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Chapter 1: Q48P (page 90)

Question: Let Σ={0,1}and let

D={w|wcontainsanequalnumberofoccurrencesofthesubstrings01and10}.

Thus101∈D because 101 contains a single 01 and a single 10, but 1010∉Dbecause 1010 contains two 10 s and one .01 Show that D is a regular language.

Short Answer

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Answer:

Regular language and its deterministic finite machine are shown below.

Step by step solution

01

Regular language

A language is regular if it can be expressed in terms of regular expression.A regular expression can also be described as a sequence of pattern that defines a string. Regular expressions are used to match character combinations in strings.

02

Deterministic finite machine for regular language.

For the string which containsΣ=0,1 where101∈Dbecause 101 contains a single 01 and a single 10 , but 1010∉Dbecause 1010 contains two 10 s and one 01.

Hence, D=w|wcontainsanequalnumberofoccurrencesofthesubstrings01and10is a regular language and deterministic finite machine is possible.

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Most popular questions from this chapter

Let Bn={ak|k is a multiple of n}Show that for eachn⩾1n, the language Bis regular.

In the traditional method for cutting a deck of playing cards, the deck is arbitrarily split two parts, which are exchanged before reassembling the deck. In a more complex cut, called Scarne’s cut, the deck is broken into three parts and the middle part in placed first in the reassembly. We’ll take Scarne’s cut as the inspiration for an operation on languages. For a language A, let CUT(A)={yxz|xyz∈A}.

a. Exhibit a languageB for whichCUT(B)≠CUT(CUT(B)).

b. Show that the class of regular languages is closed under CUT.

Question: The following are the state diagrams of two DFAs , M1 and M2 . Answer the following questions about each of these machines.

a. What is the start state ?

b. What is the set of accept states ?

c. What sequence of states does the machine go through on input aabb ?

d. Does the machine accept the string aabb ?

e. Does the machine accept the string ε ?

Let B and C be languages over ∑=0,1. Define

B¬C={w^IB|  for some y^IC, strings w and y contain equal numbers of 1s}

Show that the class of regular languages is closed under the←1operation.

Let M=(Q,Σ,δ,q0,F)be a DFA and let be a state of Mcalled its “home”. A synchronizing sequence for M and h is a string s∈Σ∗whereδ(q,s)=hforeveryq∈Q. (Here we have extended to strings, so thatδ(q,s) equals the state where M ends up when M starts at state q and reads input s .) Say that M is synchronizable if it has a synchronizing sequence for some state h . Prove that if M is a k-state synchronizable DFA, then it has a synchronizing sequence of length at mostk3 . Can you improve upon this bound?
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