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Chapter 0: Q47P (page 1)

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 .

Short Answer

Expert verified

a

b

S→XY
X→AB
Y→A1A|A1B|A1X|B1X|X1X

A→00*|ε
B→11*|ε

Step by step solution

01

Explain Context-free Grammar and Pushdown Automata

Context-free grammar generates context-free languages. The regular

languages are the subset of the context-free languages that are an identical

set of languages accepted by pushdown automata.

02

Give a PDA that recognizes B

a Consider the given,

Construct the pushdown automata P to determine the language B . The P can be

defined as (Q,∑,τ,δ,q0,F)wherelocalid="1662190593189" Q=q0,q1,q2,∑=0,1,τ=x,F=q2

The PDA is as follows,

a

03

Give a CFG that generates B .

Consider the given,

The CFG for the language is as follows,





The String S has the production of XY concatenation. The string X can consist of

any number of 1’s and 0’s that are represented by A and B . The string Y may

consist of at least only one 1 or one 1.

Therefore, the Context -free Grammar for the language B has been obtained.

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Most popular questions from this chapter

Show that if G is a CFG in Chomsky normal form, then for any stringw∈L(G) of lengthn⩾1, exactly2n-1 steps are required for any derivation of w.

Consider the problem of determining whether a Turing machine Mon an input w ever attempts to move its head left at any point during its computation on w. Formulate this problem as a language and show that it is decidable.

Consider the problem of determining whether a Turing machine M on an input w ever attempts to move its head left when its head is on the left-most tape cell. Formulate this problem as a language and show that it is undecidable.

Show that if A≤TBand B≤TC, thenA≤TC .

Convert the CFG given in Exercise 2.1 to an equivalent PDA, using the procedure given in Theorem 2.20
See all solutions

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