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Chapter 1: Q62P (page 92)

Let 危={a,b}.For each k鈮�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=危*a(危鈭�蔚)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=危*a(危鈭�蔚)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

If A is a set of natural numbers and k is a natural number greater than 1, let

Bk(A)={w|wistherepresentationinbasekofsomenumberinA}.

Here, we do not allow leading 0s in the representation of a number. For example ,B2({3,5})={11,101}and B3({3,5})={10,12}.Give an example of a set A for which B2(A)is regular butB2(A) is not regular. Prove that your example works.

Prove that for each n>0, a languageBn exists where

  1. Bnis recognizable by an NFA that hasn states, and
  2. If Bn=A1鈭�鈥�鈭�Ak, for regular languages Ai, then at least one of theAi requires a DFA with exponentially many states.

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'猢�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鈭�0*1*d).0*1+0+1*鈭�10*1

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

i).1011j).鈭�*

Question:Read the informal definition of the finite state transducer given in Exercise 1.24. Prove that no FST can output WR for every input if the input and output alphabets are {0,1}

Question:

a. Let B={1ky|y鈭�{0,1}*and ycontainsatleastk1s,fork猢�1}. Show that B is a regular language.

b. Let C={1ky|y鈭�{0,1}* and ycontainsatmostk1s,fork猢�1}. Show that C isn鈥檛 a regular language.
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