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Chapter 4: 14P (page 211)

Let 鈭={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鈥鈥is鈥鈥a鈥鈥CFG鈥over鈥0,1鈥and鈥1*鈭LG鈮鈭}is a decidable language.

Short Answer

Expert verified

The given language is a decidable language.

Step by step solution

01

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.

02

Explanation

Consider that Cis a context-free language and R is a regular language, then C鈭Ris context free. Therefore, 1*鈭LG1 鈭 鈭 L(G) is context free. The following TM decides the language of this problem.

鈥淥n input G:

1. Construct CFG H such that LH=1*鈭LG.

2. Test whether LH=鈭using the ECFGdecider R from the Theorem 4.8.

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

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

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Most popular questions from this chapter

Prove that the class of decidable languages is not closed under homomorphism

Let A be a Turing-recognizable language consisting of descriptions of Turing machines, {M1,M2,...}, where everyMiis a decider. Prove that some decidable languageDis not decided by any deciderMiwhose description appears in A. (Hint: You may find it helpful to consider an enumerator for A.)

Let A={M|M鈥is鈥鈥a鈥鈥DFA鈥鈥that鈥鈥doesn't鈥鈥accept鈥鈥any鈥鈥string鈥鈥contianing鈥鈥an鈥鈥odd鈥number鈥of鈥鈥1s}. Show that A is decidable.

Let A and B be two disjoint languages. Say that language C separates A and B if A鈯Cand B鈯C. Show that any two-disjoint co-Turing-recognizable languages are separable by some decidable language.

Say that a variable A in CFLrole="math" localid="1659808454707" G is usable if it appears in some derivation of some stringw鈭G . Given a CFGG and a variable A, consider the problem of testing whetherA is usable. Formulate this problem as a language and show that it is decidable.
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