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Chapter 4: Q28P (page 212)

Let C={{G,x}|GisaCFGxisasubstringofsomey鈭�L(G)}. Show thatC is decidable.

Short Answer

Expert verified

The languageC is decidable.

Step by step solution

01

Context free language.

Context free language is a grammar where its language or string that formed by context free grammar is supports pushdown deterministic automata.

02

The problem is decidable.

Let,C={{G,x}|GisaCFGxisasubstringofsomey鈭�L(G)}.

Showing that,Cis decidable.

Let鈥檚 the following Turing machine M= On input hG, xi,

Where, (G) is a context free grammar.

(a) Let Rbe the following RE .

R=危鈭�虫危鈭�

(b) Create the context free grammar Hsuch that,

L(H)=L(G)鈭�L(R)

(c) SubmitG to the decider for ECFG.

(d) If it accepts, then reject.

(e) Else, accept.

Mis clearly a decider since it halts. Also, Maccepts{G,x} .

If f generates some string危鈭�虫危鈭� in that is some string withx as a substring.

Hence, the languageC is decidable.

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Most popular questions from this chapter

Let C be a language. Prove that C is Turing-recognizable if a decidable language D exists such that C={x|鈭�yx,y鈭�D}.

Say that an NFA is ambiguous if it accepts some string along two different computation branches

LetAMBIGNFA={(N)|NisanambiguousNFA}.Show thatAMBIGNFA is decidable. (Suggestion: One elegant way to solve this problem is to construct a suitable DFA and then run EDFAon it.)

Let E={M|M is a DFA that accepts some string with more 1s than 0s}. Show that E is decidable. (Hint: Theorems about CFLs are helpful here.)

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.

Review the way that we define sets to be the same size in Definition 4.12 (page 203). Show that 鈥渋s the same size鈥� is an equivalence relation.
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