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Chapter 0: Q55P (page 1)

Let G1 be the following grammar that we introduced in Example

2.45. Use the DK-test to show that G1is not a DFG.
R→S|T

S→aSb|ab

T→aTbb|abb

Short Answer

Expert verified

Using the DK-test, it can be shown that G1is not a DFG.

Step by step solution

01

Explain DK-test

DK-test, is the procedure that determine that context-free grammar is

deterministic. DK is constructed in association with DFA and it accepts the

input if the input x satisfies the following conditions,

1. x is the prefix of some valid string v=xy

2. x ends with a handle of v .

02

Convert the CFG G to an equivalent PDA

Consider the grammar G1,


The language of the grammar is

,where A=and. Consider the string,and underline the handle as

follows,

aaabbb

aaSbb


asb


The leftmost reduction of the string aaabbbbbb , is as follows

aaabbbbb


aaTbbbb

aTbb

T

R

In the above two strings, the left most reduction alone was able to done. Hence, the

strings are unequal.

Therefore, form the above result, the condition of DK-test has been violated. Thus,

the grammar G1 is not a DFG.

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Most popular questions from this chapter

Let φeqbe defined as in Problem 6.10. Give a model of the sentence

This problem is inspired by the single-player game Minesweeper, generalized to an arbitrary graph. Let Gbe an undirected graph, where each node either contains a single, hidden mine or is empty. The player chooses nodes, one by one. If the player chooses a node containing a mine, the player loses. If the player chooses an empty node, the player learns the number of neighboring nodes containing mines. (A neighboring node is one connected to the chosen node by an edge.) The player wins if and when all empty nodes have been so chosen.

In the mine consistency problem, you are given a graphG along with numbers labeling some of G’s nodes. You must determine whether a placement of mines on the remaining nodes is possible, so that any node v that is labeled m has exactly m neighboring nodes containing mines. Formulate this problem as a language and show that it isNPcomplete.

Let ∑=0,1. Let B be the collection of strings that contain at least one 1 in their second half. In other words,

a. Give a PDA that recognizes B

b. Give a CFG that generates B .

Give an example of an undecidable language B, where B≤mB.

Use the result of Problem 6.21 to give a function f that is computable with an oracle for ATM, where for each n,f(n) is an incompressible string of length n.
See all solutions

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