Give an asymptotic upper bound

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Reference no: EM132136341

Assignment -

1. Let (T, ∧, ∨,', 0, 1) be a Boolean Algebra.

Define ∗ : T × T → T and o : T × T → T as follows:

x ∗ y := (x ∨ y)' x o y := (x ∧ y)'

(a) Show, using the laws of Boolean Algebra, how to define x ∗ y using only x, y, o and parentheses.

(b) Show, using the laws of Boolean Algebra, how to define x o y using only x, y, ∗ and parentheses.

Define R ⊆ T × T as follows: (x, y) ∈ R if, and only if, (x ∧ y) ∨ (x' ∧ y') = 1

(c) Show, using the laws of Boolean Algebra, that R is an equivalence relation. Hint: You may want to use the observation that if A = B = 1 then A ∧ B ∧ C = A ∧ B implies C = 1 (why?)

2. Let P F denote the set of well-formed propositional formulas made up of propositional variables, T, ⊥, and the connectives ¬, ∧, and ∨. Recall from Quiz 7 the definitions of dual and flip as functions from PF to PF:

  • dual(p) = p
  • flip(p) = ¬p
  • dual(T) = ⊥; dual(⊥) = T
  • flip(T) = T; flip(⊥) = ⊥
  • dual(¬φ) = ¬dual(φ)
  • flip(¬φ) = ¬flip(φ)
  • dual(φ ∧ ψ) = dual(φ) ∨ dual(ψ)
  • flip(φ ∧ ψ) = flip(φ) ∧ flip(ψ)
  • dual(φ ∨ ψ) = dual(φ) ∧ dual(ψ)
  • flip(φ ∨ ψ) = flip(φ) ∨ flip(ψ)

 (a) For the formula φ = p ∨ (q ∧ ¬r):

(i) What is dual(φ)?

(ii) What is flip(φ)?

(b) Prove that for all φ ∈ PF: flip(φ) is logically equivalent to ¬dual(φ).

3. Let P(n) be the proposition that: for all k, with 1 ≤ k ≤ n,

1743_figure.png

(a) Prove that P(n) holds for all n ≥ 1. (Note: it is possible to do this without using induction)

We can compute 1858_figure2.pngfrom the formula given in lectures, however this can often require computing unnecessarily large numbers. For example,1543_figure3.png= 253338471349988640 which can be expressed as a 64-bit integer, but 100! is larger than a 512-bit integer. We can, however, make use of the equation above to compute 1858_figure2.pngmore efficiently. Here are two algorithms for doing this:

613_figure4.png

Let trec(n, k) be the running time for chooseRec(n, k), and let titer(n) be the running time for chooseIter(n, k). Let Trec(n) = max0≤k≤n trec(n, k) and Titer(n) = max0≤k≤n titer(n, k) (so Trec(n) ≥ trec(n, k) for all k, and likewise for Titer(n)).

(b) Give an asymptotic upper bound for Trec(n). Justify your answer.

(c) Give an asymptotic upper bound for Titer(n). Justify your answer.

Reference no: EM132136341

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len2136341

10/9/2018 10:33:25 PM

Note: In your assignment, how you arrived at your solution is as important (if not more so) than the solution itself and will be assessed accordingly. There may be more than one way to find a solution, and your approach should contain enough detail to justify its correctness. Lecture content can be assumed to be common knowledge.

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10/9/2018 10:33:20 PM

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10/9/2018 10:33:15 PM

Be careful with giving multiple or alternative answers. If you give multiple answers, then we will give you marks only for ”your worst answer”, as this indicates how well you understood the question. Some of the questions are very easy (with the help of the lecture notes or book). You can use the material presented in the lecture or book (without proving it). You do not need to write more than necessary (see comment above). When giving answers to questions, we always would like you to prove/explain/motivate your answers. If you use further resources (books, scientific papers, the internet,...) to formulate your answers, then add references to your sources.

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