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Chapter 0: Q11P (page 1)

Let φeqbe defined as in Problem 6.10. Give a model of the sentence

Short Answer

Expert verified

Answer:

The statement is proved below.

Step by step solution

01

Turing Machine

Let us first mention φeq.

Now consider the main statement in our question:

The statement above tells about conditions of equivalence relation and less then relation.

We will define the model (A,R1'R2).

Here, A = Any universal set

R1= Equivalence Relation on A

R2= Less then Relation on

Line (1): Describe Condition of Equivalence Relation.

Line (2): Describe that for all x,y if x=y means x is not less than y.

Line (3): Describe that for all x, y if x = y , then either x< y or y <x.

Line (4): Describe that for all x,y,z if x< y and y<z then x<z

Line (5): Describe that for all x there is y, where x<y

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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

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.

Use the procedure described in Lemma 1.60to convert the following finite automata to regular expressions.

Let X=M,wM is a single-tape TM that never modifies the portion of the tape that contains the input w. Is X decidable? Prove your answer.

Modify the proof of Theorem 3.16 to obtain Corollary 3.19, showing that a language is decidable if some nondeterministic Turing machine decides it. (You may assume the following theorem about trees. If every node in a tree has finitely many children and every branch of the tree has finitely many nodes, the tree itself has finitely many nodes.)
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