Help with my computational theory homework Consider the language consisting of Turing Machines that accept at least two different strings,X = {(M) | M is a Turing Machine that accepts at least two different inputs}.That is, (M) EX if and only if |L(M)| ≥ 2.1. Show that X is Turing-recognizable.2. Give a mapping reduction ATM ≤m X, and explain why it works.3. What can we conclude about X from the fact that ATM ≤m X?

Let's approach this problem step by step: 1. Showing that X is Turing-recognizable:To show that X is…

Answered: Consider the language consisting of…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 32EQ

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Give a high-level description of a Turing machine that accepts the following language: = {#x, #x, #...#x, |x, e {0, 1} *,x, # x, for i z j}
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Exercise 2.1.2. Give an example of a set X and binary operation * on X such that (1) is associative, but not commutative. (2) is commutative, but not associative.
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(1.) Let T = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Suppose five integers are chosen from T. Show that it is not true that there must be two integers whose sum is 10 by giving a counterexample. In other words, fill in the blank with five numbers from T, no two of which have a sum of 10. (Enter your answer in set-roster notation.) (2.) Suppose five pairs of similar-looking boots are thrown together in a pile. What is the minimum number of individual boots that you must pick to be sure of getting a matched pair? Why? Since there are 5 pairs of boots in the pile, if at most one boot is chosen from each pair, the maximum number of boots chosen would be............? . It follows that if a minimum of............. boots is chosen, at least two must be from the same pair.
Prove that for any set ACR and any real number a we have m* (A + a) = m* (A).
4. Where N denotes the set of natural numbers, say that a set X of natural numbers is computably separable from a set Y of natural numbers if there is a decidable set D of natural numbers such that X ≤ D and Y ≤ (N\D). Now let T be a consistent axiomatizable extension of Robinson arithmetic Q, let A denote the set of Gödel numbers of sentences provable in T, and let B denote the set of Gödel numbers of sentences refutable in T. Is A computably separable from B? Justify your (positive or negative) answer to this question using rigorous mathematical argumentation.
4. Let W be the set of all words in the English language and let L be the set of letters {A, B, C,...,Z}. Let f: W→→L be the function that takes as input a word w and out- puts the first letter of the word. (a) Compute f({APPLE, CHERRY, STRAWBERRY}). (b) Give three elements in the set f-¹(P). (c) Is f a bijection? Use one or two sentences to explain your answer.
1. Consider the sentence 3xVy(A(x,y) → (B(y) ^ C (@))) (a) Build a model, using the natural numbers N {0,1, 2, · ..·} as the universe, where this sentence is true. (b) Build a model, using the natural numbers as the universe, where this sentence is false. (c) Is this sentence logically valid? Why or why not?
Suppose that a number x is to be selected from the real line R, and let A, B, and C be the eventsrepresented by the following subsets of R, where the notation {x : −−−} denotes the set containingevery point x for which the property presented following the colon is satisfied:A = {x : 1 ≤ x ≤ 5}B = {x : 3 < x ≤ 7}C = {x : x ≤ 0} Describe each of the following events as a set of real numbers:(a) Ac(b) A ∪ B(c) B ∩ C c(d) (A ∪ B) ∩ C Problem 1.2Toss a coin 4 times. Let A denote the event that a head is obtained on the first toss, and let Bdenote the event that a head is obtained on the fourth toss. Is A ∩ B empty? Problem 1.3 Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up.Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of theequalities (Ac ∩ Bc)c = A ∪ B and (Ac ∪ Bc)c = A ∩ B and verify that the equalities hold. Problem 1.4 We are given that P(A) = 0.55, P(B) = 0.35, and P(A∪B) = 0.75. Determine…
3. Show that any language A is recognizable if and only if A

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