A. Rex

If the data is collected, then regardless of supposed "confidentiality", it can be shared by the government agency intentionally, or it can be hacked, or it can be leaked by a politically-motivated actor. Examples: The Census Bureau provided the military with information such as names and addresses from the 1940 Census to help identify and incarcerate Japanese Americans. The Office of Personnel Management data breach (attributed to China) included millions of fingerprints. Charles Littlejohn stole and leaked thousands of IRS records (but was punished for only one instance).

@StevenGubkin: I don't know which papers Andreas Blass refers to, but such a proof of the Lagrange inversion theorem can be found in "Functional composition patterns and power series reversion" by George N. Raney (1960). pdfs.semanticscholar.org/3792/… You can find discussion of this in Flajolet & Sedgewick in item "44 Conjugacy principle and cycle lemma" and the rest of its page (where they refer to "Polish notation" as "Łukasiewicz codes") at algo.inria.fr/flajolet/Publications/book061023.pdf#page=81

Remark: The cross product is bilinear, or linear in each argument separately. It is not linear in the inputs considered together.

One excellent writeup of this correspondence appears in the book "Games, Puzzles, and Computation" by Robert Hearn and Erik Demaine. (This is the former's thesis.) To summarize: "There is a fundamental connection between the notions of game and of computation" and "games should be thought of as a valid and important model of computation, just as Turing machines are".

One can also run this correspondence in reverse. If you have a formal language L, then you can come up with a scheme for "splitting" the strings in the language into pairs. Then one can interpret the first element of the pair as the "program" and the second element of the pair as the "input". Then this becomes a (complete) semantic description of a model of computation or a programming language. You can ask all sorts of questions about the power of this model of computation.

This correspondence fully captures the semantics of the programming language or computational model, and thus considerations of computability. (It is insufficient, as written, to talk about time/space complexity.) Turing explicitly goes through the process of constructing this language in his 1936/1937 paper "On Computable Numbers, with an Application to the Entscheidungsproblem".

@PeterTaylor, regarding our earlier discussion about "languages": programming languages and computational models are in fact very strongly related to formal languages. Specifically, if you have a programming language or computational model C, then it corresponds to the formal language of pairs (c,x) where c is an instance ("program" or "machine") of the programming language, x is an input string, and c accepts ("returns true") x when run.

@SriotchilismO'Zaic: 1. You are mistaken, because the question says to include (verification) code. 2. Both the verification code required by the question, and the search code one has to write to even come up with a single answer to this question are much more involved than a 18-line proof in ring theory, which doesn't even have loops or conditional statements. 3. The meta post you link probably doesn't literally apply, but the one I linked to in the comments above does.

@SriotchilismO'Zaic: Just to clarify, even though the amount of meta-programming required to submit an answer to this question is much greater than the amount of programming required to submit an answer to the "(-a) × (-a) = a × a" question, it doesn't count as programming-related because it's meta-programming?

@PeterTaylor: I modified the question to require participants to include verification code. I also further added some discussion to the end of the question regarding why this is a coding challenge. I think the question should be reopened.

@KevinCruijssen: A couple days ago, I added a more prominent link to the word list for this challenge. Just now, I added some discussion to my question that addresses your other point regarding the record NYT grid. I think you already understood the clarification in my earlier comment, but this is more fleshed out.

I was considering asking on Meta about questions like this one (and I still might), but then I found that the issue has already been raised (obviously..!), and it seems that the consensus is questions like these are allowed: codegolf.meta.stackexchange.com/a/9604/49643

@PeterTaylor: How about if the question was changed to say that submissions have to include verification code?

@RobinRyder: that's a fine suggestion. The main issue I see is that's potentially much harder. An analog would be if the Tetris in Conway's Game of Life question took, as input, the rules of the cellular automaton.

@PeterTaylor: I didn't, and I'm not being flippant. Deciding whether a crossword grid (of arbitrary size) is fillable with words from CSW is almost certainly NP-complete. In other words, it (probably) encodes Boolean logic. I feel relatively confident in asserting that as a computational model, "crosswords with words in CSW" is similar to Boolean circuits. (This is the same as Conway's Game of Life on a finite grid.) What distinction are you drawing between Conway's Game of Life and crossword puzzles?

@PeterTaylor: and this question asks for an implementation of a simpler spec in a programming language called "crosswords with words in CSW".

@SriotchilismOZaic: can you comment on your vote to close this question? Your own highest-voted question does not require coding, and I think some people DID do it without coding: codegolf.stackexchange.com/q/143822

@PeterTaylor and others who voted to close: I believe this question is in fact most appropriate for this site. Consider as a similar example literally the highest-voted question on this site: "Build a working game of Tetris in Conway's Game of Life". That question does not require that you write any code, and in principle a non-programmer could win that challenge. Of course, in practice, you're not going to be able to do so without writing (lots of!) code. Similarly here, I don't think it's possible for a human to verify the baseline is correct without a program.

@KevinCruijssen: suppose all of the words in that crossword were in CSW. It would be invalid because, for example, the bottom-right letter can be changed from a D to an R while keeping everything a word. This would mean it is not fillable in a unique way. Does that help clarify?

@KevinCruijssen: there are many links in the post to plain text files, including this one: boardgames.stackexchange.com/a/38386/5407

@SriotchilismO'Zaic: 1. You are mistaken, because the question says to include (verification) code. 2. Both the verification code required by the question, and the search code one has to write to even come up with a single answer to this question are much more involved than a 18-line proof in ring theory, which doesn't even have loops or conditional statements. 3. The meta post you link probably doesn't literally apply, but the one I linked to in the comments above does.

@SriotchilismO'Zaic: Just to clarify, even though the amount of meta-programming required to submit an answer to this question is much greater than the amount of programming required to submit an answer to the "(-a) × (-a) = a × a" question, it doesn't count as programming-related because it's meta-programming?

@PeterTaylor: I modified the question to require participants to include verification code. I also further added some discussion to the end of the question regarding why this is a coding challenge. I think the question should be reopened.

@KevinCruijssen: A couple days ago, I added a more prominent link to the word list for this challenge. Just now, I added some discussion to my question that addresses your other point regarding the record NYT grid. I think you already understood the clarification in my earlier comment, but this is more fleshed out.

I was considering asking on Meta about questions like this one (and I still might), but then I found that the issue has already been raised (obviously..!), and it seems that the consensus is questions like these are allowed: codegolf.meta.stackexchange.com/a/9604/49643

@PeterTaylor: How about if the question was changed to say that submissions have to include verification code?

@RobinRyder: that's a fine suggestion. The main issue I see is that's potentially much harder. An analog would be if the Tetris in Conway's Game of Life question took, as input, the rules of the cellular automaton.

@PeterTaylor: I didn't, and I'm not being flippant. Deciding whether a crossword grid (of arbitrary size) is fillable with words from CSW is almost certainly NP-complete. In other words, it (probably) encodes Boolean logic. I feel relatively confident in asserting that as a computational model, "crosswords with words in CSW" is similar to Boolean circuits. (This is the same as Conway's Game of Life on a finite grid.) What distinction are you drawing between Conway's Game of Life and crossword puzzles?

@PeterTaylor: and this question asks for an implementation of a simpler spec in a programming language called "crosswords with words in CSW".

@SriotchilismOZaic: can you comment on your vote to close this question? Your own highest-voted question does not require coding, and I think some people DID do it without coding: codegolf.stackexchange.com/q/143822

@PeterTaylor and others who voted to close: I believe this question is in fact most appropriate for this site. Consider as a similar example literally the highest-voted question on this site: "Build a working game of Tetris in Conway's Game of Life". That question does not require that you write any code, and in principle a non-programmer could win that challenge. Of course, in practice, you're not going to be able to do so without writing (lots of!) code. Similarly here, I don't think it's possible for a human to verify the baseline is correct without a program.

@KevinCruijssen: suppose all of the words in that crossword were in CSW. It would be invalid because, for example, the bottom-right letter can be changed from a D to an R while keeping everything a word. This would mean it is not fillable in a unique way. Does that help clarify?

@KevinCruijssen: there are many links in the post to plain text files, including this one: boardgames.stackexchange.com/a/38386/5407

@ClementC. and kdog: That post has 13 upvotes and 0 downvotes.

Is there some characterization of which $X$ and $Y$ have this property, or some other implications between this and anything else, ... ?

I have an unrelated question: suppose I have two nonnegative symmetric matrices $X$ and $Y$, and for all $0 \le p \le 1$, I have $X^p \succeq Y^p$ (ie, $X^p - Y^p$ is positive semi-definite) -- where here by $X^p$ I mean \emph{entrywise} $p$th powers.

Nothing enlightening.

$\phi(u,w_1)$ or else there are problems with things being pure tensors at all. Etc etc, with seemingly at most $u_1$, $u_2$, $w_1$, and $w_2$ needing to be considered.

Anyway, continuing like "if $\phi(u,w_1) \ne 0$ and $s=1$, then $\phi(u,w_2)$ must be a multiple of

So I'm trying something like $s u\otimes w_1 + t u\otimes w_2 \mapsto s\phi(u,w_1) \otimes w_1 + t\phi(u,w_2)\otimes w_2$, but on the other hand, the LHS is $u \otimes (sw_1 + tw_2)$ so the RHS equals $\phi(u,sw_1 + tw_2) \otimes (sw_1 + tw_2)$.

I want that it doesn't depend on $w$.

It's linear in $u$ because F is an endomorphism.

Err, make that $u_1 \otimes w_2$.

I'm trying things like: If $u_1\otimes w_1$ goes to $u'_1 \otimes w_1 \ne 0$, then where can $u_2\otimes w_2$ go?

By actually trying to prove it. It seems to me that if you analyze what happens to $u_1 \otimes w$ and $u_2 \otimes w$ (and perhaps $w_1$ and $w_2$ as well) that things only work out if $u'$ doesn't depend on $w$.

My intuition now suggests that your desired result is true.

Okay, so you don't mean "for any u, for any u'", you do mean what I said the first time, right?

(That was an accident, sorry.)

@Bib

What kind of quantification is on u and u'? "For all w in W, for all u in U, there exists u' in U"?