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Chapter 0: Q20E (page 1)

For each of the following languages, give two strings that are members and two strings that are not members—a total of four strings for each part. Assume the Σ=a,balpha-alphabet in all parts.

a.a*b*b.aba*bc.a*∪b*d.aaa*e.Σ*aΣ*bΣ*aΣ*f.aba∪babg.(ε∪a)bh.(a∪ba∪bb)Σ*

Short Answer

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Step by step solution

01

To Concern the Two Strings

There are two strings which are concerned with members, and some of the two strings are concerned with non-members. In the question, there are some calculations we have to solve, and they're related to the step-2 answer also.

02

To Explain the Given Expression

The image is given below:

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Most popular questions from this chapter

a). Let C be a context-free language and R be a regular language. Prove that the languageC∩Ris context free.

b). Let A= { w|w∈{a,b,c}*andwcontains equal numbers of a’s,b’s,andc’s}. Use part(a) to show that A is not a CFL

Show that the set of incompressible strings contains no infinite subset that is Turing-recognizable.

LetAbe the set{x,y,z}andBbe the set{x,y}.

  1. IsAa subset ofB?
  2. IsBa subset ofA?
  3. What isA∪B?
  4. What isA∩B?
  5. What isA×B?
  6. What is the power set ofB ?

For each let Ƶm = {0, 1, 2, . . . , m − 1}, and let = (Ƶm, +, ×) be the model whose universe is Ƶm and that has relations corresponding to the + and × relations computed modulo m. Show that for each m, the theory Th is decidable.

Question:Consider the algorithm MINIMIZE, which takes a DFA as input and outputs DFA .

MINIMIZE = “On input , where M=(Q,Σ,δ,q0,A) is a DFA:

1.Remove all states of G that are unreachable from the start state.

2. Construct the following undirected graph G whose nodes are the states of .

3. Place an edge in G connecting every accept state with every non accept state. Add additional edges as follows.

4. Repeat until no new edges are added to G :

5. For every pair of distinct states q and r of and every a∈Σ :

6. Add the edge (q,r) to G if δq,a,δr,a is an edge of G .

7. For each state q,let[q] be the collection of statesq={r∈Q|noedge joins q and r in G }.

8.Form a new DFA M'=Q',Σ,δ',q'0,A'where

Q'={[q]|q∈Q}(ifq=r,onlyoneofthemisinQ'),δ'(q,a)=[δq,a]foreveryq∈Qanda∈Σ,q00=[q0],andA0={[q]|q∈A}

9. Output ( M')”

a. Show that M and M' are equivalent.

b. Show that M0 is minimal—that is, no DFA with fewer states recognizes the same language. You may use the result of Problem 1.52 without proof.

c. Show that MINIMIZE operates in polynomial time.
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