## Can deductive inference and formal proof be unified this way?

Philosophy of Logic – Reexamining the Formalized Notion of Truth https://philpapers.org/archive/OLCPOL.pdf

## Can a monster with multiattack use this ability if they are missing a limb?

For example, could a Giant Ape with one arm still use Multiattack even though it is listed as “two fist attacks”?

## Why does this trick to derive the formula for $[A^n,B]$ in terms of repeated commutators work so well?

It is a known result that, given generically noncommuting operators $$A,B$$, we have $$A^n B=\sum_{k=0}^n \binom{n}{k} \operatorname{ad}^k(A)(B) A^{n-k}.\tag A$$ This can be proved for example via induction with not too much work.

However, while trying to get a better understanding of this formula, I realised that there is a much easier way to derive it, at least on a formal, intuitive level.

### The trick

Let $$\hat{\mathcal S}$$ and $$\hat{\mathcal C}$$ (standing for “shift” and “commute”, respectively) denote operators that act on expressions of the form $$A^k D^j A^\ell$$ (denoting for simplicity $$D^j\equiv\operatorname{ad}^j(A)(B)$$) as follows:

\begin{align} \hat{\mathcal S} (A^k D^j A^\ell) &= A^{k-1} D^j A^{\ell+1}, \ \hat{\mathcal C} (A^{k-1} D^{j+1} A^\ell) &= A^{k-1} D^j A^{\ell+1}. \end{align} In other words, $$\hat{\mathcal S}$$ “moves” the central $$D$$ block on the left, while $$\hat{\mathcal C}$$ makes it “eat” the neighboring $$A$$ factor.

It is not hard to see that $$\hat{\mathcal S}+\hat{\mathcal C}=\mathbb 1$$, which is but another way to state the identity $$A[A,B]=[A,B]A+[A,[A,B]].$$ Moreover, crucially, $$\hat{\mathcal S}$$ and $$\hat{\mathcal C}$$ commute. Because of this, I can write

$$A^n B=(\hat{\mathcal S}+\hat{\mathcal C})^n (A^n B)=\sum_{k=0}^n\binom{n}{k} \hat{\mathcal S}^{n-k} \hat{\mathcal C}^{k}(A^n B),$$ which immediately gives me (A) without any need for recursion or other tricks.

### The question

Now, this is all fine and dandy, but it leaves me wondering as to why does this kind of thing work? It looks like I am somehow bypassing the nuisance of having to deal with non-commuting operations by switching to a space of “superoperators”, in which the same operation can be expressed in terms of commuting “superoperators”.

I am not even sure how one could go in formalising this “superoperators” $$\hat{\mathcal S},\hat{\mathcal C}$$, as they seem to be objects acting on “strings of operators” more than on the elements of the operator algebra themselves.

Is there a way to formalise this way of handling the expressions? Is this a well-known method in this context (I had never seen it but I am not well-versed in this kinds of manipulations)?

## Is this string approach some way better

I usually store strings as const before use them inside a for loop, I don’t know why but someone told me that type the string inside the loop is creating the same string every iteration, is that true? I mean:

private const string ADDRESS = "/User/Data/data";  private string[] Example() {     var userData = new string[data.Length];      for (int i = 0; i < data.Length; i++)     {         userData[i] = xmlDocument.DocumentElement.SelectSingleNode(ADDRESS + i);     }      return userData; } 

VS

private string[] Example() {     var userData = new string[data.Length];      for (int i = 0; i < data.Length; i++)     {         userData[i] = xmlDocument.DocumentElement.SelectSingleNode("/User/Data/data" + i);     }      return userData; } 

## Can anyone find the logic in this [on hold]

Can anyone try to find the connection between the inputs and outputs, they are logically connected (https://i.stack.imgur.com/umA46.jpg)

## Can I visit outside Incheon Airport with this conditions?

1. I have 17 hours layover for my CEB-AUH flight via Incheon
2. I am holding an active Working/Resident visa in the United Arab Emirates valid until 2020
3. I have an active US Tourist Visa valid until 2028

## Is this an easter egg of Google Docs?

So today is Friday for me and I notice something –

tgif- Thank God it’s Friday

Is it an easter egg or some kind of actual parameter?

## Meaning of strange Pointer in assembly code – is this a vector table address?

Apologies if this is a silly question, but I’m confused. I’m working on a reverse engineering assignment. While looking at a disassembled dll of possibly malicious code, I found these lines:

push nsize                  ;  makes sense push offset Security_Attr   ;  makes sense push 80h                    ;  this address does not make sense push offset read_buf        ;  makes sense call CreatePipe             ;  makes sense 

This is calling a Windows function called CreatePipe. “80h” should point to a buffer that the pipe writes to. The value just seems way too small! Is this address pointing to the user_interrupt section of the vector table? If so, is this pipe overwriting user_interrupt handlers in the vector table?

Any pointers are appreciated.

## What information about a monster can a PC expect to know (be told really) if this is the 1st ever encounter with this type of monster? [duplicate]

• How much do I tell new players about new monsters? 7 answers

As an example, I’m considering vulnerabilities, resistances and immunities. If a pc is unfamiliar with a monster and asks for this information, is it reasonable to provide this.
Another way to ask this maybe is: Is there information that is typically not provided on a 1st ever encounter?