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Give state diagrams of NFAs with the specified number of states recognizing each of the following languages. In all parts, the alphabet is {0,1}.

a. The language{w|wendswith00}with three states

b.The language of Exercise 1.6c with five states

c.The language of Exercise 1.6l with six states

d. The language {0} with two states

e. The language 0*1*0+ with three states

f. The language1*(001+)* with three states

g. The language{ε}with one state

h. The language 0*with one state

Short Answer

Expert verified

a.

b.

c.

d.

e.

f.

g.

h.

Step by step solution

01

Explain State diagram.

A state diagram is used in computer science and related fields to describe the behavior of system. A state diagram represents the transition from one state to another based on the inputs given.

02

(a) Give state diagrams of NFAs with the specified number of states

Considering the given language, relatively limited automata must have three states that take strings from all across the letters. Let M be the NFA that recognizes L . The state diagram of M is as follows:

03

(b) Give state diagrams of NFAs with the specified number of states 

Considering the given language, relatively limited automata must have five states that take strings from all across the letters. Let M be the NFA that recognizes L . The state diagram of M is as follows :

04

(c) Give state diagrams of NFAs with the specified number of states

Consider the languages,

L1={w|w c´Ç²Ô³Ù²¹¾±²Ô²õ²¹²Ô±ð±¹±ð²Ô²Ô³Ü³¾²ú±ð°ù´Ç´Ú0s}L2={w|w c´Ç²Ô³Ù²¹¾±²Ô²õ e³æ²¹³¦³Ù±ô²â t·É´Ç 1's}

Let M1andM2be the NFAs that recognizes the languages L1andL2respectively.

The language Lis defined as L=L1∪L2 and the NFA Mrecognizes the language L.

The state diagram of M1 is as follows,

The state diagram of M2 is as follows,

Therefore, the state diagram of M is as follows,

05

(d) Give state diagrams of NFAs with the specified number of states 

Considering the given language, relatively limited automata must have two states that take strings from all across the letters. Let M be the NFA that recognizes L . The state diagram of M is as follows :

06

(e) Give state diagrams of NFAs with the specified number of states

Considering the given language, relatively limited automata must have three states that take strings from all across the letters. Let M be the NFA that recognizes L . The state diagram of M is as follows :

07

(f) Give state diagrams of NFAs with the specified number of states 

Considering the given language, relatively limited automata must have three states that take strings from all across the letters. Let M be the NFA that recognizes L . The state diagram of M is as follows :

08

(g) Give state diagrams of NFAs with the specified number of states

Considering the given language, relatively limited automata must have one state that take strings from all across the letters. Let M be the NFA that recognizes L . The state diagram of M is as follows :

09

(h) Give state diagrams of NFAs with the specified number of states 

Considering the given language, relatively limited automata must have one state that take strings from all across the letters. Let M be the NFA that recognizes L . The state diagram of M is as follows :

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Most popular questions from this chapter

Question: Let B be the set of all infinite sequences over {0 , 1}. Show that B is uncountable using a proof by diagonalization.

Question: Each of the following languages is the intersection of two simpler languages. In each part, construct DFAs for the simpler languages, then combine them using the construction discussed in footnote 3 (page 46) to give the state diagram of a DFA for the language given. In all parts,∑=a,b.

a.{w|w\kern1pt has  at  least  three  a's  and  at  least  two  b's}b.w|w\kern1pt has  exactly  two  a's  and  at  least  two  b'sc.w|w\kern1pt has  even number  of  a's  and  one or  two  b'sd.{w|w\kern1pt has  even number  of  a's  and eachais followed by at least  one b}e.w|w\kern1pt starts  with  an  a  and  has at most one  bf.w|w\kern1pt has an odd number ofa's  and ends with a  bg.w|w\kern1pt has even length and an odd number ofa's

Myhill–Nerode theorem. Refer to Problem 1.51 . Let L be a language and let X be a set of strings. Say that X is pairwise distinguishable by L if every two distinct strings in X are distinguishable by L. Define the index of L to be the maximum number of elements in any set that is pair wise distinguishable by L . The index of L may be finite or infinite.

a. Show that if L is recognized by a DFA with k states, L has index at most k.

b. Show that if the index of L is a finite number K , it is recognized by a DFA with k states.

c. Conclude that L is regular iff it has finite index. Moreover, its index is the size of the smallest DFA recognizing it.

In certain programming languages, comments appear between delimiters such as #and #/. Let Cbe the language of all valid delimited comment strings. A member of Cmust begin with #/and end with #/but have no intervening #/. For simplicity, assume that the alphabet for Cis ∑={a,b,/,#}.

a. Give aDFA that recognizes C.

b. Give a regular expression that generates C.

Use the construction given in Theorem 1.39 to convert the following two nondeterministic finite automata to equivalent deterministic finite automata.

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