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context-free grammar structural induction

$10-30 USD

Cancelled
Posted almost 9 years ago

$10-30 USD

Paid on delivery
(1) Let G = ({ S, A }, Σ, R, S) be the context-free grammar with Σ = { a, b } and rules R = S → SS | ASa | aA, A → bA | SA | a. (a) Prove by structural induction that every string generated by G (from the start symbol S, that is) has an even number ≥ 2 of a’s. Show the analysis of each production. (b) Is the grammar comprehensive for this property? If you say yes, prove it by induction on strings; if you say no, find a string x /∈ L(G) that has an even number of a’s—and strengthen the “meanings” of the variables used in your proof of (a) to demonstrate clearly that x is excluded by the grammar. (2) Define the grammar G with one variable E and terminal alphabet Σ = {a, b} by the following rules: E → e(empty string)| aE | aEbE This is the same as the grammar in problem (2) of problem set 5 except we are more prosaically using a, b in place of “spear” and “dragon.” Define A = {x ∈ {a, b}∗: (∀i : 0 ≤ i ≤ |x|)diff(x, i) ≥ 0}, where diff(x, i) = #a(x1 · · · xi) − #b(x1 · · · xi). Prove that L(G) = A. For the “L(G) ⊆ A” part it is OK to give an informal proof even talking about “spears” and “dragons”—the intent of “Structural Induction” is to make design personal and intuitive. But for the “A ⊆ L(G)” part, the proof should be technical and phrased in terms of how the diff(x, i) function might behave for a given string x.
Project ID: 7413795

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3 proposals
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Active 9 yrs ago

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3 freelancers are bidding on average $31 USD for this job
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Hi, Your project sounds very interesting and I believe that I'll be able to perform the task perfectly. I have over 15 years of experience as Physics and Math teacher, plus, as of lately, I also work as online Math tutor for Android app (Snapqu) where students send a snapshot of their question and I answer them. Should you have any questions about me, my work or my experience, I'll be happy to answer. In the meantime, see my reviews and kindly consider my proposal. Kindest regards. Milan
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hello. I saw your description and attached files. I understand it and can do it . I have done several project like this. I'm an expert in Discreet Math, Data Structures and Algorithms. And I know Java and C/C++ well. I'm interested this project. I want to discuss with you about this project. If it's possible,please contact me and explain more detail. I wait your good reply. Bye.
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I had learnt fafl last semester so i can handle this project in time and make sure that there are no flaws
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Hi, How do you do ? I came across your problems 2 hrs ago and started solving them immediately . I am submitting my proposal just after finishing the solutions. I did a Theory of Computation course in my 2nd year as an undergrad following Sipser's and Kozen's books and I am very much enthusiastic about this subject. The first problem part (a) I solved by induction and (b) I solved using an unavoidable logic. For problem 2 I solved using techniques very similar to the famous Balanced Parenthesis Problem. I can surely help you by providing my solutions to you for a minimal amount and a review from you afterwards. Thank You.
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sheridan, United States
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