https://ocw.cs.pub.ro/ppcarte/ 2026-04-17T15:38:56+03:00 https://ocw.cs.pub.ro/ppcarte/ https://ocw.cs.pub.ro/ppcarte/lib/tpl/bootstrap3/images/favicon.ico text/html 2024-10-07T11:04:06+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab01&rev=1728288246&do=diff 1. Deterministic Finite Automata 2.1. DFA practice * Double-circle = final state * Hanging arrow entering state = initial state * Sink state = a state that is not final; once reached, there is no transition that leaves it, thus the DFA will reject; there is a transition that loops in this state for each possible character$ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of ones} \} $$ 3k \overset{*2}{\longrightarrow} 3k $$ 3k + 1 \overset{*2}{\longrightarrow} 3k + 2 $$ 3k + 2 \over… text/html 2024-10-12T01:40:45+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab02&rev=1728686445&do=diff 2. Programming introduction - Python Python3 is a multi-paradigm language, that combines the imperative style with the functional style. It also supports Object-Oriented programming, albeit in a limited way. Python3 offers a few programming constructs that: text/html 2024-10-22T08:13:37+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab03&rev=1729574017&do=diff 3. Regular Expressions 3.1. Natural Language / DFA $ \rightarrow $ Regex conversion For each of the exercises from DFA Seminary 1 write a regex describing the same language. 3.1.1. $ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of ones} \} $ $ 0^*10^*(10^*10^*)^*$ 3.1.2. The language of binary words which contain exactly two ones$ 0^*10^*10^*$$ (0 \cup 1)^*1$$ (0 \cup 1)^*00101(0 \cup 1)^*$$ A=\{ 0^{2k} \mid k \geq 1 \}$$ B = \{0, \epsilon \}$$ AB = ? $$ A = \{ 0^n 1^n \mid n \ge… text/html 2024-10-29T09:27:30+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab04&rev=1730186850&do=diff 4. Regex Representation in Python In this laboratory we will be using Python3 classes to introduce a possible representation of regular expressions inside code and ways to work with those Regexes. We will start with a base class Regex that will function as an interface for future descendents:$ \rightarrow $$ \rightarrow $$ \rightarrow $ text/html 2024-11-04T11:58:33+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab05&rev=1730714313&do=diff 5. Regex to NFA to DFA conversions Thompson's algorithm is used to convert from a Regex to an NFA. Subset construction is used to convert from an NFA to an equivalent DFA. Thompson's algorithm applies the following rules recursively: Exercises $ (1 \cup \varepsilon)(0^*1)^*0^* $$ (0\cup 01) \cup 1(10^*)^* \cup \varepsilon $$ (((00 \cup 11)^*1)^*0)^* $$ e^* $ text/html 2024-11-12T00:59:26+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab06&rev=1731365966&do=diff 5. Minimal DFAs Consider the following DFA (blue states are final): 5.1. Equivalence between states 5.1.1. Identify a pair of states which are indistinguishable. 5.1.2. 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.$ E1 = ((ab^*a)^+b)^* $$ E2 = (a(b\mid aa)^*ab)^* $$ (1\cup\epsilon)(00^*1)^*0^* $$ (10\cup 0)^*(01 \cup 1)^*(0 \cup \epsilon) $$ D_n $$ L(a^+b^*) $\begin{align*} &D_n = (K_n, \Sigm… text/html 2024-11-18T21:00:48+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab07&rev=1731956448&do=diff 7. Closure properties 7.1.1. (Solved during lecture) Define the reversal of a language $ L $ as $ rev(L) = \{ w \in \Sigma^* | rev(w) \in L \}$, where $ rev(c_1c_2 \dots c_n) = c_nc_{n - 1} \dots c_1$ , with $ c_i \in \Sigma, 1 \leq i \leq n $. Show that reversal is a closure property. 7.1.2. Suppose $ A$ is a DFA. Write another DFA which accepts only those words $ w$ such that both $ w \in L(A)$ and $ reverse(w) \in L(A)$$ A^R$$ reverse(L(A))$$ A$$ A^R$$ L \subseteq \Sigma^* $$ c \in \Sigm… text/html 2024-11-28T01:33:58+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab08&rev=1732750438&do=diff 8. Proving languages are not regular Pumping Lemma $\exists n \in \mathbb{N}$$ \forall w\ \ \text{s.t.}\ \ |w| \ge n $$ w = xyz $$ |xy| \le n $$ y \neq \varepsilon $$ \forall k \ge 0, w_{k} = xy^{k}z \in L$ Complement of Pumping Lemma $\forall n \in \mathbb{N}$$\exists w_{n} \in L $$ |w| \ge n $$ w_{n} $$ \forall x, y, z \in \Sigma^* $$ w_{n} = xyz $$ |xy| \le n $$ y \neq \varepsilon $$\exists k \ge 0 $$ w_{n} = xy^{k}z \notin L $not 8.1. The pumping lemma 8.1.1. Show that the pumping lemm… text/html 2024-12-02T11:33:24+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab09&rev=1733132004&do=diff 9. Push Down Automata 9.1. Writing PDAs Find PDAs that accept each of the languages below, both by empty stack and by final state. Identify which approach feels more suitable for each language. 9.1.1. $ L = \{\: w \in \{A,B\}^* \ | \:w \text{ is a palindrome}\} $. 9.1.2. $ L = \{ A^{m} B^{m+n} C^{n} \ | \: n, m \geq 0 \} $ 9.1.3. $ L = \{w \in \{a, b\}^* | \#_a(w) = \#_b(w) \} $ 9.1.4. $ L = \{w \in \{a, b\}^* | \#_a(w) \neq \#_b(w) \} $ 9.1.5. $ L = \{a^ib^jc^k | i = j \lor j = k \} $ … text/html 2024-12-16T19:34:39+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab10&rev=1734370479&do=diff 9. Context-Free Grammars 9.1. Generating a CF language Write CF grammar for the following languages. For each grammar, make sure it is not ambiguous. (Start with any CF grammar that accepts L. Then write another non-ambiguous grammar for the same language).$ L = \{\: w \in \{a,b\}^* \ | \:w \text{ is a palindrome}\} $$ S \leftarrow aSa\ |\ bSb \ |\ a \ |\ b \ |\ \epsilon $$ L = \{ a^{m} b^{m+n} c^{n} \ | \: n, m \geq 0 \} $$ S \leftarrow AB $$ A \leftarrow aAb\ |\ \ \epsilon $$ B \leftarrow bB… text/html 2026-01-18T17:55:26+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab11&rev=1768751726&do=diff 11. Closure properties for Context-Free Languages 11.0. Lexer Spec Given the following specs, construct the lexer DFA as presented in Lecture 14: * PAIRS: $ (10 | 01)* $ * ONES: $ 1+ $ * NO_CONSEC_ONE: $ (1 | \epsilon)(01 | 0)* $ Separate the following input strings into lexemes: a) 010101$ L_1 = \{ a^{n} b^{n} c^{m} \ | \: n, m \in \mathbb{N} \} $$ L_2 = \{ a^{m} b^{n} c^{n} \ | \: n, m \in \mathbb{N} \} $$ \Rightarrow L_1 \bigcap L_2 = \{ a^{n} b^{n} c^{n} \ | \: n \in \mathbb{N} \… text/html 2025-01-12T16:28:47+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:lab12&rev=1736692127&do=diff 12. Pumping Lemma for Context-Free Languages and Recap Pumping Lemma for CFL$ L $$ n \geq 1$ $ w\in L $$ |w|\geq n $$ w = uxyzw $ $$ \begin{align} & |xz| \geq 1 \\ & |xyz| \leq n \\ & ux^kyz^kw \in L \ \ \ \ \ \forall k \geq 0 \end{align} 12.1. Finding the class of languages For each of the following languages, find the most restrictive class of languages that it belongs to (regular, context-free but not regular, not context-free). Prove the language belongs to that class by finding a generat… text/html 2024-11-03T22:49:56+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:proiect&rev=1730666996&do=diff Deadline etapa 1: 21 nov 2024 23:55 Schelet etapa 1 Proiect Proiectul este structurat in trei teme al caror obiectiv final reprezinta implementarea unui lexer in Python. Ce este un lexer? Un lexer este un program care, pe baza unei specificatii, imparte un sir de caractere in subsiruri, si identifica carei $ \Delta$ text/html 2024-10-20T12:25:24+03:00 https://ocw.cs.pub.ro/ppcarte/doku.php?id=lfa:2024:team&rev=1729416324&do=diff Limbaje Formale și Automate 2024-2025 Echipa Curs * Matei POPOVICI Seminarii * Cătălin-Mihail CHIRU * Alexandru TOADER * Ștefan STEREA Orar Curs Frecvență Zi Interval orar Profesor săptămânal Miercuri 10:00-12:00 Matei POPOVICI săptămâni impare