https://ocw.cs.pub.ro/ppcarte/ 2026-04-25T14:45:14+03:00 https://ocw.cs.pub.ro/ppcarte/ https://ocw.cs.pub.ro/ppcarte/lib/tpl/bootstrap3/images/favicon.ico text/html 2024-10-06T21:24:45+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2023&rev=1728239085&do=diff Structura curs universitar LFA 2022-2023 * Echipa * Regulament * Lectures * 1. ??? * Labs * 1. Programming introduction - Python * 2. DFA and Regex intro * 3. Regex practice * 4. Regex to DFA * 5. DFA minimisation * 6. DFA to Regex * 7. Closure properties * 8. Pumping lemma * 9. Context-Free Languages * 10. CFG to PDA; Lexers * Proiect * Coding Style * Bonus text/html 2025-09-30T16:37:39+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024&rev=1759239459&do=diff Structura curs universitar LFA 2024-2025 * 1. DFA intro * 2. Programming introduction - Python * 3. Regular Expressions * 4. Regex Representation in Python * 5. Regex to NFA to DFA conversions * 6. DFA Minimisation * 7. Closure properties * 8. Pumping lemma * 9. Push Down Automata * 10. Context-Free Grammars * 11. Closure properties for Context-Free Languages text/html 2017-11-15T14:25:00+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:cfg&rev=1510748700&do=diff Context-Free Grammars Motivation Regular expressions are insufficient in describing the structure of complicated languages (e.g. programming languages). We recall our previously-used example: simple arithmetic expressions with parentheses. <expr> ::= <atom> | <expr> + <expr> | (<expr>) $ G=(V,\Sigma,R,S)$$ V$$ \Sigma\subseteq V$$ R$$ (V\setminus\Sigma)\times V^*$$ V\setminus\Sigma$$ V^*$$ V$$ R$$ S\in V\setminus\Sigma$$ G$$ V=\{S\}$$ \Sigma=\{a,b\}$$ R=\{S\rightarrow aSb, S\rightarrow \eps… text/html 2018-11-07T10:55:06+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:cfgpda&rev=1541580906&do=diff CFG - PDA equivalence We prove that languages generated by Context-Free Grammar coincide with those accepted by PDAs. The proof is done in two steps. CFG to PDA Given a Context-Free grammar $ G=(V,\Sigma,R,S)$ , we build a PDA $ M=(K,\Sigma,\Gamma,\Delta,q_0,Z_0,F)$ which accepts precisely $ L(G)$ . The main idea is:$ \alpha A \beta \Rightarrow_G \alpha \gamma \beta$$ G$$ M$$ S\rightarrow AB$$ A\rightarrow aB \mid bA$$ B\rightarrow bB\mid \epsilon$$ S\Rightarrow AB \Rightarrow bAB \Rightarr… text/html 2018-02-02T18:33:15+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:cfl-prop&rev=1517589195&do=diff Properties of Context-Free Languages Chomsky Normal Form Definition (CNF): $ G$Chomsky Normal Formeach * $ A\rightarrow BC$ where $ A,B,C$ are non-terminals * $ A\rightarrow a$ where $ A$ is a non-terminal and $ a$ is a terminal useless We also prove that: Proposition (CNF): $ L$$ L\neq \emptyset$$ \epsilon\not\in L$$ G$Chomsky Normal Form$ L(G)=L$$ G$$ A\rightarrow B_1 \ldots a\ldots B_n$$ B_i$$ a$$ A\rightarrow B_1\ldots C\ldots B_n$$ C\rightarrow a$$ A\rightarrow B_1 \ldot… text/html 2020-10-19T15:32:25+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:dfa-old&rev=1603110745&do=diff Deterministic automata Motivation In the last lecture, we have shown that, for each regular expression $ e$ , we can construct an NFA $ M$ , such that $ L(e) = L(M)$ . While NFAs are a computational instrument for establishing acceptance, they are not an ideal one. Nondeterminism is difficult to translate per. se. in a programming language. It is also quite inefficient. We illustrate this via our previous example, below. Recall our NFA representation in Haskell:$ w$$ nfa$$ w\in L(nfa)$$ s'$$ … text/html 2020-10-19T15:36:32+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:dfa&rev=1603110992&do=diff Deterministic automata Definition (DFA): Deterministic Finite Automata$ (K,\Sigma,\delta,q_0,F)$$ K$$ \Sigma$$ \Delta : K \times \Sigma \rightarrow K$total Definition (Configuration): configuration$ K\times\Sigma^*$ Definition (One-step move): $ \vdash_M \subseteq (K\times \Sigma^*) \times (K\times \Sigma^*)$one-step * $ (q,w) \vdash_M (q',w')$ iff $ w=cw'$ for some symbol $ c$ of the alphabet, and $ \delta(q,c)=q'$ $ \vdash_M^*$reflexive and transitive closure$ \vdash_M$$ (q,w)… text/html 2021-09-28T13:21:22+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:dictionaries&rev=1632824482&do=diff Dictionaries and hashing A review over hashtables Python dictionaries are hashtable implementations. In short, the instruction: val = d[x] will: * apply the a function $ hash$ on object x, to obtain a bucket $ b$ , ($ hash(x)=b$ ) * buckets are just collections of key-value pairs (often implemented as arrays). Next we search (in linear time) for key $ b$$ hash$ text/html 2022-12-15T11:30:43+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exam&rev=1671096643&do=diff Examen LFA 2021-2022 TBA Examen LFA 2020-2021 Examenul la cursul de Limbaje Formale si Automate valorează 5 puncte. Pentru promovarea examenului este necesar 50% din punctaj (2.5 puncte). Cele 5 puncte vor fi obținute în cadrul a doua probe: * text/html 2021-01-13T10:09:20+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:examen-optional&rev=1610525360&do=diff Examen optional 2020-2021 Miercuri 27 ianuarie, de la ora 10:00, in cadrul ultimului curs de LFA, vom organiza un examen-interviu (oral) optional, la care studentii se pot inscrie pentru a degreva examenul oral din sesiune. Cu acordul studentilor, vom inregistra o parte din interviuri, astfel incat procedura sa poata fi transparenta pentru toti studentii. Cateva observatii vizavi de acest examen: text/html 2021-01-18T16:24:27+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exercise-sheet-1-solution&rev=1610979867&do=diff Exercise sheet I solution $ \{a, b\}$ 1. Definiți câte un AFD pentru limbajul șirurilor care conțin: 1.a. cel puțin doi $ b$ 1.b. exact doi $ b$ 1.c. cel mult doi $ b$ Solutie: 2. Scrieți un AFD peste alfabetul $ \{0, 1\}$, care recunoaște reprezentări binare ale numerelor divizibile cu 5. E.g. 0, 101, 1010, 1111, $ w=1010$$ 5k$$ 5k+1$$ 5k+2$$ 5k+3$$ 5k+4$$ 5k$$ k$$ q$$ n = 5k+r$$ b$$ 2*n + b$$ 10k + 2r + b$$ r$$ (2r + b)$$ 5$$ \vdots 1$$ \vdots 2$$ \varepsilon$$ \varepsilon$$ N=(K,\S… text/html 2020-11-05T11:14:29+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exercise-sheet-1&rev=1604567669&do=diff Exercise sheet I $ \{a, b\}$ 1. Definiți câte un AFD pentru limbajul șirurilor care conțin: * cel puțin doi $ b$ * exact doi $ b$ * cel mult doi $ b$ 2. Scrieți un AFD peste alfabetul $ \{0, 1\}$, care recunoaște reprezentări binare ale numerelor divizibile cu 5. E.g. 0, 101, 1010, 1111, $ \varepsilon$$ \varepsilon$$ |E|$$ E$$ \varepsilon$$ \emptyset$$ \cup$$ *$$ E$$\begin{equation} |E| \ge \max_{w \in L(E)}{|w|}\end{equation} text/html 2021-01-18T19:29:30+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exercise-sheet-2-solution&rev=1610990970&do=diff Solutii pentru exercise sheet 2 1. Fie $ fst(L) = \{ w \in \Sigma^\star \mid \exists x \in \Sigma^\star, a.î. wx \in L\ și\ |x| = |w| \}$. Arătați că $ fst$ este o proprietate de închidere pentru limbajele regulate. Solutie: Pentru a intelege mai bine $ fst(L)$ , il putem aplica pe un limbaj finit, dar si pe unul infinit: * $ fst(\{A,BAAB,AB\} = \{BA,A\}$ * $ fst(L(A^*B^+)) = \{B,A,AA,BB,AB, \ldots \}$ Conditia $ \mid x\mid = \mid w\mid$ ne obliga sa identificam $ L$$ L(A^*B^+)$$ L(B… text/html 2020-12-04T23:39:06+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exercise-sheet-2&rev=1607117946&do=diff 1. Fie $ fst(L) = \{ w \in \Sigma^\star \mid \exists x \in \Sigma^\star, a.î. wx \in L\ și\ |x| = |w| \}$. Arătați că $ fst$ este o proprietate de închidere pentru limbajele regulate. 2. Fie $ L_3 = \{ w \in \{0, 1\}^* \mid \#_{01}(w) = \#_{10}(w) \}$ (i.e. cuvintele conțin același număr de secvențe “01” și “10”). Demonstrați că $ L_3$ este regulat. 3. Demonstrați că $ L = \{ a^nb^mc^{n-m} \mid n \ge m \ge 0 \}$ nu este un limbaj regulat, fără a folosi izomorfisme.$ L$$ L_1$$ L_2$$ L_1 \cap L_… text/html 2021-01-26T10:00:38+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exercise-sheet-3-solution&rev=1611648038&do=diff Folosiți lema de pompare pentru a arăta că următoarele limbaje nu sunt independente de context: 1. $ L_1 = \{ 0^{2^n} \mid n \ge 0 \}$ Solutie: Alegem (tentativ) $ w_n = 0^{2^n}$ . Conform lemei de pompare pentru LIC, cuvantul $ w_n$ este de forma $ xyzuv$ , unde $ \mid yzu \mid \leq n$ si $ yu \neq \epsilon$ . Cautam o valoare pentru $ i$ astfel incat $ xy^izu^iv \not\in L_1$$ \#\alpha$$ \alpha$$ \#x + i\#y + \#z + i\#u + \#v \neq 2^k$$ k$$ i>0$$ \#x + i\#y + \#z + i\#u + \#v < 2^{n+1}$$ … text/html 2021-01-15T10:55:09+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exercise-sheet-3&rev=1610700909&do=diff Folosiți lema de pompare pentru a arăta că următoarele limbaje nu sunt independente de context: 1. $ L_1 = \{ 0^{2^n} \mid n \ge 0 \}$ 2. $ L_2 = \{ a^nb^mc^nd^m \mid n, m \ge 0 \}$ text/html 2021-01-28T10:54:47+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exercise-sheet-4-solution&rev=1611824087&do=diff 1. Scrieți o MT care să decidă $ L_1 = \{ 0^{2^n} \mid n \ge 0 \}$ Mașina funcționează astfel: * merge de la stânga la dreapta marcând cu X zerouri din doi în doi * dacă a găsit un singur 0 acceptă * dacă a găsit un număr impar de 0 respinge * întoarce capul de citire pe primul simbol și repetă procesul$ q_{reject}$$ q_{accept}$$ L_2 = \{ a^nb^mc^nd^m \mid n, m \ge 0 \}$$ q_{reject}$$ q_{accept}$$ q_{accept}$$ q_{accept}$ text/html 2021-01-20T15:39:24+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:exercise-sheet-4&rev=1611149964&do=diff Scrieți câte o mașină Turing (tabel de tranziție, graf sau fișier input pentru un simulator) care să decidă limbajele: 1. $ L_1 = \{ 0^{2^n} \mid n \ge 0 \}$ 2. $ L_2 = \{ a^nb^mc^nd^m \mid n, m \ge 0 \}$ text/html 2017-11-15T14:23:03+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:flex&rev=1510748583&do=diff An introduction to Flex Compiling Flex Consider the following Flex application, which counts newline, word and byte counts, in a fashion similar to the wc (word count) Unix tool. %option noyywrap %{ #include <unistd.h> int chars = 0; int words = 0; int lines = 0; %} %% [a-zA-Z]+ { words++; chars += strlen(yytext); } \n { chars++; lines++; } . { chars++; } %% int main(int argc, char **argv) { yylex(); printf("%8d%8d%8d\n", lines, words, chars); } text/html 2020-10-19T15:29:01+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:introduction&rev=1603110541&do=diff Introduction to Formal Languages and Automata Preliminaries Formal Languages are a branch of theoretical Computer Science which has applications in numerous areas. Some of them are: * compilers and compiler design * complexity theory * program verification text/html 2021-09-01T18:47:31+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:istoric-examene&rev=1630511251&do=diff Examene LFA Examen restanta LFA 10 sept 2021 1. Selectati din lista de mai jos acele expresii regulate care genereaza acelasi limbaj ca AFN-ul din diagrama. 0 fin---> <--- U 1 U 0 1 * 1*10*0 (Fals) * (0 U 1)* (Adevarat)$ L = \{0^{n}1^{n}0^n \mid n > 0\}$$ L$$ L \dot L(0^*)$$ L(0^*)\dot L$$ \{w^Rw \mid w \in \{0,1\}^*\}$$ \{0^n1^{m+k}0^p \mid m > n, k > p \}$$ L=\{a^nb^m \mid n > m > 0\}$$ S \leftarrow ABA, A\leftarrow 1A \mid \epsilon, B\leftarrow 1B \mid 1$$ … text/html 2018-10-11T09:26:56+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:jflex&rev=1539239216&do=diff An introduction to JFlex What is missing in the Haskell analyser? Our Haskell analyser is generic (independent of the language being scanned), however it lacks certain key features which are key for production development: * the regular expressions are defined as part of the code, together with the functions text/html 2018-10-02T11:20:57+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab&rev=1538468457&do=diff Lab Lab-01 - Expresii regulate Key insights: * more languages than regular expressions * we do not know how to write regular expressions for some languages (as a direct consequence of the above) * reg.exps. are unambiguous and FINITE language representations$ \Sigma=\{0, 1\}$$ L=\{011\}$$ E=011$$ E=((01)1)$$ \Sigma=\{a, b\}$$ L=\{a, b\}$$ E=a \cup b$$ E=(a \cup b)$$ \Sigma=\{0, 1\}$$ L=\{e, 0, 1, 00, 01, 10, 11, 000, \ldots\}$$ E=(0 \cup 1)^{*}$$ \Sigma=\{0, 1\}$$ L=\{010010101000010, 0… text/html 2021-10-07T11:32:48+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab01-python-intro&rev=1633595568&do=diff 1. Introduction to Python Python3 is a multi-paradigm language, that combines the imperative style with the functional style. It also supports Object-Oriented programming, albeit in a limited way. Python3 offers a few programming constructs that: text/html 2021-10-13T16:13:01+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab02-dfa&rev=1634130781&do=diff 2. Deterministic finite automata Classes in Python Python supports a limited version of Object-Oriented programming, which includes class definitions and inheritance. The concept of interface and interface implementation is absent. Hence, when inheriting a function, it is the job of the programmer to make sure a method is overloaded correctly (otherwise it is just another definition of the class).$ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of 1s} \} $ text/html 2021-10-16T18:57:19+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab03-dfa-regexp&rev=1634399839&do=diff 4. Regular expressions 4.1. Formation rules (concatenation, reunion, Kleene star) 4.1.1. $ A=\{ 0^{2k} \mid k \geq 1 \}$ $ B = \{0, \epsilon \}$ $ AB = ? $ 4.1.2. $ A = \{ 0^n 1^n \mid n \geq 1 \}$ $ B = \{ 1^n \mid n \geq 1 \} $ $ AB = ? $ $ BA = ? $ 4.1.3. $ A = \emptyset $ $ B = \{ 1^n \mid n \geq 1 \} $ $ AB = ? $ $ A^* = ? $ $ B^* = ? $ 4.2. Writing Regular Expressions 4.2.1. Write a regular expression for the language of arithmetic expressions containing +, * and… text/html 2021-10-25T11:46:09+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab03-lexers&rev=1635151569&do=diff 3. Lexers 3.1. Longest match Lexers use specs to split a string into lexemes (of a given type, called token). In this lab, the specs will be DFAs. A lexer runs DFAs on a string, searching for the longest prefix which is accepted by at least one of the DFAs. $ A_1$$ A_1$$ w$$ A_2$$ A_3$$ A_4$$ A_5$$ w$$ A$$ B$$ w$$ A$$ B$$ B$$ A$$ A$$ B$$ A_3$$ A_4$$ A_5$ text/html 2021-11-02T18:02:12+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab04-regexp-nfa&rev=1635868932&do=diff 5. Regex to DFA conversion 5.1. Nondeterministic Finite Automata 5.1.1. Consider the following NFA: What are all reachable configurations from (0,abba) ? 5.1.2. What is the accepted language of the previous NFA? 5.1.3. Write an NFA without $ \varepsilon$-transitions, which accepts the language $ L = \{abc,abd,aacd\}$$ \Sigma = \{a,b,c,d\}$$ (0,abbabbb)$$ (1 \cup \varepsilon)(00^*1)^*0^*$$ \varepsilon$$ E(q)$ text/html 2020-11-08T13:11:07+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab05-nfa-python&rev=1604833867&do=diff NFA Python Implementation Consider the following encoding of an NFA: <number_of_states> <list_of_chars_in_alphabet> <list_of_final_states> <state> <symbol> <list_of_states> Example: 9 a b 4 0 EPS 1 0 a 5 0 b 0 1 EPS 2 1 b 3 5 2 EPS 4 2 a 2 3 3 a 3 3 b 3 4 EPS 4 7 8 5 a 6 5 b 5 6 EPS 7 6 a 6 7 EPS 6 7 b 4 $ k > 0 $ text/html 2022-11-17T19:16:11+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab06-closure-properties&rev=1668705371&do=diff 7. Closure properties 7.1. Show that $(10 \cup 0)^*(1 \cup \epsilon)$ and $(1 \cup \epsilon)(00^*1)^*0^*$ are equivalent regular expressions. Are there several strategies? * You can use an existing minimisation algorithm to find the minimal DFAs. Then, label each state from each DFA from 0 to |K|. Fix an ordering of the symbols of the alphabet. Sort the transitions by symbol. Make a textual representation of each DFA which includes the number of states and the sorted transition function. … text/html 2021-11-04T15:34:12+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab06-mindfa&rev=1636032852&do=diff 6. Minimal DFAs 6.1. Equivalence between states Consider the following DFA: ---------- ---------- 6.1.1. Identify a pair of states which are indistinguishable. 6.1.2. Identify a pair of final or non-final states which are distinguishable. The pair must be distinguished by a word different from the empty word.$ (1\cup\epsilon)(00^*1)^*0^*$$ (10\cup 0)^*(01 \cup 1)^*(0 \cup \epsilon)$ text/html 2021-12-06T12:08:02+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab07-the-pumping-lemma&rev=1638785282&do=diff 8. Proving languages are not regular 8.1. The pumping lemma 8.1.1. Show that the pumping lemma holds for finite languages. 8.1.2.* Find a language which is not regular for which the pumping lemma holds. 8.2. Languages which are not regular Show that each of the languages from the list below is not regular.$ L = \{ \: A^n B^m \: | \: 0 \leq n \leq m \: \} $$ L = \{ \: w \in \{A,B\}^* \: | \: \#A(w) = \#B(w) \: \} $$ L = \{(01)^n(10)^n \mid n > 0 \} $$ L = \{ \: w \in \{A,B\}^* \: | \: \te… text/html 2020-12-04T22:08:53+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab08-push-down-automata&rev=1607112533&do=diff Push down automata Exercise 1. Write PDAs for the following languages: 1.1. $ L = \{\: w \in \{A,B\}^* \ | \:w \text{ is a palindrome}\} $ 1.2. $ L = \{\: w \in \{A,B\}^* \ | \: \text{#}A(w) \neq \: \text{#}B(w) \} $ 1.3. $ L = \{ A^{m} B^{m+n} C^{n} \ | \: n, m \geq 0 \} $ 1.4. $ L = \{ A^{n}B^{n}C^{m}D^{m} | \: \ n,m \geq 0 \} \cup \{ A^nB^mC^mD^n | \: \ n,m \geq 0 \} $ 1.5. $ L = \{ A^{i}B^{j}C^{k} | \: \text{ i=j or j=k} \} $ Exercise 2. Acceptance by empty stack. Consider the foll… text/html 2021-11-30T11:05:46+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab09-cfl&rev=1638263146&do=diff 9. Context-Free Languages 9.1. Accepting and generating a CF language Write a PDA as well as a CF grammar for the following languages. Specify - for each PDA, the way it accepts (by empty-stack or by final state). For each grammar, make sure it is not ambiguous. (Start with any CF grammar that accepts L. Then write another non-ambiguous grammar for the same language).$ L = \{\: w \in \{A,B\}^* \ | \:w \text{ is a palindrome}\} $$ L = \{ A^{m} B^{m+n} C^{n} \ | \: n, m \geq 0 \} $$ L = \{w \in … text/html 2020-11-27T15:32:14+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab09-grammars&rev=1606483934&do=diff Grammars Writing Grammars 1. Write grammars for the following languages: 1.1. $ L = \{a^{m}b^{m + n}c^{n} | n, m \geq 0 \} $ 1.2. $ L = \{a^ib^jc^k | i = j \lor j = k \} $ 1.3. $ L = \{w \in \{a, b\}^* | \#_a(w) \neq \#_b(w) \} $ Ambiguity 2. Which of the following grammars are ambiguous? Justify each answer. Modify the grammar to remove ambiguity, wherever the case. 2.1. $ S \leftarrow aA | A $ $ A \leftarrow aA | B $$ B \leftarrow bB | \epsilon $$ S \leftarrow AS | \epsilon $$ A \… text/html 2020-12-08T09:12:57+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab10-cfl-closures&rev=1607411577&do=diff Context Free Languages Context-Free Grammar to Pushdown Automata 1. Consider the following CFG: $ S \leftarrow X\ |\ Y $ $ X \leftarrow YXY\ |\ 0X\ |\ 0 $ $ Y \leftarrow YY\ |\ 1\ |\ \epsilon $ 1.1 Write a PDA which accepts L(G) 1.2 Write a sequence of derivations which yeilds $ S\ \Rightarrow\ 110X1Y $ . What is it's corresponding sequence of transitions in the PDA?$ (p, 111100, XZ0) \mapsto^* (p, e, Z0) $$ \alpha $$ \alpha_1 \ldots \alpha_n $$ L = \{a^{n}b^{2n}c^{2m}d^{m}\ |\ n, m \… text/html 2021-12-14T16:31:07+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab10-lexer&rev=1639492267&do=diff 10. Writing a parser for a CF language 10.1. A basic functional structure for a parser Consider the following language encoding expressions: * $ S \leftarrow M \mid M + S$ * $ M \leftarrow A \mid A * M$ * $ A \leftarrow 0 \mid 1 \mid (S)$ 10.1.1. Implement an AST for expressions. * A parser is a function which takes a string and has text/html 2020-12-13T22:22:44+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab11-cfl-pumping-lemma&rev=1607890964&do=diff Context Free Languages Pumping Lemma 1. Pumping Lemma Prove that the following languages are not context-free. 1.1. $ L=\{a^nb^{2n}c^n \ | \ n \geq 0\} $ 1.2. $ L=\{a^n \ | \ n \text{ is a prime number}\} $ 1.3. $ L=\{ww \ | \ w \in \{0,1\}^*\} $ 2. Simple parser based on PDAs. 2.1. Write a PDA which accepts the language described below: <expr> ::= <expr> + <expr> | <expr> * <expr> | (<expr>) | <number> <number> ::= [0-9]+ $\texttt{Expr}$$\texttt{Plus}$$\texttt{Mult}$$\texttt{Par}$$\t… text/html 2023-01-12T18:00:40+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab11-cfprop&rev=1673539240&do=diff L11. Laborator recapitulativ 11.1. Fie $ L$ un limbaj descris de o expresie regulata ce nu contine operatorul Kleene Star. Care afirmatie este adevarata? * $ L$ este regulat * $ L$ nu este regulat * $ L$ este independent de context * $ L$ este infinit * $ \overline{L}$$ \mathcal{L}$$ \epsilon$$ \mathcal{L}\subsetneq LR$$ LR$$ A$$ E_1$$ \overline{L(A)}$$ E_2$$ L(A)$$ L(E_1E_2) = \emptyset$$ E_1$$ E_2$$ L(E_1 \cup E_2) = \Sigma^*$$ L(E_1) \subsetneq L(E_2)$$ w = c_1c_2\ldots c_n$$ … text/html 2021-01-11T10:54:26+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab12&rev=1610355266&do=diff Mașina Turing <http://www.jflap.org/> <https://turingmachinesimulator.com/> 1. Scrieți o Mașină Turing care primește ca input un șir binar arbitrar și îl mută mai la dreapta cu două poziții. 101110110100001##... ##101110110100001##... 2. Scrieți o Mașină Turing care incrementează la nesfârșit un contor binar. Cel mai semnificativ bit este cel mai din $ L = \{ w\_w \mid w \in \{0, 1\}^* \}$ text/html 2021-01-17T23:41:16+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab13&rev=1610919676&do=diff Limbaje decidabile 1. Descrieți un algoritm care decide limbajul $ GEN_{finite} = \{ \langle \mathcal{G} \rangle \mid \mathcal{G} \text{ e GIC} \land L(\mathcal{G}) \text{ finit} \}$ 2. Demonstrați că mulțimea limbajelor decidabile e închisă în raport cu următoarele operații: * complement * reuniune * interesecție * concatenare * star “”“ text/html 2021-01-24T19:12:26+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:lab14&rev=1611508346&do=diff Limbaje nedecidabile 1. Demonstrați că limbajul $ REG_{TM} = \{ <M> \mid L(M) \text{ este regulat } \}$ este nedecidabil. 2. Demonstrați că limbajul $ R_{TM} = \{ <M> \mid w \in L(M) \iff w^R \in L(M) \}$ este nedecidabil ($ w^R$ este inversul cuvântului $ w$). 3. Demonstrați că relația $ \leq_M$ e tranzitivă (i.e. $ A \leq_M B$ și $ B \leq_M C \implies A \leq_M C$). 4. Demonstrați că limbajul $ EMPTY_{TM} = \{ <M> \mid L(M) = \emptyset \} $ este nedecidabil.$ HALT$$ ALL_{TM} = \{ <M> \mid L… text/html 2018-07-20T12:00:48+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:languages&rev=1532077248&do=diff Languages and Regular Expressions Motivation Regular expressions are a means for defining tokens (roles) during the lexical phase. Token definition is instrumental in the development of parser generators (e.g. ANTLR). Let $ \Sigma$ be an alphabet, i.e. a finite set whose elements we call $ \{a,b,c\}$$ \Sigma$$ \emptyset$$ c\in\Sigma$$ e$$ e'$$ (ee')$$ ee$$ (e\cup e')$$ e$$ (e)^*$$ (A\cup B)(a\cup b)^*(0 \cup 1)^*$$ (a \cup b)^*$$ (0 \cup 1)^*$$ 01^*\cup 1$$ union(kleene(concat(0,1)),1)$$ (01… text/html 2020-10-19T15:38:03+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:nfa-old&rev=1603111083&do=diff Nondeterministic Automata Motivation In the previous lecture we have investigated the semantics of regular expressions and saw that how we can determine the language accepted by, e.g. $ (A\cup B)(a\cup b)*(0 \cup 1)*$ . However, it is not straightforward to compute whether a given word $ w$$ L(e)$$ w \in L((A\cup B)(a\cup b)*(0 \cup 1)*)$$ 0\in set$$ 1\in set$$ 2\in set$$ 3\in set$$ Q_0$$ Q'$$ Q''$$ Q$$ Q'$$ M=(K,\Sigma,\Delta,q_0,F)$$ K$$ \Sigma$$ \Delta$$ K \times \Sigma^* \times K$$ q_0\in … text/html 2020-10-19T15:38:52+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:nfa&rev=1603111132&do=diff Nondeterministic Automata Motivation In the previous lecture we have investigated the semantics of regular expressions and saw that how we can determine the language accepted by, e.g. $ (A\cup B)(a\cup b)*(0 \cup 1)*$ . However, it is not straightforward to compute whether a given word $ w$$ L(e)$$ M=(K,\Sigma,\Delta,q_0,F)$$ K$$ \Sigma$$ \Delta$$ K \times \Sigma^* \times K$$ q_0\in K$$ F\subseteq K$$ K=\{q_0,q_1,q_2\}$$ \Sigma=\{0,1\}$$ \Delta=\{(q_0,0,q_0),(q_0,1,q_0),(q_0,0,q_1),(q_1,1,q_2)… text/html 2018-02-02T18:33:50+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:parser&rev=1517589230&do=diff Writing parsers with ANTLR What is ANTLR (4.0) ANTLR (ANother Tool for Language Recognition) is widely used parser-generator. The input for ANTLR is a grammar. The output is a set of files implementing a: * lexer * a parser together with a walker text/html 2017-11-15T14:26:37+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:pda&rev=1510748797&do=diff Push-down Automata Introduction Push-down automata (or PDAs) are acceptors of Context-Free languages. Informally, a push-down automata is equipped with a simplistic type of memory - a stack. Thus, a PDA is able to push or pop symbols, however it is unable to directly $ M=(K,\Sigma,\Gamma,\Delta,q_0,Z_0,F)$$ K$$ \Sigma$$ \Gamma$$ \Sigma$$ \Delta\subseteq K\times \Sigma^*\times \Gamma^*\times K\times\Gamma^*$$ (q,c,\gamma,q',\gamma')\in\Delta$$ q$$ c$$ \gamma$$ q'$$ \gamma'$$ \gamma=a$$ a$$ \gam… text/html 2019-10-31T13:56:46+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:prop&rev=1572523006&do=diff Pumping Lemma Motivation Consider the language defined via BNF (Bakus-Naur Form) as follows: * <expr> ::= <expr> + <expr> | (<expr> + <expr>) | <atom> which describes simple arithmetic expressions with parentheses. Let us ignore the construction rules for atoms (let $ M_a$$ M_a$$ k\in\mathbb{N}$$ k$$ k$$ L$$ n$$ L$$ w\in L$$ n$$ w = xyz$$ \mid xy \mid \leq n$$ y \neq \epsilon$$ k > 0$$ xy^kz \in L$$ x$$ y$$ z$$ L$$ L$$ n$$ M$$ L$$ \mid w \mid \geq \mid K\mid$$ M$$ M$$ L$$ L$$ L = {0^i1^i \m… text/html 2018-07-23T14:47:50+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:reg&rev=1532346470&do=diff Regular languages Definition In the previous lectures, we have introduced regular expressions, NFAs and DFAs as finite representations for languages, and showed the following links between them. * $ E \rightarrow NA$ - each regular expression $ e$ can be transformed to a $ M$$ L(e) = L(M)$$ NA \rightarrow DA$$ M$$ M'$$ L(M) = L(M')$$ L$$ L=L(E)$$ E$$ LR\subset 2^{\Sigma^*}$$ L(NFA)\subset 2^{\Sigma^*}$$ L(DFA)\subset 2^{\Sigma^*}$$ LR \subseteq L(NFA) \subseteq L(DFA)$$ L(DFA) \subseteq L… text/html 2022-10-04T13:41:34+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:team&rev=1664880094&do=diff Limbaje Formale și Automate 2021-2022 Echipa Curs * Matei POPOVICI Seminarii * 332CB Alexandra UDRESCU * 335CB Mihai UDUBASA * 334CB Mihai CALITESCU * 331CB Alexandru ILIE * 336CB Bogdan DEAC * 333CB Matei POPOVICI Orar Curs Frecevență Zi Interval orar Profesor săptămânal Miercuri 10:00-12:00 Matei POPOVICI săptămâni pare