Prove and discuss the following reductions.
Walk through the proof to show that the problem of proving the language of a Turning Machine is a context-free language is undecidable. (Do not use Rice’s theorem as a black box and note that this is not the same problem as Theorem 5.13 in the textbook.)
Given α1,…,αm,β1,…,βmα1,…,αm,β1,…,βm: Construct the following CFG G=(V,Σ,R,S)G=(V,Σ,R,S): V={S,S1,S2}V={S,S1,S2}, R={S→S1|S2,S1→α1S1σ1|⋯|αmS1σm|α1σ1|⋯|αmσm,S2→β1S2σ1|⋯|βmS2σm|β1σ1|⋯|βmσm}R={S→S1|S2,S1→α1S1σ1|⋯|αmS1σm|α1σ1|⋯|αmσm,S2→β1S2σ1|⋯|βmS2σm|β1σ1|⋯|βmσm} (where σiσi are new characters added to the alphabet, e.g., σi=i–σi=i_).
If the language is ambiguous, then there is a derivation of some string ww in two different ways. Supposing, wlog, that the derivations both start with the rule S→S1S→S1, reading the new characters backwards until they end makes sure there can only be one derivation, so that's not possible. Hence, we see that the only ambiguity can come from one S1S1 and one S2S2 'start'. But then, taking the substring of ww up to the beginning of the new characters, we have a solution to the PCP (since the strings of indices used after those points match).
Similarly, if there is no ambiguity, then the PCP cannot be solved, since a solution would imply an ambiguity that just follows S⇒S1⇒∗ασ~S⇒S1⇒∗ασ~ and S⇒S2⇒∗βσ~S⇒S2⇒∗βσ~, where α=βα=β are strings of matching αα's and ββ's (since the σ~σ~'s match).
Hence, we've reduced to PCP, and since that's undecidable, we're done.
Prove and discuss the following reductions. Walk through the proof to show that the problem of...
Problem 2. (Undecidable) and prove it (33 points) Formulate the following problem as a language is undecidable Given a Turing machine M determine whether L(M) is context-free Hint: you can reduce the ATM problem to this problem, as we did for the REGULARTM language problem that we discussed in class.
SUBJECT:THEORY OF COMPUTATION CAN SOMEONE PLEASE HELP ME I HAVE POSTED IT REPEATEDLY AND I KEEP GEETING INCOMPLETE / INCORRECT ANSWER . I WILL GIVE YOU A HIGH REVIEW IF YOU HELP ME AND IT IS DONE PROPERLY ! Note: Please show/explain all cases clearly for the pumping lemma and describe how your Turing machine works for each state transition. Problem 1: Non-context-free languages and Tining Machine Models B5] context-free: 쉑: Use the pumping lemma for context-free languages to show...
(Automata): prove using the pumping lemma that the following language is not context-free: where: ; b)using closure properties and the previous proof, show that the following language is not context free language: Really need your help with this, it is important for the test. please explain what you to do so i can study it throughly. thank you very much! Labc be...bc2m de fefefnghqhq.h 1, т > п> о >0; > т,п, о 0; /12, ...j2n0; k1, k2,.. k, >...
Check My Work еВook Problem Walk-Through Black-Scholes Model Assume that you have been given the following information on Purcell Industries call options: Strike price of option $12 Current stock price $13 Time to maturity of option 6 months Risk-free rate 6% Variance of stock return = 0.14 1 0.54821 N(di) 0.70823 d2 0.28363 N(d2) 0.61165 According to the Black-Scholes option pricing model, what is the option's value? Do not round intermediate calculations. Round your answer to the nearest cent. Use...
In the following problem, we will work through a proof of an important theorem of arithmetic. Your job will be to read the proof carefully and answer some questions about the argument. Theorem (The Division Algorithm). For any integer n ≥ 0, and for any positive integer m, there exist integers d and r such that n = dm + r and 0 ≤ r < m. Proof: (By strong induction on the variable n.) Let m be an arbitrary...
Advanced Linear Algebra (bonus problem) 1. (This question guides you through a different proof of part of the Decomposition Theorem. So you are not allowed to use the Decomposition Theorem when answering this question.) Let F be a field and V an n-dimensional F-vector space for n > I. Let θ E End(V) be a linear transformation and α E F an eigenvalue of. Recall that the generalised α-eigenspace of θ is a) Suppose that 0 υ Ε να and...
Subject: Proof Writing (functions) In need of help on this proof problem, *Prove the Following:* Here are the definitions that we may need for this problem: 1) Let f: A B be given, Let S and T be subsets of A Show that f(S UT) = f(s) U f(T) Definition 1: A function f from set A to set B (denoted by f: A+B) is a set of ordered Pairs of the form (a,b) where a A and b B...
I can't get it right ebook Problem Walk-Through 0.3 A stock's returns have the following distribution: Demand for the Probability of This Rate of Return If Company's Products Demand Occurring This Demand Occurs weak 0.1 (38) Below average (10) Average Above average 0.3 0.1 0.1 63 1.0 1.0 Assume the risk-free rate is 2%. Calculate the stock's expected return, standard deviation, coefficient of variation, and Sharpe ratio. Do not round intermediate ca culations. Round your answers to two decimal places....
(12 pts) Solve each of the following Do not only state your solution- Show how you obtained it. That is, if you use substitution, you must present the complete inductive proof that your solution is correct. If you obtained the solution from the tree. Note that you are to prove matching upper and lower bounds recurrences using substitution or a recursion tree. use a recursion tree, show the recursion tree and discuss how you (а) Т(п) — 4T (п/2) +...
Prove that mutual exclusion is satisfied by the following algorithm for the critical section problem: (image attached) Note that the if statement is atomic. By "if statement is atomic" it means that lines p2 and q2 are either executed in their entirety once execution begins. No interleaving is possible for example, between the "if" test and the assignment of wantp/wantq. You must prove the appropriate program invariant with the help of induction. It's okay to prove lemmas through program invariant-induction...