exercises/src/ex-01/ex/dfa.tex new file mode 100644 index 0000000..38b2ca6 --- /dev/null +++ b/info/exercises/src/ex-01/ex/dfa.tex @@ -0,0 +1,774 @@ + + +\begin{exercise}{} + + For each the following languages, construct an NFA \(\mathcal{A}\) that + recognizes them, i.e. \(L(\mathcal{A})\) = \(L_i\): + \begin{enumerate} + \item \(L_1\): binary strings divisible by 3 + \item \(L_2\): binary strings divisible by 4 + \item \(L_3\): \(\{(w_1 \oplus w_2) \mid w_1 \in L_1 \land w_2 \in L_2 \land |w_1| = |w_2|\}\) + \end{enumerate} + where \(\oplus\) is the bitwise-xor operation on binary strings. + + \begin{solution} + \begin{enumerate} + \item The language of binary strings divisible by 3. We need two + observations to construct this automaton: + \begin{enumerate} + \item If the automaton has consumed a binary string \(s\) with decimal + value, say, \(val(s) = n\), then we can determine the decimal value of + the string after reading one more character as either \(val(s0) = 2n\) + or \(val(s1) = 2n + 1\). + \item The set of strings is finite, but it is sufficient to know only + the value of the string \emph{modulo 3} to determine if it is divisible. + \end{enumerate} + + We construct the automaton \(\mathcal{A}_1 = (Q, \Sigma, \delta, q_{init}, F)\) + where: + \begin{itemize} + \item \(Q = \{q_{init}, q_0, q_1, q_2\}\), representing the initial + state (empty word has no value), and the states corresponding to the + values \(0, 1, 2\) modulo 3. + \item \(\Sigma = \{0, 1\}\), as required. + \item \(\delta = \{(q_i, 0, q_j) \mid 2i \mod 3 = j\} \cup \{(q_i, 1, + q_j) \mid (2i + 1) mod 3 = j\} \cup \{(q_{init}, 0, q_0), (q_{init}, 1, + q_1)\}\), i.e., there is a transition from \(q_i\) to \(q_j\) if, as the + currently known value modulo 3 is \(i\), on reading \(0\) the next value + is \(j = 2i \mod 3\). We use the fact that \(2n \mod 3 = 2 (n \mod 3) + \mod 3\). The case for reading \(1\) is similar. The translations from + the initial state are to the states corresponding to the values \(0, 1\) + modulo 3. + + For example, if we have read ``1101'', with decimal value \(13\), we + must be in state \(q_1\), as \(13 \mod 3 = 1\). On reading a \(0\), we + have the string ``11010'' with decimal value \(26\), and \(26 \mod 3 = + 2\), so we transition to \(q_2\). + + The full automaton is below. + \item \(F = \{q_0\}\) as we accept that words that are divisible by 3, + and are hence equal to 0 modulo 3. + \end{itemize} + + The automaton is: + \begin{center} + \begin{tikzpicture}[node distance = 2cm, on grid] + \node[state, initial] (qi) {\(q_{init}\)}; + \node[state, accepting, right of = qi] (q0) {\(q_0\)}; + \node[state, above right of = q0] (q1) {\(q_1\)}; + \node[state, below right of = q1] (q2) {\(q_2\)}; + + \draw[->] + (qi) edge node[above] {\(0\)} (q0) + (qi) edge node[above] {\(1\)} (q1) + % + (q0) edge[loop below] node[below] {\(0\)} (q0) + (q0) edge node[above] {\(1\)} (q1) + % + (q1) edge[bend right] node[below] {\(0\)} (q2) + (q1) edge[bend left] node[below] {\(1\)} (q0) + % + (q2) edge node[above] {\(0\)} (q1) + (q2) edge[loop below] node[below] {\(1\)} (q2) + ; + \end{tikzpicture} + \end{center} + + \item The language of binary strings divisible by 4. The construction is + similar to the one above, now with 5 states. + \begin{center} + \begin{tikzpicture}[node distance = 2cm, on grid] + \node[state, initial] (qi) {\(q_{init}\)}; + \node[state, accepting, right of = qi] (q0) {\(q_0\)}; + \node[state, above of = q0] (q1) {\(q_1\)}; + \node[state, right of = q1] (q2) {\(q_2\)}; + \node[state, right of = q0] (q3) {\(q_3\)}; + + \draw[->] + (qi) edge node[above] {\(0\)} (q0) + (qi) edge node[above] {\(1\)} (q1) + % + (q0) edge[loop below] node[below] {\(0\)} (q0) + (q0) edge node[left] {\(1\)} (q1) + % + (q1) edge node[above] {\(0\)} (q2) + (q1) edge[bend right] node[above] {\(1\)} (q3) + % + (q2) edge node[above] {\(0\)} (q0) + (q2) edge[bend left] node[above] {\(1\)} (q1) + % + (q3) edge node[right] {\(0\)} (q2) + (q3) edge[loop below] node[below] {\(1\)} (q3) + ; + \end{tikzpicture} + \end{center} + + \item To compute the bitwise-xor of two strings, we must compute a product + automaton. To accept a word \(w\), there must exist \(w_1, w_2\) such that + \(w_1 \in L_1\), and \(w_2 \in L_2\). + + We do not explicitly construct the automaton, but present an argument. + First, consider the truth table for xor: + + \begin{center} + \begin{tabular}{c c | c} + \(b_1\) & \(b_2\) & \(b_1 \oplus b_2\) \\ + \hline + 0 & 0 & 0 \\ + 0 & 1 & 1 \\ + 1 & 0 & 1 \\ + 1 & 1 & 0 \\ + \end{tabular} + \end{center} + + Notably, given a xor result, we cannot exactly determine the input bits. + In essence, we construct an automaton that, given a string, tries to + simulate the two input automata in parallel non-deterministically on all + possible pairs of input strings. If any of them are accepted, that means + we found a pair of strings that, one, are accepted by the two original + automata, and two, have the input string as their bitwise-xor. + + Formally, the automaton \(\mathcal{A}_3 = (Q, \Sigma, \delta, q_{init}, + F)\) has: + + \begin{itemize} + \item \(Q = Q_1 \times Q_2\), where \(Q_1\) and \(Q_2\) are the state + sets of \(\mathcal{A}_1\) and \(\mathcal{A}_2\). + \item \(\Sigma = \{0, 1\}\) as before. + \item \(q_{init} = (q_{1, init}, q_{2, init})\) where \(q_{1, init}\) + and \(q_{2, init}\) are the initial states of \(\mathcal{A}_1\) and + \(\mathcal{A}_2\). + \item \(F = F_1 \times F_2\) similarly. + \item \(\delta\) is constructed as follows: for a pair of states \((q_1, + q_2)\), on reading a \(0\), we look at the truth table of xor; two input + pairs \((0, 0)\) and \((1, 1)\) could have produced this result bit. + Hence, we add transitions for both automata simultaneously, \((((q_1, + q_2), 0, (q_1', q_2')))\) corresponding to possible inputs \((0, 0)\) if + \(\delta_1(q_1, 0, q_1')\) and \(\delta_2(q_2, 0, q_2')\), and similarly + \(((q_1, q_2), 0, (q_1', q_2'))\) corresponding to possible inputs \((1, + 1)\) if \(\delta_1(q_1, 1, q_1'')\) and \(\delta_2(q_2, 1, q_2'')\). + + The case for reading a \(1\) is similar, with possible input pairs \((0, + 1)\) and \((1, 0)\). + \end{itemize} + + \end{enumerate} + \end{solution} + +\end{exercise} + +% \begin{exercise}{} +% \todo{Suggestion: From Kozen H2.3, recognizing with errors; regular language modulo n Hamming distance is regular} +% \end{exercise} + +\begin{exercise}{} + Give a verbal and a set-notational description of the language accepted by + each of the following automata. You can assume that the alphabet is \(\Sigma = + \{a, b\}\). + + \begin{enumerate} + \item \(\mathcal{A}_1\) + \begin{center} + \begin{tikzpicture}[node distance = 2cm, on grid] + \node[state, initial] (q0) {\(q_0\)}; + \node[state, accepting, right of = q0] (q1) {\(q_1\)}; + \node[state, right of = q1] (q2) {\(q_2\)}; + + \draw[->] (q0) edge node[above] {\(a\)} (q1) + (q1) edge node[above] {\(a\)} (q2) + (q2) edge[loop above] node[above] {\(a, b\)} (q2) + (q0) edge[loop above] node[above] {\(b\)} (q0) + (q1) edge[loop above] node[above] {\(b\)} (q1) + ; + \end{tikzpicture} + \end{center} + \item \(\mathcal{A}_2\) + \begin{center} + \begin{tikzpicture}[node distance = 2cm, on grid] + \node[state, accepting, initial] (q0) {\(q_0\)}; + \node[state, accepting, right of = q0] (q1) {\(q_1\)}; + \node[state, right of = q1] (q2) {\(q_2\)}; + + \draw[->] (q0) edge node[below] {\(a\)} (q1) + (q1) edge[bend right] node[above] {\(b\)} (q0) + (q2) edge[loop above] node[above] {\(a, b\)} (q2) + (q0) edge[loop above] node[above] {\(b\)} (q0) + (q1) edge node[above] {\(a\)} (q2) + ; + \end{tikzpicture} + \end{center} + \end{enumerate} + + + \begin{solution} + \begin{enumerate} + \item + As regular expression: \(b^*ab^*\), this is the language of words that + contain exactly one \(a\). In set-notation: + \begin{equation*} + \{w \mid \exists! i.\; 0 \leq i \leq |w| \land w_{(i)} = a\} + \end{equation*} + + \item + As generalized regular expression (with complement): \((\Sigma^* aa + \Sigma^*)^c\). Without complement: \((b^*(ab^*)^*)^*\). This is the language + of words that contain no consecutive pair of \(a\)'s. In set-notation: + \begin{equation*} + \{w \mid \forall i.\; 0 \leq i < |w| \land w_{(i)} = a \implies (i + 1 \geq |w| \lor w_{(i + 1)} \neq a)\} + \end{equation*} + \end{enumerate} + \end{solution} + +\end{exercise} + +% \begin{exercise}{} +% Construct a DFA for the following languages: + +% \begin{enumerate} +% \item the set of strings over \(\Sigma = \{0, 1\}\) such that the number +% of 0's is a multiple of 2, and the number of 1's is a multiple of 3. +% \note{\cite[HW 1.1.c]{kozen2007automata}} +% \item the set of strings over \(\Sigma = \{0, 1\}\) that, as binary +% numerals, have decimal value 42. \note{\cite[Ex 2.1.a]{mogensen2010basics}} +% \end{enumerate} + +% For the cases above: + +% \begin{enumerate} +% \item is the complement of each language regular as well? +% \item is their union regular? +% \item is their intersection regular? +% \item is their difference regular? +% \end{enumerate} + + +% \begin{solution} +% The DFA constructions: + +% \paragraph*{The set of strings over \(\Sigma = \{0, 1\}\) such that the +% number of 0's is a multiple of 2, and the number of 1's is a +% multiple of 3.} The DFA states count the number of 0's and 1's modulo +% 2 and 3 respectively. The transitions correspond to incrementing +% modulo 2 and 3. A state \(q_{n, m}\) represents the set of words +% such that the number of 0's \(= n \mod 2\) and the number of 1's \(= +% m \mod 3\). + +% The state \(q_{0, 0}\) is the desired initial \emph{and} +% final state (why?). + +% The transitions are: +% \begin{align*} +% q_{0, 0} \xrightarrow{0} q_{1, 0} & & q_{0, 0} \xrightarrow{1} q_{0, 1} \\ +% q_{0, 1} \xrightarrow{0} q_{1, 1} & & q_{0, 1} \xrightarrow{1} q_{0, 2} \\ +% q_{0, 2} \xrightarrow{0} q_{1, 2} & & q_{0, 2} \xrightarrow{1} q_{0, 0} \\ +% q_{1, 0} \xrightarrow{0} q_{0, 0} & & q_{1, 0} \xrightarrow{1} q_{1, 1} \\ +% q_{1, 1} \xrightarrow{0} q_{0, 1} & & q_{1, 1} \xrightarrow{1} q_{1, 2} \\ +% q_{1, 2} \xrightarrow{0} q_{0, 2} & & q_{1, 2} \xrightarrow{1} q_{1, 0} +% \end{align*} + +% In this manner, DFA states represent a \emph{finite partitioning} of the +% set of words \(\Sigma^*\), and the transitions define how these +% partitions correspond to each other. + +% \textbf{Extra}: This is made more apparent in the algorithms for DFA +% minimization and equivalence checking and makes DFAs as interesting as +% they are theoretically. See +% \href{https://en.wikipedia.org/wiki/Myhill%E2%80%93Nerode_theorem}{Myhill-Nerode Theorem} +% or +% \href{https://en.wikipedia.org/wiki/Brzozowski_derivative}{Brzozowski derivatives} +% on Wikipedia. + +% \paragraph*{The set of binary strings whose value is less than 42}. + +% First, let's attempt to describe this set as a regular expression. The +% binary representation of 42 is \(101010_2\). We write the expression in +% a natural way, decomposing case by case looking at the bits: + +% \begin{tikzpicture}[node distance = 3cm, on grid] +% \node (leading) {\(0^*\)}; +% \node[below left = 1cm of leading] (leadinglabel) {Leading zeroes}; + +% \node[above right of = leading] (b50) {\(0 \,\Sigma^5\)}; +% \node[below = 0.5cm of b50] (b50label) {\(< 32\)}; + +% \node[below right of = leading] (b51) {\(1\)}; +% \node[below = 0.5cm of b51] (b51label) {\(\geq 32\)}; + +% \node[above right of = b51] (b40) {\(1 \,\Sigma^4\)}; +% \node[below = 0.5cm of b40] (b40label) {\(\geq 48\)}; + +% \node[below right of = b51] (b41) {\(0\)}; +% \node[below = 0.5cm of b41] (b41label) {\(< 48\)}; + +% \node[right of = b41] (dots) {\(\ldots\)}; + +% \path[->] (leading) edge node[above left] {read 5th bit as 0} ( b50) +% (leading) edge (b51) +% (b51) edge (b40) +% (b51) edge node[above right] {read 4th bit as 0, continue} (b41) +% (b41) edge (dots) +% ; +% \end{tikzpicture} + +% And finally collapse this diagram into an expression: +% % +% \begin{gather*} +% L_{\leq 42} = 0^*(0\,\Sigma^5 \mid 1(0(0\,\Sigma^3 \mid 1(0(0\,\Sigma^1 \mid 10))))) +% \end{gather*} + +% The `quick terminating' branches read \(010101\) (one's complement of +% \(42_{10}\)) and the `continuing' branches read \(101010\), i.e. +% \(42_{10}\). This is just a binary search for the number 42! + +% The DFA for this regular expression (not provided) looks nearly +% identical to the diagram above. \(L_{< 42}\) requires only a minor +% change at the end. + +% This can be done more generally, given any binary string, constructing a +% DFA recognizing all binary strings numerically smaller than it. + +% \vspace{1em} + +% For the regularity on applying operations: + +% \begin{enumerate} +% \item the complements are regular. In the DFA, change all final +% states to non-final, and all previously non-final states to final. +% This is an NFA accepting the complement of the original language. +% \item the union is regular. Take the new NFA \((Q_1 \cup Q_2, +% \Sigma, \delta_1 \cup \delta_2, S_1 \cup S_2, F_1 \cup F_2)\) where +% \(A_1 = (Q_1, \Sigma, \delta_1, S_1, F_1)\) and \(A_2 = (Q_2, +% \Sigma, \delta_2, S_2, F_2)\) are the individual DFAs for the +% languages above. This is the `trivial' union of the automata, and +% accepts exactly the words that are in one or both of the languages. +% \item the intersection is regular. The construction is a bit more +% involved. Intuitively, in the way that the union NFA allows +% accepting runs from either automaton, the intersection NFA must +% simulate the runs of both automata together. Given two automata +% \(A_1 = (Q_1, \Sigma, \delta_1, S_1, F_1)\) and \(A_2 = (Q_2, +% \Sigma, \delta_2, S_2, F_2)\) representing the two languages, define +% the intersection NFA \(A_\cap = (Q_1 \times Q_2, \Sigma \times +% \Sigma, \delta, S_1 \times S_2, F_1 \times F_2)\), where \(\delta\) +% is defined as +% \begin{gather*} +% \delta \subseteq (Q_1 \times Q_2) \times (\Sigma \times \Sigma) \times (Q_1 \times Q_2) \\ +% \delta((q_1, q_2), (a_1, a_2), (q_1', q_2')) \iff \delta_1(q_1, a_1, q_1') \land \delta_2(q_2, a_2, q_2')~. +% \end{gather*} +% Thus, \(A_\cap\) operates simultaneously on a state from \(A_1\), +% and a state from \(A_2\). There is a transition from \((q_1, q_2)\) +% to \((q_1', q_2')\) when reading the letters \((a_1, a_2)\) +% simultaneously if the transitions on \emph{both} sides are defined. + +% Discuss and convince yourself that the language of \(A_\cap\) is +% precisely the intersection of the languages of \(A_1\) and \(A_2\). + +% \item the difference is regular. It follows from the complement and +% intersection regularities as \(L_1 \setminus L_2 = L_1 \cap \bar +% L_2\). +% \end{enumerate} + +% This has some interesting consequences! Consider the predicate \(x \geq +% 42 \land x > 14\). We can construct DFAs representing the individual +% conditions, and to represent the logical-and, take their intersection. +% This gives us a way to solve some arithmetic constraints using automata! + +% \textbf{Extra}: Automata (of different kinds) have been extensively +% studied as ways to solve many classes of constraints. Regarding +% arithmetic, there is the seminal result by B\"uchi, Weak Second-order +% Arithmetic and Finite Automata \cite{buchi1960weak}. +% \end{solution} +% \end{exercise} + +% \begin{exercise}{} +% \todo{Optionally, ask them to construct the NFA? But it's too simple} + +% Consider the following NFA, \(A\): +% % +% \begin{center} +% \begin{tikzpicture}[node distance = 2cm, on grid] +% \node[state, initial] (q0) {\(q_0\)}; +% \node[state, right of = q0] (q1) {\(q_1\)}; +% \node[state, right of = q1] (q2) {\(q_2\)}; +% \node[state, accepting, right of = q2] (q3) {\(q_3\)}; + +% \draw[->, loop] (q0) edge node[above] {\(a, b\)} (q0); + +% \path[->] (q0) edge node[above] {\(a\)} (q1) +% (q1) edge node[above] {\(a\)} (q2) +% (q2) edge node[above] {\(a\)} (q3) +% ; + +% \end{tikzpicture} +% \end{center} + +% Describe the language \(L(A)\). Is \(A\) a DFA (barring missing +% transitions)? If not, determinize it using the subset construction seen in +% class +% % +% \footnote{This is called subset / product / powerset construction by +% different authors. Wikipedia for quick reference: +% \url{https://en.wikipedia.org/wiki/Powerset_construction}}. + + +% \begin{solution} +% \(L(A)\) is the set of strings over \(\{a, b\}\) ending in three a's. As +% a regular expression, \(L(A) = (a \mid b)^*aaa\). + +% It is not deterministic due to the two outgoing transitions from \(q_0\) +% labelled \(a\). + +% For the subset construction, we define the power set automaton \(A' = +% (2^Q, \Sigma, \delta', S', F')\), where \(Q = \{q_0, q_1, q_2, q_3\}\). + +% The set of initial states \(S'\) only contains the singleton +% \(\{q_0\}\), as we start in solely the initial state \(q_0\). So, \(S' = +% \{\{q_0\}\}\). + +% The set of final states \(F'\) is any set that contains a final state, +% here only \(q_3\). Thus, \(F' = \{\{s \in 2^Q \mid q_3 \in s\}\}\). This +% corresponds to `angelic' executions, i.e., a word is accepted if there +% is \emph{a} run of it that leads to a final state. + +% We draw the set of states \(2^Q\). The states are represented compactly +% due to space constraints: \(s_{23}\) represents the set \(\{q_2, +% q_3\}\), for example. The empty set is marked by \(s_\emptyset\). +% % +% \begin{center} +% \begin{tikzpicture}[node distance = 2cm, on grid] + +% \node[state] (q0000) {\(s_\emptyset\)}; +% \node[state, right of = q0000] (q0001) {\(s_0\)}; +% \node[state, right of = q0001] (q0011) {\(s_{01}\)}; +% \node[state, right of = q0011] (q0010) {\(s_1\)}; + +% \node[state, below of = q0000] (q0100) {\(s_2\)}; +% \node[state, right of = q0100] (q0101) {\(s_{02}\)}; +% \node[state, right of = q0101] (q0111) {\(s_{012}\)}; +% \node[state, right of = q0111] (q0110) {\(s_{01}\)}; + +% \node[state, accepting, below of = q0100] (q1100) {\(s_{23}\)}; +% \node[state, accepting, right of = q1100] (q1101) {\(s_{023}\)}; +% \node[state, accepting, right of = q1101] (q1111) {\(s_{0123}\)}; +% \node[state, accepting, right of = q1111] (q1110) {\(s_{123}\)}; + +% \node[state, accepting, below of = q1100] (q1000) {\(s_3\)}; +% \node[state, accepting, right of = q1000] (q1001) {\(s_{03}\)}; +% \node[state, accepting, right of = q1001] (q1011) {\(s_{013}\)}; +% \node[state, accepting, right of = q1011] (q1010) {\(s_{13}\)}; + +% \node[above = 1.5cm of q0001] (start) {start}; +% \draw[->] (start) -- (q0001); + +% \end{tikzpicture} +% \end{center} + +% % not relevant anymore +% % The states are drawn as a +% % \href{https://en.wikipedia.org/wiki/Karnaugh_map}{Karnaugh Map} on 4 +% % bits for consistency. Each adjacent cell differs by only `one bit'. +% % + +% Finally, we add the transitions corresponding to the outgoing +% transitions \(t_0 - t_5\). We define the new transition relation as +% % +% \begin{gather*} +% \forall\; s, s' \in 2^Q.\; \delta'(s, a, s') \iff s' = \cup \{p \mid \exists\; q \in s.\; \delta(q, a, p)\} +% \end{gather*} +% Example: what is the transition from the set \(\{q_0, q_2\}\) on \(a\)? +% From \(q_0\) we can transition to \(q_0\) (by \(t_0\)) or to \(q_1\) (by +% \(t_2\)), and from \(q_2\), we can only transition to \(q_3\) (by +% \(t_4\)). Thus, we have the transition \(\{q_0, q_2\} \xrightarrow{a} +% \{q_0, q_1\} \cup \{q_3\}\), or in the notation as above, \(s_{02} +% \xrightarrow{a} s_{013}\). We do this for every state and every letter +% (\(16 \times 2\)). + +% The resulting automaton is too complicated to draw fully while +% maintaining any readability! Below, we draw all the relevant +% transitions. Any missing transitions go to the dead state +% \(s_\emptyset\) (marked by the input `several'). +% % +% % \begin{center} +% % \begin{tikzpicture}[node distance = 2cm, on grid] + +% % \node[state] (q0000) {\(s_\emptyset\)}; +% % \node[state, right of = q0000] (q0001) {\(s_0\)}; +% % \node[state, right of = q0001] (q0011) {\(s_{01}\)}; +% % \node[state, right of = q0011] (q0010) {\(s_1\)}; + +% % \node[state, below of = q0000] (q0100) {\(s_2\)}; +% % \node[state, right of = q0100] (q0101) {\(s_{02}\)}; +% % \node[state, right of = q0101] (q0111) {\(s_{012}\)}; +% % \node[state, right of = q0111] (q0110) {\(s_{12}\)}; + +% % \node[state, accepting, below of = q0100] (q1100) {\(s_{23}\)}; +% % \node[state, accepting, right of = q1100] (q1101) {\(s_{023}\)}; +% % \node[state, accepting, right of = q1101] (q1111) {\(s_{0123}\)}; +% % \node[state, accepting, right of = q1111] (q1110) {\(s_{123}\)}; + +% % \node[state, accepting, below of = q1100] (q1000) {\(s_3\)}; +% % \node[state, accepting, right of = q1000] (q1001) {\(s_{03}\)}; +% % \node[state, accepting, right of = q1001] (q1011) {\(s_{013}\)}; +% % \node[state, accepting, right of = q1011] (q1010) {\(s_{13}\)}; + +% % \node[above = 1.5cm of q0001] (start) {start}; +% % \draw[->] (start) -- (q0001); + +% % \path[->, color = red] +% % (q0001.45) edge[out=0, in=90, loop] node[above] {\calpha} (q0001.45) +% % (q0011.45) edge[out=0, in=90, loop] node[above] {\calpha} (q0011.45) +% % (q0101.45) edge[out=0, in=90, loop] node[above] {\calpha} (q0101.45) +% % (q0111.45) edge[out=0, in=90, loop] node[above] {\calpha} (q0111.45) +% % (q1111.45) edge[out=0, in=90, loop] node[above] {\calpha} (q1111.45) +% % (q1101.45) edge[out=0, in=90, loop] node[above] {\calpha} (q1101.45) +% % (q1001.45) edge[out=0, in=90, loop] node[above] {\calpha} (q1001.45) +% % (q1011.45) edge[out=0, in=90, loop] node[above] {\calpha} (q1011.45) +% % ; + + +% % \path[->, color = blue] +% % (q0001.225) edge[out=180, in=270, loop] node[below left] {\cbeta} (q0001.225) +% % (q0011.225) edge[out=180, in=270, loop] node[below left] {\cbeta} (q0011.225) +% % (q0101.225) edge[out=180, in=270, loop] node[below left] {\cbeta} (q0101.225) +% % (q0111.225) edge[out=180, in=270, loop] node[below left] {\cbeta} (q0111.225) +% % (q1111.225) edge[out=180, in=270, loop] node[below left] {\cbeta} (q1111.225) +% % (q1101.225) edge[out=180, in=270, loop] node[below left] {\cbeta} (q1101.225) +% % (q1001.225) edge[out=180, in=270, loop] node[below left] {\cbeta} (q1001.225) +% % (q1011.225) edge[out=180, in=270, loop] node[below left] {\cbeta} (q1011.225) +% % ; + + +% % \path[->, color = OliveGreen] +% % (q0001) edge[] node[below] {\cgamma} (q0011) +% % (q0011.315) edge[in=-90, out=0, loop] node[below] {\cgamma} (q0011.315) +% % (q0101) edge[] node[below] {\cgamma} (q0111) +% % (q0111.315) edge[in=-90, out=0, loop] node[below] {\cgamma} (q0111.315) +% % (q1111.315) edge[in=-90, out=0, loop] node[below] {\cgamma} (q1111.315) +% % (q1101) edge[] node[below] {\cgamma} (q1111) +% % (q1001) edge[] node[below] {\cgamma} (q1011) +% % (q1011.315) edge[in=-90, out=0, loop] node[below] {\cgamma} (q1011.315) +% % ; + +% % \path[->, color = purple] +% % (q0010) edge[] node[right] {\ceta} (q0110) +% % (q0011) edge[] node[right] {\ceta} (q0111) +% % (q0110.135) edge[out=90, in=180, loop] node[above] {\ceta} (q0110.135) +% % (q0111.135) edge[out=90, in=180, loop] node[above] {\ceta} (q0111.135) +% % (q1010) edge[] node[right] {\ceta} (q1110) +% % (q1011) edge[] node[right] {\ceta} (q1111) +% % (q1110.135) edge[out=90, in=180, loop] node[above] {\ceta} (q1110.135) +% % (q1111.135) edge[out=90, in=180, loop] node[above] {\ceta} (q1111.135) +% % ; + +% % \path[->, color = orange] +% % (q0100) edge[] node[left] {\cpi} (q1100) +% % (q0101) edge[] node[left] {\cpi} (q1101) +% % (q0110) edge[] node[left] {\cpi} (q1110) +% % (q0111) edge[] node[left] {\cpi} (q1111) +% % (q1100.0) edge[in=0, out=90, loop] node[above right] {\cpi} (q1100.0) +% % (q1101.0) edge[in=0, out=90, loop] node[above right] {\cpi} (q1101.0) +% % (q1110.0) edge[in=0, out=90, loop] node[above right] {\cpi} (q1110.0) +% % (q1111.0) edge[in=0, out=90, loop] node[above right] {\cpi} (q1111.0) +% % ; + +% % \end{tikzpicture} +% % \end{center} + +% % \begin{center} +% % \begin{tikzpicture}[node distance = 2cm, on grid] + +% % \node[state] (q0000) {\(s_\emptyset\)}; +% % \node[state, right of = q0000] (q0001) {\(s_0\)}; +% % \node[state, right of = q0001] (q0011) {\(s_{01}\)}; +% % \node[state, right of = q0011] (q0010) {\(s_1\)}; + +% % \node[state, below of = q0000] (q0100) {\(s_2\)}; +% % \node[state, right of = q0100] (q0101) {\(s_{02}\)}; +% % \node[state, right of = q0101] (q0111) {\(s_{012}\)}; +% % \node[state, right of = q0111] (q0110) {\(s_{12}\)}; + +% % \node[state, accepting, below of = q0100] (q1100) {\(s_{23}\)}; +% % \node[state, accepting, right of = q1100] (q1101) {\(s_{023}\)}; +% % \node[state, accepting, right of = q1101] (q1111) {\(s_{0123}\)}; +% % \node[state, accepting, right of = q1111] (q1110) {\(s_{123}\)}; + +% % \node[state, accepting, below of = q1100] (q1000) {\(s_3\)}; +% % \node[state, accepting, right of = q1000] (q1001) {\(s_{03}\)}; +% % \node[state, accepting, right of = q1001] (q1011) {\(s_{013}\)}; +% % \node[state, accepting, right of = q1011] (q1010) {\(s_{13}\)}; + +% % \node[above = 1.5cm of q0001] (start) {start}; +% % \draw[->] (start) -- (q0001); + +% % % (q0000) +% % % (q0001) +% % % (q0010) +% % % (q0011) +% % % (q0100) +% % % (q0101) +% % % (q0110) +% % % (q0111) +% % % (q1000) +% % % (q1001) +% % % (q1010) +% % % (q1011) +% % % (q1100) +% % % (q1101) +% % % (q1110) +% % % (q1111) + +% % \path[->] +% % (q0000) edge[loop] node[above] {\(a\)} (q0000) +% % (q0001) edge[] node[above] {\(a\)} (q0011) +% % (q0010) edge[] node[above] {\(a\)} (q0100) +% % (q0011) edge[] node[above] {\(a\)} (q0111) +% % (q0100) edge[] node[above] {\(a\)} (q1000) +% % (q0101) edge[] node[above] {\(a\)} (q1011) +% % (q0110) edge[] node[above] {\(a\)} (q1100) +% % (q0111) edge[] node[above] {\(a\)} (q1111) +% % (q1000) edge[] node[above] {\(a\)} (q0000) +% % (q1001) edge[] node[above] {\(a\)} (q0011) +% % (q1010) edge[] node[above] {\(a\)} (q0100) +% % (q1011) edge[] node[above] {\(a\)} (q0111) +% % (q1100) edge[] node[above] {\(a\)} (q1000) +% % (q1101) edge[] node[above] {\(a\)} (q1011) +% % (q1110) edge[] node[above] {\(a\)} (q1100) +% % (q1111) edge[loop] node[above] {\(a\)} (q1111) +% % ; + +% % \path[->] +% % (q0000) edge[loop] node[right] {\(b\)} (q0000) +% % (q0001) edge[] node[right] {\(b\)} (q0001) +% % (q0010) edge[] node[right] {\(b\)} (q0000) +% % (q0011) edge[] node[right] {\(b\)} (q0001) +% % (q0100) edge[] node[right] {\(b\)} (q0000) +% % (q0101) edge[] node[right] {\(b\)} (q0001) +% % (q0110) edge[] node[right] {\(b\)} (q0000) +% % (q0111) edge[] node[right] {\(b\)} (q0001) +% % (q1000) edge[] node[right] {\(b\)} (q0000) +% % (q1001) edge[] node[right] {\(b\)} (q0001) +% % (q1010) edge[] node[right] {\(b\)} (q0000) +% % (q1011) edge[] node[right] {\(b\)} (q0001) +% % (q1100) edge[] node[right] {\(b\)} (q0000) +% % (q1101) edge[] node[right] {\(b\)} (q0001) +% % (q1110) edge[] node[right] {\(b\)} (q0000) +% % (q1111) edge[loop] node[right] {\(b\)} (q0001) +% % ; + + + +% % \end{tikzpicture} +% % \end{center} + +% \begin{center} +% \begin{tikzpicture}[node distance = 2cm, on grid] + +% \node[state] (q0000) {\(s_\emptyset\)}; +% \node[state, right of = q0000] (q0001) {\(s_0\)}; +% \node[state, right of = q0001] (q0011) {\(s_{01}\)}; +% \node[state, right of = q0011] (q0010) {\(s_1\)}; + +% \node[state, below of = q0000] (q0100) {\(s_2\)}; +% \node[state, right of = q0100] (q0101) {\(s_{02}\)}; +% \node[state, right of = q0101] (q0111) {\(s_{012}\)}; +% \node[state, right of = q0111] (q0110) {\(s_{12}\)}; + +% \node[state, accepting, below of = q0100] (q1100) {\(s_{23}\)}; +% \node[state, accepting, right of = q1100] (q1101) {\(s_{023}\)}; +% \node[state, accepting, right of = q1101] (q1111) {\(s_{0123}\)}; +% \node[state, accepting, right of = q1111] (q1110) {\(s_{123}\)}; + +% \node[state, accepting, below of = q1100] (q1000) {\(s_3\)}; +% \node[state, accepting, right of = q1000] (q1001) {\(s_{03}\)}; +% \node[state, accepting, right of = q1001] (q1011) {\(s_{013}\)}; +% \node[state, accepting, right of = q1011] (q1010) {\(s_{13}\)}; + +% \node[above = 1.5cm of q0001] (start) {start}; +% \draw[->] (start) -- (q0001); + +% \node[left = 1.5cm of q0000] (several) {several}; +% \draw[->] (several) -- (q0000); + +% % (q0000) +% % (q0001) +% % (q0010) +% % (q0011) +% % (q0100) +% % (q0101) +% % (q0110) +% % (q0111) +% % (q1000) +% % (q1001) +% % (q1010) +% % (q1011) +% % (q1100) +% % (q1101) +% % (q1110) +% % (q1111) + +% \path[->] +% (q0000.north) edge[out=45, in=135, loop] node[above] {\(a, b\)} (q0000.north) +% (q0001) edge[] node[above] {\(a\)} (q0011) +% (q0010) edge[] node[above] {\(a\)} (q0100) +% (q0011) edge[] node[right] {\(a\)} (q0111) +% (q0100) edge[bend right] node[left] {\(a\)} (q1000) +% (q0101) edge[] node[above] {\(a\)} (q1011) +% (q0110) edge[] node[above] {\(a\)} (q1100) +% (q0111) edge[] node[right] {\(a\)} (q1111) +% (q1001) edge[] node[above] {\(a\)} (q0011) +% (q1010) edge[] node[above] {\(a\)} (q0100) +% (q1011) edge[bend right] node[right] {\(a\)} (q0111) +% (q1100) edge[] node[right] {\(a\)} (q1000) +% (q1101) edge[] node[above] {\(a\)} (q1011) +% (q1110) edge[bend left] node[below] {\(a\)} (q1100) +% (q1111.315) edge[out=270, in=0, loop] node[right] {\(a\)} (q1111.315) +% ; + +% \path[->] +% (q0001.45) edge[out=0, in=90, loop] node[right] {\(b\)} (q0001.45) +% (q0011.north) edge[bend right] node[above] {\(b\)} (q0001.north) +% (q0101) edge[] node[right] {\(b\)} (q0001) +% (q0111) edge[] node[right] {\(b\)} (q0001) +% (q1001) edge[bend left] node[left] {\(b\)} (q0001) +% (q1011) edge[] node[right] {\(b\)} (q0001) +% (q1101) edge[bend left = 45] node[above left] {\(b\)} (q0001) +% (q1111) edge[] node[right] {\(b\)} (q0001) +% ; +% \end{tikzpicture} +% \end{center} + +% After removing the remaining dead states and transitions for visibility, +% we can see the DFA: + +% \begin{center} +% \begin{tikzpicture}[node distance = 2cm, on grid] +% \node[state, initial] (q0) {\(s_0\)}; +% \node[state, right of = q0] (q1) {\(s_{01}\)}; +% \node[state, right of = q1] (q2) {\(s_{012}\)}; +% \node[state, accepting, right of = q2] (q3) {\(s_{0123}\)}; + +% \draw[->, in=180, out=90, loop] (q0.135) edge node[above] {\(b\)} (q0.135); +% \draw[->, in=45, out=-45, loop] (q3.0) edge node[right] {\(a\)} (q3.0); + +% \path[->] (q0) edge node[above] {\(a\)} (q1) +% (q1) edge node[above] {\(a\)} (q2) +% (q2) edge node[above] {\(a\)} (q3) +% ; +% \path[->] (q1) edge[bend left] node[below left] {\(b\)} (q0) +% (q2) edge[bend right] node[above] {\(b\)} (q0) +% (q3) edge[bend left] node[below] {\(b\)} (q0) +% ; + +% \end{tikzpicture} +% \end{center} + +% This is an `eager' version of our original NFA. It accepts the same +% language. However, since it cannot be `non-deterministic' by +% definition, every time it comes across an \(a\), it must `guess' that it +% is the first of three ending \(a\)'s, and backtrack if it is not the +% case. + +% % Although equally powerful, the DFA more accurately captures how a +% % human-written parser would operate in practice on this language. + +% \end{solution} +% \end{exercise} diff --git a/info/exercises/src/ex-01/ex/languages.tex b/info/exercises/src/ex-01/ex/languages.tex new file mode 100644 index 0000000..6ae0821 --- /dev/null +++ b/info/exercises/src/ex-01/ex/languages.tex @@ -0,0 +1,140 @@ + +\section{Languages and Automata} + +\begin{exercise}{} + + For each of the following expressions on sets of words, match them to a + property \(P\) that characterizes them, i.e., the language is exactly the set + of words \(\{ w \ \mid P(w) \}\). + + Languages: + \begin{enumerate} + \item \(\{a,ab\}^*\) + \item \(\{aa\}^* \cup \{aaa\}^*\) + \item \(a^+b^+\) + \end{enumerate} + + Predicates in set notation: + \begin{enumerate} + \renewcommand{\theenumi}{\Alph{enumi}} + \item \(\{w \mid \forall i. 0 \le i \le |w| \land w_{(i)} = b \implies (i > 0 \land w_{(i - 1)} = a)\}\) % 1 + \item \(\{w \mid \forall i. w_{(i)} = b \implies w_{(i - 1)} = a\}\) % wrong + \item \(\{w \mid \exists i. 0 < i < |w| \land w_{(i)} = b \land w_{(i - 1)} = a\}\) % wrong + \item \(\{w \mid (|w| = 0 \mod 2 \lor |w| = 0 \mod 3) \land \forall i. 0 \leq i < |w| \implies w_{(i)} = a\}\) % 2 + \item \(\{w \mid \forall i. 0 \le i \le |w| \land w_{(i)} = a \implies w_{(i + 1)} = b\}\) % wrong + \item \(\{w \mid \exists i. 0 < i < |w| - 1 \land + (\forall y. 0 \leq y \leq i \implies w_{(y)} = a) \land (\forall y. i < y < |w| \implies w_{(y)} = b) \}\) % 3 + \end{enumerate} + + + + \begin{solution} + + \(1 \mapsto A, 2 \mapsto D, 3 \mapsto F\). + + We prove the first case as an example. + + \begin{equation*} + \{a,ab\}^* = \{w \mid \forall i. 0 \le i \le |w| \land w_{(i)} = b \implies (i > 0 \land w_{(i - 1)} = a)\} + \end{equation*} + + We must prove both directions, i.e. that \(\{a,ab\}^* \subseteq \{w \mid + P(w)\}\) and that \(\{w \mid P(w)\} \subseteq \{a,ab\}^*\). + + \noindent + \textbf{Forward}: \(\{a,ab\}^* \subseteq \{w \mid P(w)\}\): + + We must show that for all \(w \in \{a,ab\}^*\), \(P(w)\) holds. For any \(i + \in \naturals\), given that \(0 \le i \le |w| \land w_{(i)} = b\), we need + to show that \(i > 0 \land w_{(i - 1)} = a\). + + From the definition of \(*\) on sets of words, we know that there must exist + \(n < |w|\) words \(w_1, \ldots, w_n \in \{a, ab\}\) such that \(w = w_1 + \ldots w_n\). The index \(i\) must be in the range of one of these words, + i.e. there exist \(1 \leq m \leq n\) and \(0 \leq j < |w_m|\) such that + \(w_{(i)} = w_{m(j)}\). + + We know that \(w_{(i)} = b\) and \(w_{m} \in \{a, ab\}\) by assumption. The + case \(w_m = a\) is a contradiction, since it cannot contain \(b\). Thus, + \(w_m = ab\). We know that \(w_{(i)} = w_{m(j)} = b\), so \(j = 1\). Thus, + \(w_{(i - 1)} = w_{m(j - 1)} = w_{m(0)} = a\), as required. Since \(i - 1 + \geq 0\), being an index into \(w\), \(i > 0\) holds as well. Hence, + \(P(w)\) holds. + + \noindent + \textbf{Backward}: \(\{w \mid P(w)\} \subseteq \{a,ab\}^*\): + + We must show that for all \(w\) such that \(P(w)\) holds, \(w \in + \{a,ab\}^*\). We know by definition of \(*\) again, that \(w \in \{a, + ab\}*\) if and only if there exist \(n < |w|\) words \(w_1, \ldots, w_n \in + \{a, ab\}\) such that \(w = w_1 \ldots w_n\). We attempt to show that if + \(P(w)\) holds, then \(w\) admits such a decomposition. + + We proceed by induction on the length of \(w\). + + \noindent + \textit{Induction Case \(|w| = 0\)}: The empty word has a decomposition \(w = + \epsilon\) (with \(n = 0\) in the decomposition). QED. + + \noindent + \textit{Induction Case \(|w| = 1\)}: The word \(w\) is either \(a\) or \(b\). We know + that \(P(w)\) holds, so \(w = a\) (why?). The decomposition is \(w = a\), + with \(n = 1\) and \(w_1 = a\). QED. + + \noindent + \textit{Induction Case \(|w| > 1\)}: + + Induction hypothesis: for all words \(v\) such that \(|v| < |w|\) and + \(P(v)\) holds, \(v\) admits a decomposition into words in \(\{a, ab\}\), + and thus \(v \in \{a, ab\}^*\). + + We need to show that if \(P(w)\) holds, then \(w\) admits such a + decomposition as well. Split the proof based on the first two characters of + \(w\). There are four possibilities. We give the name \(v\) to the rest of + \(w\). + + \begin{enumerate} + \item \(w = aav\): \(P(w)\) holds, so \(\forall i. 0 \le i \le |w| \land + w_{(i)} = b \implies (i > 0 \land w_{(i - 1)} = a)\). In particular, we + can restrict to \(i > 1\) as + + \begin{equation*} + \forall i. 2 \le i \le |w| \land w_{(i)} = b \implies (i > 0 \land w_{(i - 1)} = a) + \end{equation*} + + but \(w_{(i)}\) for \(i \geq 2\) is simply \(v_{(i - 2)}\). Rewriting: + + \begin{equation*} + \forall i. 2 \le i \le |w| \land v_{(i - 2)} = b \implies (i > 0 \land v_{(i - 3)} = a) + \end{equation*} + + Finally, since the statement holds for all \(i\), we can replace \(i\) by + \(i + 2\) without loss of generality, using \(|v| = |w| - 2\): + + \begin{equation*} + \forall i. 0 \le i \le |v| \land v_{(i)} = b \implies (i > 0 \land v_{(i - 1)} = a) + \end{equation*} + + This is precisely the statement \(P(v)\), so by the induction hypothesis, + \(v\) has a decomposition into words in \(\{a, ab\}\), \(v = v_1\ldots + v_m\) for some \(m < |v|\) and \(v_i \in \{a, ab\}\). + + We can now construct a decomposition for \(w\), \(w = w_1\ldots w_{m+2}\) + such that \(w_1 = a\), \(w_2 = a\), and \(w_{i + 2} = v_i\) for \(1 \le i + \le m\). Since \(m < |v|\) and \(|v| = |w| - 2\), \(m + 2 < |w|\). QED. + + \item \(w = aab\): by the same argument as the previous case, \(v\) has a decomposition + into words in \(\{a, ab\}\), \(v = v_1\ldots v_m\) for some \(m < |v|\) + and \(v_i \in \{a, ab\}\). + + We can similarly construct a decomposition for \(w\), \(w = w_1\ldots + w_{m+1}\) such that \(w_1 = ab\) and \(w_{i + 1} = v_i\) for \(1 \le i \le + m\). Since \(m < |v|\) and \(|v| = |w| - 2\), in particular \(m + 1 < + |w|\). QED. + + \item \(w = bav\) or \(w = bbv\): \(P(w)\) cannot hold (set \(i = 0\)), so + the statement is vacuously true. + \end{enumerate} + \end{solution} + +\end{exercise} diff --git a/info/exercises/src/ex-01/ex/lexer.tex b/info/exercises/src/ex-01/ex/lexer.tex new file mode 100644 index 0000000..cf5ece9 --- /dev/null +++ b/info/exercises/src/ex-01/ex/lexer.tex @@ -0,0 +1,154 @@ + +% flavour text for lexer constructions + +\section{Lexing} + +% In the lectures, we have seen how to manually construct a lexer for small +% regular expressions. We often use tools that generate lexers from regular +% expressions. You will see one such tool, Silex, while building the Amy lexer. + +% % about automata for lexing +% Lexing frameworks process the description of tokens for a given language, and +% may use a variety of techniques to construct the final lexer. The result is a +% program that accepts a string and returns a list of tokens. One way to do this +% automatically is by constructing and composing automata. + +Consider a simple arithmetic language that allows you to compute one arithmetic +expression, construct conditionals, and let-bind expressions. An example program +is: + +\begin{lstlisting} + let x = 3 in + let y = ite (x > 0) (x * x) 0 in + (2 * x) + y +\end{lstlisting} + +The lexer for this language must recognize the following tokens: + +\begin{align*} + \texttt{keyword}: &\quad \texttt{let} \mid \texttt{in} \mid \texttt{ite}\\ + \texttt{op}: &\quad \texttt{+} \mid \texttt{-} \mid \texttt{*} \mid \texttt{/} \\ + \texttt{comp}: &\quad \texttt{>} \mid \texttt{<} \mid \texttt{==} \mid \texttt{<=} \mid \texttt{>=} \\ + \texttt{equal}: &\quad \texttt{=} \\ + \texttt{lparen}: &\quad \texttt{(} \\ + \texttt{rparen}: &\quad \texttt{)} \\ + \texttt{id}: &\quad letter \cdot (letter \mid digit)^* \\ + \texttt{number}: &\quad digit^+ \\ + \texttt{skip}: &\quad \texttt{whitespace} +\end{align*} + +For simplicity, \(letter\) is a shorthand for the set of all English lowercase +letters \(\{a - z\}\) and \(digit\) is a shorthand for the set of all decimal +digits \(\{0 - 9\}\). + +% \todo{if we allow an \texttt{ite} keyword with operators, we can ask them how +% chained operators would be parsed, eg: \texttt{<===>=<===}. Is this interesting?} + +\begin{exercise}{} + For each of the tokens above, construct an NFA that recognizes strings matching + its regular expression. + + \begin{solution} + The construction is similar in each case, following translation of regular + expressions to automata. For example: + + \begin{itemize} + \item \texttt{keyword}: \texttt{let} $\mid$ \texttt{in} $\mid$ \texttt{ite} + \begin{center} + \begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto] + \node[state,initial] (q_0) {$q_0$}; + % + \node[state] (ql_1) [above right=of q_0] {$q_l$}; + \node[state] (ql_2) [right=of ql_1] {$q_e$}; + \node[state,accepting] (ql_3) [right=of ql_2] {$q_{let}$}; + % + \node[state] (qin_1) [right=of q_0] {$q_{i1}$}; + \node[state] (qin_2) [right=of qin_1] {$q_{in}$}; + % + \node[state] (qite_1) [below right=of q_0] {$q_{i2}$}; + \node[state] (qite_2) [right=of qite_1] {$q_t$}; + \node[state,accepting] (qite_3) [right=of qite_2] {$q_{ite}$}; + % + \path[->] + (q_0) edge node {\texttt{l}} (ql_1) + (ql_1) edge node {\texttt{e}} (ql_2) + (ql_2) edge node {\texttt{t}} (ql_3) + % + (q_0) edge node {\texttt{i}} (qin_1) + (qin_1) edge node {\texttt{n}} (qin_2) + % + (q_0) edge node {\texttt{i}} (qite_1) + (qite_1) edge node {\texttt{t}} (qite_2) + (qite_2) edge node {\texttt{e}} (qite_3) + ; + \end{tikzpicture} + \end{center} + + \item \texttt{id}: \texttt{letter} $\cdot$ (\texttt{letter} $\mid$ \texttt{digit})$^*$ + \begin{center} + \begin{tikzpicture}[shorten >=1pt,node distance=3cm,on grid,auto] + \node[state,initial] (q_0) {$q_0$}; + % + \node[state] (q1) [accepting, right=of q_0] {$q_1$}; + % + \path[->] + (q_0) edge node {\texttt{letter}} (q1) + (q1) edge[loop above] node {\texttt{letter}} (q1) + (q1) edge[loop below] node {\texttt{digit}} (q1) + ; + \end{tikzpicture} + \end{center} + \end{itemize} + + The other cases are similar. + \end{solution} +\end{exercise} + +A lexer is constructed by combining the NFAs for each of the tokens in parallel, +assuming maximum munch. The resulting token is the first NFA in the token order +that accepts a prefix of the string. Thus, tokens listed first have higher +priority. We then continue lexing the remaining string. You may assume that the +lexer drops any \texttt{skip} tokens. + +\begin{exercise}{} + + For each of the following strings, write down the sequence of tokens that + would be produced by the constructed lexer, if it succeeds. + + \begin{enumerate} + \item \texttt{let x = 5 in x + 3} + \item \texttt{let5x2} + \item \texttt{xin} + \item \texttt{==>} + \item \texttt{<===><==} + \end{enumerate} + + \begin{solution} + \begin{enumerate} + \item \texttt{[keyword("let"), id("x"), equal, number("5"), keyword("in"), id("x"), op("+"), number("3")]} + \item \texttt{[id("let"), number("5"), id("x2")]} + \item \texttt{[id("xin")]} + \item \texttt{[comp("=="), op(">")]} + \item \texttt{[comp("<="), comp("=="), op(">"), comp("<="), equal("=")]} + \end{enumerate} + \end{solution} + +\end{exercise} + + +\begin{exercise}{} + Construct a string that would be lexed differently if we ran the NFAs in parallel + and instead of using token priority, simply picked the longest match. + + \begin{solution} + There are many possible solutions. The key is to notice which tokens have + overlapping prefixes. + + An example is \texttt{letx1in}, which would be lexed as + \texttt{[keyword("let"), id("x1"), keyword("in")]} if we check acceptance in + order of priority, but as \texttt{[id("letx1in")]} if we run them in + parallel. + \end{solution} +\end{exercise} + + \ No newline at end of file diff --git a/info/exercises/src/ex-01/main.tex b/info/exercises/src/ex-01/main.tex new file mode 100644 index 0000000..cf01061 --- /dev/null +++ b/info/exercises/src/ex-01/main.tex @@ -0,0 +1,30 @@ +\documentclass[a4paper]{article} + +\input{../macro} + +% \printanswers + +\title{CS 320 \\ Computer Language Processing\\Exercises: Weeks 1 and 2} +\author{} +\date{February 28, 2025} + +\begin{document} +\maketitle + + % languages as sets + \input{ex/languages.tex} + + % regex + + % automata + \input{ex/dfa.tex} + + % regex to automata + + % constructing lexers + \input{ex/lexer.tex} + +\bibliographystyle{plain} +\bibliography{../biblio} + +\end{document} diff --git a/info/exercises/src/macro.tex b/info/exercises/src/macro.tex new file mode 100644 index 0000000..24c8c53 --- /dev/null +++ b/info/exercises/src/macro.tex @@ -0,0 +1,114 @@ +% macro.tex +% common to all exercises + +\usepackage[dvipsnames]{xcolor} +\usepackage{amsmath, amssymb} +\usepackage{xspace} +\usepackage[colorlinks]{hyperref} +\usepackage{tabularx} + +% for drawing +\usepackage{tikz} +\usetikzlibrary{automata, arrows, shapes, positioning} + +\usepackage{forest} + +\usepackage{bussproofs, bussproofs-extra} + +% for code +\usepackage{listings} + +\definecolor{dkgreen}{rgb}{0,0.6,0} +\definecolor{gray}{rgb}{0.5,0.5,0.5} +\definecolor{mauve}{rgb}{0.58,0,0.82} + +\lstdefinestyle{scalaStyle}{ + frame=tb, + language=scala, + aboveskip=3mm, + belowskip=3mm, + showstringspaces=false, + columns=flexible, + basicstyle=\small\ttfamily, + numbers=none, + numberstyle=\tiny\color{gray}, + keywordstyle=\color{blue}, + commentstyle=\color{dkgreen}, + stringstyle=\color{mauve}, + frame=none, + breaklines=true, + breakatwhitespace=true, + tabsize=2, +} + +\lstset{style=scalaStyle} + +% lstinline in math mode +% https://tex.stackexchange.com/a/127018 + +\usepackage{letltxmacro} +\newcommand*{\SavedLstInline}{} +\LetLtxMacro\SavedLstInline\lstinline +\DeclareRobustCommand*{\lstinline}{% + \ifmmode + \let\SavedBGroup\bgroup + \def\bgroup{% + \let\bgroup\SavedBGroup + \hbox\bgroup + }% + \fi + \SavedLstInline +} + +% for exercises + +\newcounter{exercisenum} +\newcommand{\theexercise}{\arabic{exercisenum}} + +\usepackage{marginnote} +\newenvironment{exercise}[1]{\refstepcounter{exercisenum}\paragraph*{Exercise \theexercise}{\reversemarginpar\marginnote{#1}}}{} + +\usepackage{verbatim} + +\usepackage{ifthen} +\newboolean{showanswers} +\setboolean{showanswers}{false} +\newcommand{\printanswers}{\setboolean{showanswers}{true}} +\newcommand\suppress[1]{} + +\newcommand\ite[3]{\ifthenelse{\boolean{#1}}% +{#2\suppress{#3}}% +{\suppress{#2}#3}} + + +\newenvironment{solution}{% + \ite{showanswers}{\paragraph*{Solution}}{\expandafter\comment} +}{% + \ite{showanswers}{\hfill\(\square\)}{\expandafter\endcomment} +} + +\newcommand{\note}[1]{\hfill #1} + +\sloppy + +%% actual macros + +% meta +\newcommand{\todo}[1]{\textcolor{red}{[Todo: #1]}} + +\newcommand{\fbowtie}{\mathrel \blacktriangleright \joinrel \mathrel \blacktriangleleft} + +\newcommand{\easy}{\ensuremath{\bigstar}\xspace} +\newcommand{\hard}{\small\ensuremath{\fbowtie}\xspace} + +% real +\newcommand{\naturals}{\mathbb{N}} +\newcommand{\booleans}{\mathbb{B}} + +% church booleans +\newcommand{\tru}{\lstinline|true|\xspace} +\newcommand{\fls}{\lstinline|false|\xspace} + +% proof systems +\newcommand{\natded}{\mathcal{N}} % natural deduction + -- GitLab