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Chapter 1: Q3E (page 83)

Question : The formal description of a DFA M is q1,q2,q3,q4,q5,u,d,未,q3,q3, where 未 is given by the following table. Draw the state diagram of this machine.

Short Answer

Expert verified

Answer:

The state diagram is

Step by step solution

01

Explain the given DFA

The specified DFA M is defined as follows:

M=(Q,鈭�,未,q3,F)Q={q1,q2,q3,q4,q5}a={u,d}未=Transition

The transition of the given DFA is given as follows:

02

Step 2: Draw the state diagram of the given machine

Assume that the initial state and the final state is q3, and represent it by double circle. The transitions are made from one state to another based on the input symbol and the data from the table given. The state diagram for the DFA M is as follows:

Therefore, the state diagram for the machine M has been obtained.

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Most popular questions from this chapter

A finite state transducer (FST) is a type of deterministic finite automaton whose output is a string and not just accept or reject. The following are state diagrams of finite state transducers T1andT2.

Each transition of an FST is labeled with two symbols, one designating the input symbol for that transition and the other designating the output symbol. The two symbols are written with a slash, I, separating them. In T1, the transition from q1鈥�toq2has input symbol 2 and output symbol 1. Some transitions may have multiple input鈥搊utput pairs, such as the transition in T1from q1to itself. When an FST computes on an input string w, it takes the input symbols w1路路路wnone by one and, starting at the start state, follows the transitions by matching the input labels with the sequence of symbols w1路路路wn=w. Every time it goes along a transition, it outputs the corresponding output symbol. For example, on input 2212011, machine T1enters the sequence of states q1,q2,q2,q2,q2,q1,q1,q1and produces output 1111000. On input abbb, T2outputs 1011. Give the sequence of states entered and the output produced in each of the following parts.

a. T1on input011

b. T1on input211

c. T1on input121

d. T1on input0202

e. T2on input b

f. T2on input bbab

g. T2on input bbbbbb

h. T2on input localid="1663158267545" 蔚

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�*

We define the avoids operation for languages A and B to be

AavoidsB={w|w鈭�Aandwdoesn鈥�tcontainanystringinBasasubstring}.

Prove that the class of regular languages is closed under the avoids operation.

Let M=(Q,危,未,q0,F)be a DFA and let be a state of Mcalled its 鈥渉ome鈥�. A synchronizing sequence for M and h is a string s鈭埼ｂ垪where未(q,s)=hforeveryq鈭�Q. (Here we have extended to strings, so that未(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?

Let A be any language. Define DROP-OUT(A)to be the language containing all strings that can be obtained by removing one symbol from a string in A. Thus,DROP-OUT(A)={xz|xyz鈭�A鈥�鈥�where鈥�鈥�x,z鈭�鈭�*,y鈭�鈭�} . Show that the class of regular languages is closed under the DROP-OUT operation. Give both a proof by picture and a more formal proof by construction as in Theorem 1.47.
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