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Michael Sipser Edition 3 Exercise 4 Question 29 (Page No. 212)
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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.
michaelsipser
theoryofcomputation
contextfreegrammars
decidability
proof
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Oct 18, 2019
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Theory of Computation
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Lakshman Patel RJIT

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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 18, 2019
in
Theory of Computation
by
Lakshman Patel RJIT

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michaelsipser
theoryofcomputation
contextfreelanguages
contextfreegrammars
decidability
proof
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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 18, 2019
in
Theory of Computation
by
Lakshman Patel RJIT

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michaelsipser
theoryofcomputation
contextfreegrammars
decidability
proof
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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

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michaelsipser
theoryofcomputation
contextfreegrammars
turingrecognizablelanguages
decidability
proof
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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, 2019
in
Theory of Computation
by
Lakshman Patel RJIT

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michaelsipser
theoryofcomputation
contextfreegrammars
turingmachine
decidability
proof
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