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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.
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Theory of Computation
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Michael Sipser Edition 3 Exercise 5 Question 32 (Page No. 241)
Prove that the following two languages are undecidable. $OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$. $PREFIXFREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefixfree}\}$.
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Oct 20
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3
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.)
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Oct 19
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Michael Sipser Edition 3 Exercise 5 Question 2 (Page No. 239)
Show that $EQ_{CFG}$ is coTuringrecognizable.
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Oct 17
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Michael Sipser Edition 3 Exercise 5 Question 1 (Page No. 239)
Show that $EQ_{CFG}$ is undecidable.
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Oct 17
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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.
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Oct 17
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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.
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Oct 17
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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 $E_{CFG}$.)
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Oct 17
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Theory of Computation
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9
Michael Sipser Edition 3 Exercise 2 Question 59 (Page No. 160)
If we disallow $\epsilon$rules in CFGs, we can simplify the DKtest. 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 DKtest iff it is a DCFG.
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Oct 13
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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 DKtest to show that $G_{1}$ is not a DCFG. $R \rightarrow S \mid T$ $S \rightarrow aSb \mid ab$ $T \rightarrow aTbb \mid abb$
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Oct 13
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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 DKtest to show that G is a DCFG. Describe a DPDA that recognizes L(G).
asked
Oct 13
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Theory of Computation
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12
Michael Sipser Edition 3 Exercise 2 Question 52 (Page No. 159)
Show that every DCFG generates a prefixfree language.
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Oct 13
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13
Michael Sipser Edition 3 Exercise 2 Question 51 (Page No. 159)
Show that every DCFG is an unambiguous CFG.
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Oct 13
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14
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 $B$.
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Oct 13
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15
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.
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Oct 12
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16
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 $T_{ij}$.
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Aug 20
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Compiler Design
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17
Ullman (Compiler Design) Edition 2 Exercise 4.4 Question 9 (Page No. 232)
Every language that has a contextfree 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 YoungerKasami (or CYK) algorithm is based on dynamic programming. ... a_{l}a_{2}\cdot\cdot\cdot a_{n} is in the language?
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Aug 20
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Compiler Design
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18
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.
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Aug 17
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19
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 $+\ast aaa$ ... $bfactor\:\rightarrow\:not\:bfactor\mid (bexpr)\mid true\mid false$
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contextfreegrammars
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ambiguous
descriptive
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1
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20
Ullman (Compiler Design) Edition 2 Exercise 4.2 Question 1 (Page No. 206)
Consider the contextfree 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.
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compilerdesign
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ambiguous
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21
Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 6 (Page No. 52)
Construct a contextfree grammar for roman numerals.
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Jul 26
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Compiler Design
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contextfreegrammars
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22
Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 4 (Page No. 51  52)
Construct unambiguous contextfree grammars for each of the following languages. In each case show that your grammar is correct. Arithmetic expressions in postfix notation. Leftassociative lists of identifiers separated by commas. Right ... $(d)$.
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Jul 26
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23
Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 3 (Page No. 51)
Which of the grammars are ambiguous? $S\rightarrow 0S1 \mid 01$ $S\rightarrow +SS \mid SS \mid a$ $S\rightarrow S(S)S \mid \epsilon$ $S\rightarrow aSbS \mid bSaS \mid \epsilon$ $S\rightarrow a \mid S+S \mid SS \mid S^{\ast} \mid (S)$
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Jul 26
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ambiguous
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24
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. $S\rightarrow 0S1 \mid 01$ $S\rightarrow +SS \mid SS \mid a$ $S\rightarrow S(S)S \mid \epsilon$ $S\rightarrow aSbS \mid bSaS \mid \epsilon$ $S\rightarrow a \mid S+S \mid SS \mid S^{\ast} \mid (S)$
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25
Ullman (Compiler Design) Edition 2 Exercise 2.2 Question 1 (Page No. 51)
Consider the contextfree 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.
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26
Ullman (TOC) Edition 3 Exercise 9.5 Question 1 (Page No. 418)
Let $L$ be the set of (codes for) contextfree 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.
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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) $L=$ {$w:n_a(w)+n_b(w)\neq n_c(w)$} .
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Peter Linz Edition 4 Exercise 7.4 Question 8 (Page No. 204)
Let G be a contextfree grammar in Greibach normal form. Describe an algorithm which, for any given k, determines whether or not G is an LL (k) grammar.
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29
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 contextfree language.
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30
Peter Linz Edition 4 Exercise 7.4 Question 5 (Page No. 204)
Show that any LL grammar is unambiguous.
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