Question: A homomorphism is a function f : from one alphabet to strings over another alphabet. We can extend f to operate on

A homomorphism is a function f : Σ → Γ from one alphabet to strings over another alphabet. We can extend f to operate on strings by defining f(w) = f(w₁)f(w₂) · · · f(wn), where w = w₁w₂ · · ·w_n and each w_i ∈ Σ. We further extend f to operate on languages by defining f(A) = {f(w)| w ∈ A}, for any language A.

a. Show, by giving a formal construction, that the class of regular languages is closed under homomorphism. In other words, given a DFA M that recognizes B and a homomorphism f, construct a finite automaton M₀ that recognizes f(B). Consider the machine M0 that you constructed. Is it a DFA in every case?

b. Show, by giving an example, that the class of non-regular languages is not closed under homomorphism.

Step by Step Solution

★★★★★

3.61 Rating (180 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a Yes the class of regular languages is closed under homomorphism To construct a finite ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Introduction theory computation Questions!

(a) Prove that a homomorphism is onto if and only if its rank equals the dimension of its codomain. (b) Conclude that a homomorphism between vector spaces with the same dimension is one-to-one if and...

(a) Suppose f is a one-to-one differentiable function and its inverse function f-1 is also differentiable. Use implicit differentiation to show that provided that the denominator is not 0. (b) If...

(a) Let(R, +, )and(S, , ) be rings with zero elements ZR and Zs, respectively. If f: R S is a ring homomorphism, let K = [a R | f(a) = Zs} Prove that K is an ideal of R. (K is called the kernel of...

A homomorphism is a function f : from one alphabet to strings over another alphabet. We can extend f to operate on strings by defining f(w) = f(w1)f(w2) f(wn), where w = w1w2 wn and each wi . We...

A homomorphism is a function f : ' from one alphabet to strings over another alphabet. We can extend f to operate on strings by defining f ( w ) = f ( w 1 ) f ( w 2 ) d o t s f ( w n ) , where w = w...

A bomomorphism is a function f : l o n g r i g h t a r r o w * from one alphabet to strings over another alphabet. We can extend f to operate on strings by defining f ( w ) = f ( w 1 ) f ( w 2 ) c d...

Microkernel operating systems aim to address perceived modularity and reliability issues in traditional "monolithic" operating systems. (i) Describe the typical architecture of a microkernel...

In this question you will be asked to reflect on a project you have been involved in or observed, in which a design evolved, or could have evolved, through applying a theory of user behaviour. You...

In a Hopfield neural network configured as an associative memory, with all of its weights trained and fixed, what three possible behaviours may occur over time in configuration space as the net...

re Regular Languages and Finite Automata (a) Let L be the set of all strings over the alphabet {a, b} that end in a and do not contain the substring bb. Describe a deterministic finite automaton...

Mike Ray and Tony Farnham had formed a small engineering company in 1977, Farnray Tools, that made small wooden-handled garden tools such as trowels and hand forks. Mike and Tony had started in a...

I. M. Aruban has a sandwich shop in a downtown business district. Several of his customers have said that they would purchase from his shop more often if he offered a delivery service. I. M. is...

54. LO.10 The MGP General Partnership was created on January 1 of the current year by having Melinda, Gabe, and Pat each contribute $10,000 cash to the partnership in exchange for a one-third...

What publisher was the first to understand the commercial value of citators? What company currently owns this citator service?

Describe in detail how to swap two nodes x and y (and not just their contents) in a singly linked list L given references only to x and y. Repeat this exercise for the case when L is a doubly linked...

Describe in detail an algorithm for reversing a singly linked list L using only a constant amount of additional space.

The number of operations executed by algorithms A and B is 8nlogn and 2n 2 , respectively. Determine n 0 such that A is better than B for n n 0 .

Pittman Company is a small but growing manufacturer of telecommunications equipment. The company has no sales force of its own; rather, it relies completely on independent sales agents to market its...

answer the question in english.. iv. Additional Information: QUESTION 1 (30 marks) En Adam, an accountant at Ehsan Manufacturing Sdn Bhd (EMSB). He is preparing financial reports for the year ending...

On January 1, 2021, Displays Incorporated had the following account balances: Accounts Debit Credit Cash $ 32,000 Accounts receivable 29,000 Supplies 35,000 Inventory 65,000 Land 237,000 Accounts...