## Proving a Language is not Regular using Myhill-Nerode Theorem

I have the language $$L$$ $$=$$ {$$a$$ $$=$$ $$b$$ $$+$$ $$c | a$$, $$b$$, $$c$$ are binary integers, and $$a$$ is the sum of $$b$$ and $$c$$}, with alphabet $$\sum =\left\{0,\:1,\:+,\:=\right\}$$. I need to prove that this language is not regular using the Myhill-Nerode Theorem.

I am able to prove it using the pumping lemma by using $$a$$ $$=$$ $$111$$ $$=$$ $$b$$ $$+$$ $$c$$ $$=$$ $$100$$ $$+$$ $$11$$. This string has a pumping length of 2. I divide the string into $$x$$ $$=$$ $$1$$, $$y$$ $$=$$ $$1$$ and $$z$$ $$=$$ 1=100+11. So if I pump $$y$$ with $$i$$ $$=$$ $$2$$, I change $$y$$ to $$11$$. Therefore, the string after pumping is $$1111=100+11$$, which is not true, leading to a contradiction that violates the pumping lemma.

But I am not sure how I would approach this using the Myhill-Nerode theorem. I know the idea is to find an infinite set in $$L$$ that is pairwise distinguishable. But I am not sure which set I could choose. Could I use the same sets as the ones I used with the pumping lemma or would it have to be something different?

Thank you!

## Bayes theorem and randomized algorithms

Are there any randomized algorithms that make use of Bayes theorem? where are they used and why?

## Proving a language is not regular using the Myhill-Nerode Theorem

I have to prove that the following languages are not regular using the Myhill-Nerode Theorem.

$$1)$$ $$\left\{0^n1^m0^n\left|m,\:n\:\ge 0\right|\right\}$$

$$2)$$ $$\left\{w\left|w\:element\:of\:\left\{0,\:1\right\}^{star}\:is\:not\:a\:palindrome\right|\right\}$$

For the first question, I did the following:

I considered the set $$\left\{0^n1^m\left|m,\:n\:\ge 0\right|\right\}$$. To prove that this set is pairwise distinguishable by the original language, I said that for all $$m$$ and $$n$$, $$0^n1^m$$ is distinguishable from all previous $$0^i1^m,\:0\:\le i\le n-1$$ because there exists a $$z=0^n$$ such that $$0^n1^mz$$ is an element of the original language but $$0^i1^mz,\:0\le i\le n-1$$ is not an element of the original language.

I first want to ask whether this was indeed the correct way to do the proof?

I am also quite confused for the second question as I can’t even seem to find a string that is part of the language but is a palindrome. All the strings other than the empty string in that language are not palindromes. So I am quite confused on how to approach the problem.

Any help would be highly appreciated!

## This theorem: L2= { W E {a I b} * : no prefix of w starts with b}

L2= { W E {a I b} * : no prefix of w starts with b} = { W E {a, b} * : the first character of w is a} U {e}

Why is it in union with an empty string? If an empty string from b can also be a prefix.

## Does the master theorem applies to this recurrence?

The recurrence:

$$T(n) = pT(n/q) + \log n$$

for p < q and p >= 2.

So, I’ve figured out it would fall into case 1, since we have $$n^{log_{q}p} = n^r$$, for $$0, which would mean that $$f(n) = \log n$$ would be dominated by a polynomial factor, i.e $$\log n = O(n^{r-e})$$ for $$0 > e < r$$. But according to my professor, this is wrong, and, even though it’s dominated, it isn’t by a polynomial factor. So, who’s right after all?

## Impossible recurrence relation help with masters theorem, plus evaluation of complexity

I had an exercise to solve a recurrence relation in my exam, I think it was a tricky question but I am not 100% sure.

The recurrence was $$T(n) = 2*T(n) + \sqrt{n} +42$$, it was specifically asked to be solved using the master theorem, and I wrote it cannot be solved using master theorem because it requires for b>1. Do you think is correct?

After that I had a small piece of code to evaluate, after a look at it I decided it was $$O(n^3)$$

    for i in range(1,n):       j = i+1       while(j/n<=n):         k=1         while(k<=n):             constantFunction(4) # a function that executes in constant time             k=k+3         j=j+1 

In my opinion the function going from 1 to n, makes the while(j/n<=n) always be <=n as $$n-> infinity$$, which makes the other while inside execute until k<=n, k will grow fast and soon that while loop makes me wonder, will it stop ? I mean let’s assume n goes to infinity, then k will grow, but not as much as n, so it’ll be $$O(n^3)$$. Am I right?

Thank you

## solve T (n) = 2nT (n/2) + nn b) 2T (n/4) + n0.51 by master theorem in design and analysis of algorithm

T (n) = 2nT (n/2) + nn b) 2T (n/4) + n0.51 apply master theorem in design and analysis of algorithms

## How to find running time complexity of divide and conquer method without Master Theorem

Oh yes. There is nothing better than an apple for breakfast. Milk goes great with it, but not everyone can drink it.

## Recurrence relation (not solvable by the master theorem)

Consider the following recursion: $$\begin{cases} T(n) = 2T(\frac{n}{2}) + \frac{n}{\log n} &n > 1 \ O(1) &n = 1 \end{cases}$$.

The master theorem doesn’t work, as the exponent of $$\log n$$ is negative. So I tried unfolding the relation and finally got the equation: $$T(n) = n[1 + \frac{1}{\log(\frac{n}{2})} + \frac{1}{\log(\frac{n}{4})} + … + \frac{1}{\log(2)}]$$.

I do not know how to simplify (inequalities to use???) from here. A trivial method would be to assume that all reciprocal of the log terms are $$< \frac{1}{\log(2)}$$, and since there are $$\log n$$ terms, the summation of all the reciprocal-log terms is $$< \frac{\log n }{\log(2)} = \log_2 n$$, which gives $$T(n) = O(n \log n)$$. However this is a very poor approximation, as by the master theorem we can check that the time complexity for the recursive relation $$T(n) = 2T(\frac{n}{2}) + n$$ is $$O(n \log n)$$. Can someone find a tighter correct upper bound?

## What Makes A TM undecidable (using Recursion Theorem)

PROOF :We assume that Turing machine H decides ATM for the purpose of obtaining a contradiction. We construct the following machine B.

B =“On input w:

1. Obtain, via the recursion theorem, own description ⟨B⟩.
2. Run H on input ⟨B, w⟩.
3. Do the opposite of what H says. That is, accept if H rejects and reject if H accepts.”

I can not understand point 3 I want to know its logic

Does this have same logic as the halting problem where if we get a yes it loops forever so it does not halt and if it does not halt we get a no then it halts?