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Chapter 1: Q62P (page 92)

Let Σ={a,b}.For each k≥1, let role="math" localid="1660750960062" Dkbe the language consisting of all strings that have at least one a among the last k symbols. Thus Dk=Σ*a(Σ∪ε)k-1.Describe a DFA with at most k+1states that recognizes Dk in terms of both a state diagram and a formal description.

Short Answer

Expert verified

Deterministic finite automaton for this language is given below.

Step by step solution

01

Deterministic finite automaton.

Deterministic finite automata (or DFA) are finite state machines that accept or reject strings of characters by parsing them through a sequence that is uniquely determined by each string.

02

Deterministic finite automaton and its state diagram.

Dkbe the language consisting of all strings that have at least one'a' among lastk symbols.

Fig: Deterministic finite automaton

ThusDk=Σ*a(Σ∪ε)k-1., and a DFA deterministic finite automaton with at mostk+1 states which accepts this language.

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Most popular questions from this chapter

  1. Show that ifis a DFA that recognizes languageB, swapping the accept and non accept states inyields a new DFA recognizing the complement ofB. Conclude that the class of regular languages is closed under complement.
  2. Show by giving an example that ifM is an NFA that recognizes language C swapping the accept and non accept states in Mdoesn’t necessarily yield a new NFA that recognizes the complement of C. Is the class of languages recognized by NFAs closed under complement? Explain your answer.

Question: Let Σ={1,#}and let

Y={w|w=x1#x2#···#xkfork⩾0,eachxi∈1*,andxi≠xjfori≠j}.

Prove that Y is not regular.

The pumping lemma says that every regular language has a pumping length P , such that every string in the language can be pumped if it has length p or more. If P is a pumping length for language A, so is any length p'⩾pThe minimum pumping length for A is the smallest p that is a pumping length for A . For example, if A=01*, the minimum pumping length is 2.The reason is that the string s=0is in A and has length 1 yet s cannot be pumped; but any string A in of length 2 or more contains a 1 and hence can be pumped by dividing it so that x=0,y=1,andzis the rest. For each of the following languages, give the minimum pumping length and justify your answer.

a).0001*b).0*1*c).001∪0*1*d).0*1+0+1*∪10*1

role="math" localid="1660797009042" e).(01)*f).∈g).1*01*01*h).10(11*0)*

i).1011j).∑*

If A is any language, let A12− be the set of all first halves of strings in A so that ,

A12-={x|forsomey,|x|=|y|andxy∈A}.

Show that if A is regular, then so is A12−

Prove that every NFA can be converted to an equivalent one that has a single accept state.
See all solutions

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