Recent questions tagged context-free-grammars - GATE Overflow

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ISRO2020-39
The language which is generated by the grammar $S \rightarrow aSa|bSb|a|b$ over the alphabet of $\{a,b\}$ is the set of Strings that begin and end with the same symbol All odd and even length palindromes All odd length palindromes All even length palindromes

asked Jan 13 in Theory of Computation by Satbir Boss (24.1k points) | 114 views

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Michael Sipser Edition 3 Exercise 5 Question 36 (Page No. 242)
Say that a $CFG$ is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{CFG} = \{\langle G \rangle \mid \text{G is a minimal CFG}\}$. Show that $MIN_{CFG}$ is $T-$recognizable. Show that $MIN_{CFG}$ is undecidable.

asked Oct 20, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 79 views

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Michael Sipser Edition 3 Exercise 5 Question 32 (Page No. 241)
Prove that the following two languages are undecidable. $OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$. $PREFIX-FREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefix-free}\}$.

asked Oct 20, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 20 views

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Michael Sipser Edition 3 Exercise 5 Question 21 (Page No. 240)
Let $AMBIG_{CFG} = \{\langle G \rangle \mid \text{G is an ambiguous CFG}\}$. Show that $AMBIG_{CFG}$ is undecidable. (Hint: Use a reduction from $PCP$ ... $a_{1},\dots,a_{k}$ are new terminal symbols. Prove that this reduction works.)

asked Oct 19, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 5 views

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Michael Sipser Edition 3 Exercise 5 Question 2 (Page No. 239)
Show that $EQ_{CFG}$ is co-Turing-recognizable.

asked Oct 17, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 12 views

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Michael Sipser Edition 3 Exercise 5 Question 1 (Page No. 239)
Show that $EQ_{CFG}$ is undecidable.

asked Oct 17, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 5 views

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Michael Sipser Edition 3 Exercise 4 Question 31 (Page No. 212)
Say that a variable $A$ in $CFL\: G$ is usable if it appears in some derivation of some string $w \in G$. Given a $CFG\: G$ and a variable $A$, consider the problem of testing whether $A$ is usable. Formulate this problem as a language and show that it is decidable.

asked Oct 17, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 10 views

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Michael Sipser Edition 3 Exercise 4 Question 29 (Page No. 212)
Let $C_{CFG} = \{\langle G, k \rangle \mid \text{ G is a CFG and L(G) contains exactly $k$ strings where $k \geq 0$ or $k = \infty$}\}$. Show that $C_{CFG}$ is decidable.

asked Oct 17, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 7 views

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Michael Sipser Edition 3 Exercise 4 Question 28 (Page No. 212)
Let $C = \{ \langle G, x \rangle \mid \text{G is a CFG $x$ is a substring of some $y \in L(G)$}\}$. Show that $C$ is decidable. (Hint: An elegant solution to this problem uses the decider for $E_{CFG}$.)

asked Oct 17, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 7 views

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Michael Sipser Edition 3 Exercise 2 Question 59 (Page No. 160)
If we disallow $\epsilon$-rules in CFGs, we can simplify the DK-test. In the simplified test,we only need to check that each of DK’s accept states has a single rule. Prove that a CFG without $\epsilon$-rules passes the simplified DK-test iff it is a DCFG.

asked Oct 13, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 6 views

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Michael Sipser Edition 3 Exercise 2 Question 55 (Page No. 159)
Let $G_{1}$ be the following grammar that we introduced in Example $2.45$. Use the DK-test to show that $G_{1}$ is not a DCFG. $R \rightarrow S \mid T$ $S \rightarrow aSb \mid ab$ $T \rightarrow aTbb \mid abb$

asked Oct 13, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 5 views

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Michael Sipser Edition 3 Exercise 2 Question 54 (Page No. 159)
Let G be the following grammar: $S \rightarrow T\dashv $ $T \rightarrow T aT b \mid T bT a | \epsilon$ Show that $L(G) = \{w\dashv \: \mid w\: \text{contains equal numbers of a’s and b’s} \}$. Use a proof by induction on the length of $w$. Use the DK-test to show that G is a DCFG. Describe a DPDA that recognizes L(G).

asked Oct 13, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 6 views

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Michael Sipser Edition 3 Exercise 2 Question 52 (Page No. 159)
Show that every DCFG generates a prefix-free language.

asked Oct 13, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 7 views

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Michael Sipser Edition 3 Exercise 2 Question 51 (Page No. 159)
Show that every DCFG is an unambiguous CFG.

asked Oct 13, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 6 views

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Michael Sipser Edition 3 Exercise 2 Question 47 (Page No. 159)
Let $\Sigma = \{0,1\}$ and let $B$ be the collection of strings that contain at least one $1$ in their second half. In other words, $B = \{uv \mid u \in \Sigma^{\ast}, v \in \Sigma^{\ast}1\Sigma^{\ast}\: \text{and} \mid u \mid \geq \mid v \mid \}$. Give a PDA that recognizes $B$. Give a CFG that generates $B$.

asked Oct 13, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 12 views

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Michael Sipser Edition 3 Exercise 2 Question 46 (Page No. 158)
Consider the following CFG $G:$ $S \rightarrow SS \mid T$ $T \rightarrow aT b \mid ab$ Describe $L(G)$ and show that $G$ is ambiguous. Give an unambiguous grammar $H$ where $L(H) = L(G)$ and sketch a proof that $H$ is unambiguous.

asked Oct 12, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 12 views

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Ullman (Compiler Design) Edition 2 Exercise 4.4 Question 10 (Page No. 233)
Show how, having filled in the table as in Question $4.4.9$, we can in $O(n)$ time recover a parse tree for $a_{1}a_{2}\cdot\cdot\cdot a_{n}$. Hint: modify the table so it records, for each nonterminal $A$ in each table entry $T_{ij}$, some pair of nonterminals in other table entries that justified putting $A$ in $T_{ij}$.

asked Aug 20, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 3 views

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Ullman (Compiler Design) Edition 2 Exercise 4.4 Question 9 (Page No. 232)
Every language that has a context-free grammar can be recognized in at most $O(n^{3})$ time for strings of length $n$. A simple way to do so,called the Cocke- Younger-Kasami (or CYK) algorithm is based on dynamic programming. ... a_{l}a_{2}\cdot\cdot\cdot a_{n} is in the language?

asked Aug 20, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 3 views

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Ullman (Compiler Design) Edition 2 Exercise 4.2 Question 3 (Page No. 207)
Design grammars for the following languages: The set of all strings of $0's$ and $1's$ such that every $0$ is immediately followed by at least one $1$. The set of all strings of $0's$ and $1's$ that are palindromes; that is, the string ... $1's$ of the form $xy$, where $x\neq y$ and $x$ and $y$ are of the same length.

asked Aug 17, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 6 views

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Ullman (Compiler Design) Edition 2 Exercise 4.2 Question 2 (Page No. 206 - 207)
Repeat Question $4.2.1$ for each of the following grammars and strings: $S\rightarrow 0S1\mid 01$ with string $000111$. $S\rightarrow +SS\mid \ast SS\mid a$ with string $+\ast aaa$ ... $bfactor\:\rightarrow\:not\:bfactor\mid (bexpr)\mid true\mid false$

asked Aug 17, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 15 views

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1 answer

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Ullman (Compiler Design) Edition 2 Exercise 4.2 Question 1 (Page No. 206)
Consider the context-free grammar:$S\rightarrow SS + \mid SS {\ast} \mid a$and the string $aa + a{\ast}$. Give a leftmost derivation for the string. Give a rightmost derivation for the string. ... for the string. Is the grammar ambiguous or unambiguous? Justify your answer. Describe the language generated by this grammar.

asked Aug 7, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 52 views

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Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 6 (Page No. 52)
Construct a context-free grammar for roman numerals.

asked Jul 26, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 7 views

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Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 4 (Page No. 51 - 52)
Construct unambiguous context-free grammars for each of the following languages. In each case show that your grammar is correct. Arithmetic expressions in postfix notation. Left-associative lists of identifiers separated by commas. Right- ... $(d)$.

asked Jul 26, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 16 views

+1 vote

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Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 3 (Page No. 51)
Which of the grammars are ambiguous? $S\rightarrow 0S1 \mid 01$ $S\rightarrow +SS \mid -SS \mid a$ $S\rightarrow S(S)S \mid \epsilon$ $S\rightarrow aSbS \mid bSaS \mid \epsilon$ $S\rightarrow a \mid S+S \mid SS \mid S^{\ast} \mid (S)$

asked Jul 26, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 26 views

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Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 2 (Page No. 51)
What language is generated by the following grammars? In each case justify your answer. $S\rightarrow 0S1 \mid 01$ $S\rightarrow +SS \mid -SS \mid a$ $S\rightarrow S(S)S \mid \epsilon$ $S\rightarrow aSbS \mid bSaS \mid \epsilon$ $S\rightarrow a \mid S+S \mid SS \mid S^{\ast} \mid (S)$

asked Jul 26, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 6 views

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Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 1 (Page No. 51)
Consider the context-free grammar $S\rightarrow SS+\mid SS^{\ast}\mid a$ Show how the string $aa+a^{\ast}$ can be generated by this grammar. Construct a parse tree for this string. What language does this grammar generate? Justify your answer.

asked Jul 26, 2019 in Compiler Design by Lakshman Patel RJIT Veteran (59.1k points) | 4 views

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Ullman (TOC) Edition 3 Exercise 9.5 Question 1 (Page No. 418)
Let $L$ be the set of (codes for) context-free grammars $G$ such that $L(G)$ contains at least one palindrome. Show that $L$ is undecidable. Hint: Reduce PCP to $L$ by constructing, from each instance of PCP a grammar whose language contains a palindrome if and only if the PCP instance has a solution.

asked Jul 26, 2019 in Theory of Computation by Lakshman Patel RJIT Veteran (59.1k points) | 11 views

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Peter Linz Edition 4 Exercise 7.4 Question 9 (Page No. 204)
Give LL grammars for the following languages, assuming $Σ =$ {$a,b, c$}. (i) $L=$ {$a^nb^mc^{n+m}:n\geq0,m\geq0$} . (ii) $L=$ {$a^{n+2}b^mc^{n+m}:n\geq0,m\geq0$} . (iii) $L=$ {$a^nb^{n+2}c^{m}:n\geq0,m\gt1$} . (iv) $L=$ {$w:n_a(w)\lt n_b(w)$} . (v) $L=$ {$w:n_a(w)+n_b(w)\neq n_c(w)$} .

asked Jun 25, 2019 in Theory of Computation by Naveen Kumar 3 Boss (15.4k points) | 35 views

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Peter Linz Edition 4 Exercise 7.4 Question 8 (Page No. 204)
Let G be a context-free grammar in Greibach normal form. Describe an algorithm which, for any given k, determines whether or not G is an LL (k) grammar.

asked Jun 25, 2019 in Theory of Computation by Naveen Kumar 3 Boss (15.4k points) | 13 views

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Peter Linz Edition 4 Exercise 7.4 Question 6 (Page No. 204)
Show that if G is an LL (k) grammar, then L (G) is a deterministic context-free language.

asked Jun 25, 2019 in Theory of Computation by Naveen Kumar 3 Boss (15.4k points) | 9 views

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