CS 3800 Homework 3 Spring 2024R

Java Python CS 3800

Homework 3

Spring 2024

1. [6 Points] In the shortlex numbering of strings over the alphabet {0, 1, 2, 3} what number corre-sponds to string 0 0 2 1 3 0 2 ?

2. [10 Points] In the shortlex numbering of strings over the alphabet {a, b, c}, what string has number 28599 ?

3. [8 Points] Suppose that S0, S1, S2, . . . is a sequence of subsets of N. Which one of the following sets is guaranteed to be different from all sets in the sequence?

(a) S = {n | n ̸∈ Sn2 }

(b) S = {n | n 2 ̸∈ Sn2 }

(c) S = {n 2 | n ̸∈ Sn2 }

(d) S = {n 2 | n 2 ̸∈ Sn}

(e) S = {n 2 | n ̸∈ Sn}

Explain why your chosen set is guaranteed not to be in the sequence S0, S1, S2, . . .

4. [9 Points] Use (a modified) diagonalization to show that for each sequence S0, S1, S2, . . . of subsets of N, there is a set S ⊆ N consisting only of even numbers such that

S ≠ S0, S1, S2, . . .

Give a formula for your set S, similar to those suggested in Problem 3.

5. [4 Points] Alice and Bob. Alice found a proof that no numbering of finite subsets of N is possible, that includes every finite subset of N; that is, for each sequence S0, S1, S2, . . . of finite subsets of N, there is some finite subset S ⊆ N that isn’t in the sequence. Here is her argument:

Consider any sequence S0, S1, S2, . . . of finite subsets of N.

1. Claim: some finite subset of N is not in the sequence.

2. Proof of the claim:

3. Define a new subset S ⊆ N by:

S = {n ∈ N | n ̸∈ Sn}

4. Thus for each n ∈ N, number n guarantees that S ≠ Sn

5. Therefore S is not a set of the sequence. □

CS 3800 Homework 3 Spring 2024R

Bob claims that there is an error in Alice’s argument. Is he right? If yes, what is the flaw? If no, does Alice’s result contradict our results in lecture?

6. [5 Points] Cantor’s Theorem states that for every function f : X → P(X) from a set X to its power set, there exists a set S ⊆ X such that S ≠ f(x) for all x ∈ X. This set S is constructed using Cantor’s diagonalization method.

Suppose that X = {a, b, c, d, e} and that f is given by

f(a) = {a, c, d, e}

f(b) = {a, d, e}

f(c) = ∅

f(d) = {a, c, d}

f(e) = {a, b, d}

What is the set S constructed by Cantor’s diagonalization method?

7. [9 Points] Consider the finite automaton whose diagram is depicted below.

(a) Describe the language recognized by this automaton.

(b) Write down this automaton’s formal (5-tuple) description. (Specify each of the 5 compo-nents.)

8. [5 Points] Draw the state transition diagram for the DFA whose formal description is

({q1, q2, q3}, {a, b}, δ, q1, {q1, q2})

where δ is the function given in the following table:

9. [6 Points]. Tint. Submission is separate from the other problems. Instructions will be posted on Piazza. Let language L consist of all binary strings in which any two successive zeros are separated by an even number of ones. For example, ε ∈ L and 101101 ∈ L and 101100111101 ∈ L but 010 ̸∈ L and 1011011101 ̸∈ L. Write a tint program for a DFA over the alphabet {0, 1} such that L is the language accepted by your DFA.

10. [0 Point] Do not submit. Exercise 1.1 page 83. The solution is in the book page 94, this is for practice only.

11. [0 Point] Do not submit. Exercise 1.2 page 83. The solution is in the book page 94, this is for practice only