Chapter 4: 14P (page 211)

Let $\sum = {0, 1}$ .Show that the problem of determining whether a CFG generates some string in $1^{}$ is decidable. In other words, show that
${<G> | G is a CFG over \{0, 1\} and 1 \cap L (G) \neq \emptyset}$ is a decidable language.

Short Answer

Expert verified

The given language is a decidable language.

Step by step solution

Decidable language

In terms of finite automata (FA), a decidable refers to a problem of testing whether a deterministic finite automata (DFA) accepts an input string. A decidable language corresponds to algorithmically solvable decision problems.

Explanation

Consider that $C$ is a context-free language and R is a regular language, then $C \cap R$ is context free. Therefore, $1 * \cap L (G)$ 1 ∗ ∩ L(G) is context free. The following TM decides the language of this problem.

“On input $<G>$ :

1. Construct CFG H such that $L (H) = 1 * \cap L (G)$ .

2. Test whether $L (H) = \emptyset$ using the $E_{C F G}$ decider R from the Theorem 4.8.

3. If R accepts, reject; if R rejects, accept.”

Therefore, the given language is decidable and it has been proven above.

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Let $\sum = {0, 1}$ .Show that the problem of determining whether a CFG generates some string in $1^{}$ is decidable. In other words, show that
${<G> | G is a CFG over \{0, 1\} and 1 \cap L (G) \neq \emptyset}$ is a decidable language.

Short Answer

Step by step solution

Decidable language

Explanation

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Let ∑={0,1}.Show that the problem of determining whether a CFG generates some string in 1*is decidable. In other words, show that {G|G is a CFG over 0,1 and 1*∩LG≠∅}is a decidable language.

Short Answer

Step by step solution

Decidable language

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

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Let $\sum = {0, 1}$ .Show that the problem of determining whether a CFG generates some string in $1^{}$ is decidable. In other words, show that
${<G> | G is a CFG over \{0, 1\} and 1 \cap L (G) \neq \emptyset}$ is a decidable language.