代写CS 3800 Spring 2024 Homework 6代做Processing

2024.09.30 - 首页 >> CS

CS 3800

Spring 2024

Homework 6

3. [14 Points] Pumping in regular languages. For each of the following languages (over Σ = {0, 1}), give its minimum pumping length and justify your answer.

(a) 1*0*

(b) ε

(d) (111)*0*

(e) {w ∈ {0, 1}* | w contains neither 000 nor 111 as substrings}

4. [10 Points] Proving that languages aren’t regular. Use the Pumping Lemma to show that the following languages over Σ = {a, b} are not regular. In each case, carefully describe the string that will be pumped and explain why pumping it leads to a contradiction.

(a) {a a a bn an | n ≥ 0}

(b) {w w | w ∈ Σ*}

(w stands for w with each occurrence of a replaced by b, and vice versa.)

5. [5 Points] Regular or not? Is the language {ai bj | 0 ≤ i ≤ j} regular or not? Carefully prove your answer. Aim for a proof that is as simple as possible. You may use any results from class.

6. ⋆ [6 Points] Halfway through. Given a language A ⊆ Σ*, we write A2/1 to denote the set of left halves of words in A. That is

A2/1 = {x ∈ Σ* | ∃y (y ∈ Σ* and |x| = |y| and xy ∈ A)}

For example, if A = {ε, a, ba, aba, aabb}, then A2/1 = {ε, b, aa}.

Prove that if A is regular, then A2/1 is also regular.

[To do so, assume that A is accepted by some DFA M = (Q, Σ, δ, s, F) and construct a DFA (or NFA) M′ = (Q′ , Σ, δ′ , s′ , F′ ) that accepts A2/1. Specify all components of M′ and explain what it does.]

7. [10 Points] Parse Trees. Consider the context-free grammar for arithmetic expressions that was discussed in class. Its rules were

E → E + T | T

T → T × F | F

F → (E) | a

Give parse trees and derivations for the following strings

(a) a

(b) a + a + a

8. [24 Points] CFGs. Give context-free grammars that generate the following languages over the alphabet Σ = {a, b}. In each case, try to minimize the number of variables and specify all components of the quadruple (V, Σ, R, S).

(a) The empty language ∅.

(b) Σ*.

(d) {ambn | 0 ≤ m ≤ n}

(e) {a a a bn an | n ≥ 0}

(f) {w | w contains at least four a’s}.

(g) {w | w ≠ ε and w begins and ends with the same symbol}.

9. [10 Points] CFGs. Construct context-free grammars that generate the languages below. For each grammar specify all components of the quadruple (V, Σ, R, S).

(a) L1 = {ai bj ck | j = i + k and i, j, k ≥ 0}

(b) L2 = {ai bj ck | i = j + k and i, j, k ≥ 0}

10. [9 Points] Right-regular grammars. Construct right-regular grammars for the following lan-guages.

(a) {w ∈ {a, b}* | the length of w is odd}

(b) {ai bj | i, j ≥ 0 and i + j is odd}

11. [12 Points] Chomsky. Use the method from class (lecture 13) to convert each of the following grammars to Chomsky normal form. Note that not all passes are always needed.

(a) S → a S a b | B

B → b b C | b b

C → ε | c C

(b) S → A B

A → a | B

B → b | A | ε