## Prove with pumping theorem that the language { \$a^n b^n b^m a^m | n ≠ m \$ } is not context free

I’m having a trouble proving it to be context free, with an example if I take w = $$a^k b^k b^{k+1} a^{k+1}$$

then it would be problematic if the partition of $$vxy$$ with $$|v| = |y|$$ was in the $$b^{k+1} a^{k+1}$$ part as I can’t get it to be equal with $$k$$ or make the amount of $$a$$ and $$b$$ different

It is also allowed to be proven with Ogden’s Lemma

## context free – Eliminating left recursion using a loop construct

I’m trying to merge a little bit of theory with a little bit of practice. I’m writing a parser-generator that generates a top-down parser based on a given grammar.

I’d like to handle left- and right recursion properly, so they would each yield the “left-recursive” and “right-recursive” tree respectively (I’m not sure if left- and rightmost derivative would be a correct term here, that would mean the whole tree). This is very important for PEG, where the associativity and order of operations is encoded in the tree. The problem is that top-down parsers can’t deal with left-recursion directly.

I know the elimination rule for left-recursion, which is given a rule in the form

$$A rightarrow A alpha_1 | dots | A alpha_n | beta_1 | dots | beta_m$$

We can rewrite it into right-recursion using the following 2 rules:

$$A rightarrow beta_1 A’ | dots | beta_m A’ \ A’ rightarrow alpha_1 A’ | dots | alpha_n A’ | varepsilon$$

This is fine for the “formal” part, but the grammar became right-recursive. To me this rule seems like it’s just using right-recursion to essentially loop parts of the grammar.

So my idea was to introduce a looping construct that would account for folding the tree left. If I introduce a new postfix operator `*` that would mean repeating the construct 0 or more times, would the following also be a valid equivalent left-recursion elimination for A?

$$A rightarrow (beta_1 | dots | beta_m) (alpha_1 | dots | alpha_n)^*$$

Note that since the repetition can occur 0 times, this should account for the “epsilon-case” too.

I have a strong suspicion that this is equivalent, as I did something very similar when manually parsing left-associative operators in math expressions. I’d like confirmation before introducing it in a general tool.

## sharepoint online – Context menu jumps

When right clicking on documents in SharePoint Online to reveal the context menu, the menu jumps for some reason which results in clicking on the wrong options. For example, if you want to click Download, the jump causes you to accidently click on Delete as shown in the video in the link below:

View post on imgur.com

Is it possible to stop this from happening?

Using fully updated Windows 10 and Edge.

## Add a JS SearchBox over a Leaflet Maps View (D7) (Context Asset Load)

I’m struggling to add a Google style Search input box over my full page Views Leaflet Map having added Leaflet-Custom-searchbox Any direction/advice would be very much welcome.
Thanks.

## 8 – How can I use “OR” in the context filters of the view?

I have a site built with Drupal 8. I created a view with 2 relationships for 2 different entities. I added to this view 2 context filter (store ID) for the 2 relations.

My problem is that View doesn’t offer “OR” for context filters. I tested the following module but it does not work :

https://www.drupal.org/project/views_contextual_filters_or

How can I use “OR” in the context filters of the view?

Can we rewrite the contextual filters or there are other solutions.

## 8 – How to use one of the 2 context filters in a view (with an OR)?

I have a view with 2 relations with the store for 2 different entities.

I want to filter the view with the store id for the 2 relationships.

I created 2 context filters in the views. You must use the first context filter OR the second.

## sharepoint online – SPFx – How to know if a user is in the context of a document set

We want to display custom menu buttons (through CommandSet) inside a document set (when the user has selected a document set in a document library and is browsing the contents of the set). We haven’t found a solution, through SPFx, which gives us information that the user is inside a document set. Neither Url or _sppageContextInfo object contains this information.

If there any way to get this information through SPFx?

## For two languages over the same alphabet that are not context free, can their intersection be context free?

Or does at least one of them have to be context free?

## real analysis – Statistical efficiency in context of neural networks?

According to https://en.wikipedia.org/wiki/Efficiency_(statistics), an estimator with statistical efficiency is one that “… needs fewer observations than a less efficient one to achieve a given performance.”

But I am reading Deep Learning by Aaron C. Courville, Ian J. Goodfellow, and Yoshua Bengio, in which they also use statistical efficiency. However, I do not see if they intend to mean the same thing with the use statistical efficiency. The example that I would like for you to look at is the following(it appears in the context of explaining the potential advantages of using convolutions in neural networks. $$m$$ is input size, $$n$$ is number of outputs, and $$k$$ is the kernel size):

In a convolutional neural net, each member of the kernel is used at
every position of the input (except perhaps some of the boundary
pixels, depending on the design decisions regarding the boundary). The
parameter sharing used by the convolution operation means that rather
than learning a separate set of parameters for every location, we
learn only one set. This does not affect the runtime of forward
propagation—it is still O(k × n)—but it does further reduce the
storage requirements of the model to k parameters. Recall that k is
usually several orders of magnitude less than m. Since m and n are
usually roughly the same size, k is practically insignificant compared
to m ×n. Convolution is thus dramatically more efficient than dense
matrix multiplication in terms of the memory requirements and
statistical efficiency.

I cannot see why using having less parameters to store and thus fewer operations to carry out in itself has anything to do with statistical efficiency as per the definition in the wiki link. Sadly, statistical efficiency is not defined in the book so must refer to some universal definition of sorts.