Explanation Given: set of n -equivalence classes of states of M. Proof: We know that the equivalence classes of R k are a refinement of the equivalence classes of R k − 1 for each positive integer k. The equivalence classes are finite sets, and finite sets cannot be refined indefinitely (the most refined they can be is for each equivalence class to contain just one state). Therefore this sequence of refinements must stabilize and remain unchanged from some point onward. It remains to show that as soon as we have R n = R n + 1 , then R n = R m for all m > n, from which it follows that R n = R * , and so the equivalence classes for these two relations will be the same. By induction, it suffices to show that if R n = R n + 1 , then R n + l = R n + 2 ...

8th Edition

ISBN: 9781260521337

Author: ROSEN

Publisher: MCG

See similar textbooks

Videos

Question

Chapter 13.3, Problem 59E

To determine

To show:

There is a nonnegative integer n such that the set of n -equivalence classes of states of M is the same as the set of ( n + 1)-equivalence classes of states of M.

Then show for this integer n, the set of n -equivalence classes of states of M equals the set of *-equivalence classes of states of M.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.3, Problem 58E

Chapter 13.3, Problem 60E

Students have asked these similar questions

Consider the following language over the alphabet Σ = {a, b} L = {w : nb(w) is even, na(w) is odd, and w does not contain the substring ba} 1. Design the minimal deterministic finite automata which recognizes L. You do not need to prove correctness nor minimality. 2. Provide regular expressions for each of the equivalence classes of ≡L. [Hint: Use your answer from 1. A conversion algorithm is not necessary.]

Let Z denote the set of integers. If m is a positive integer, we write Zm for the system of "integers modulo m." Some authors write Z/mZ for that system. For completeness, we include some definitions here. The system Zm can be represented as the set {0, 1,..., m - 1} with operations (addition) and (multiplication) defined as follows. If a, b are elements of {0, 1,..., m - 1}, define: ab the element c of {0, 1,...,m - 1} such that a +b-c is an integer multiple of m. a b = the element d of {0, 1,..., m - 1} such that ab -d is an integer multiple of m. For example, 30 4 = 2 in Z5, 303= 1 in Z4, and -1 = 12 in Z₁3. To simplify notations (at the expense of possible confusion), we abandon that new notation and write a + b and ab for the operations in Zm, rather than writing ab and a b. = Let Q denote the system of rational numbers. We write 4Z for the set of multiples of 4 in Z. Similarly for 4Z12. Consider the following number systems: Z, Q, 4Z, Z3, Z8, Z9, 4Z12, Z13. One system may be…

Compute for (1+u+u²)(1+u).

Chapter 13 Solutions

DISCRETE MATHEMATICS+ITS APPL. (LL)-W/A

Ch. 13.1 - Exercises 1-3 refer to the grammar with start...Ch. 13.1 - Exercises 1-3 refer to the grammar with start...Ch. 13.1 - Prob. 3E Ch. 13.1 - Let G=(V,T,S,P) be the phrase-structure grammar...Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Show that the grammar given in Example 5 generates...Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

There is a nonnegative integer n such that the set of n -equivalence classes of states of M is the same as the set of ( n + 1)-equivalence classes of states of M. Then show for this integer n, the set of n -equivalence classes of states of M equals the set of *-equivalence classes of states of M.

Solution Summary: The author explains that the equivalence classes of R k are a refinement of the

Videos

Want to see the full answer?

Chapter 13 Solutions