Re:
by topato on Sunday November 05, 2023 @06:39AM
(#38150250)
I'm glad to know other people are annoyed by this rather perverse style of writing. I associate it with "web business guru" / "side-hustle tutorial" / "how I made X amount of dollars in Y months", and it reeks of ultra thirdh::) xx
For a while, I
There is a reason why conjunction (AND) is also called logical multiplication, and disjunction (OR) logical addition. There is not much difference between this and:
> There is a reason why conjunction (AND) is also called logical multiplication, and disjunction (OR) logical addition.
Really? The connection between AND and multiplication is obvious, but disjunction doesn't behave like addition. (For one thing, just like AND, it destroys information, which multiplication does do and addition doesn't.) Addition would usually be XOR.
In fact, AND and XOR are what you get when you apply multiplication and addition to the integers (mod 2).
Re:
by dvt on Sunday November 05, 2023 @07:56AM
(#38150734)
It's called logical addition which is different than arithmetic addition, so it's best to not conflate the two. It's called "addition" because you "add" another propostion, creating a disjunction. And by the rules of logic, as long as the original proposition was true, the truth value is preserved no matter how many propositions you introduce (or "add").
> It's called "addition" because you "add" another propostion, creating a disjunction.
By that argument, conjunction is also called "addition". Perhaps there's a different reason?
The choice of terminology, contrasting "logical addition" with "logical multiplication", obviously indicates that logical addition is supposed to be analogous to addition. What is the analogy?
Re:
by andai on Sunday November 05, 2023 @08:37AM
(#38151005)
I don't know if this relates to formal logic, but in C a true value is defined as not zero, so all nonzero values are treated as the same value. So the only way to get a false is to OR (or add) two zeroes, making the two operations equivalent as far as boolean logic goes.
Actually, you can add 1 and -1 to get 0, which breaks the model... Hmm...
Re:
by EGreg on Sunday November 05, 2023 @09:52AM
(#38151617)
The analogy is deep. But actually the relationship is as follows:
P(A u B) = P(A) + P(B) - P(A^B)
P(A ^ B) = P(A u B) - P(A) - P(B)
And to the guy who said xor is addition — no it’s not:
P(A xor B) = P(A) + P(B) - 2 * P(A^B)
In Probability, when you have a space of outcomes, doing a Union on two disjoint events (sets of outcomes) the probabilities add. Whereas doing an Intersection on two non-disjoint events those probabilities multiply. If two events have no outcomes in common, for instance, the probability of them both occurring is zero.
When you “condition” probability on an event X happening, you restrict the space of all outcomes to that set X, and intersect every event with it.
When you condiition B[n] = A[n] given that (A[n-1] AND … A[1]) having already happened, you have a sales funnel and each step is independent of the previous, so the probabilities multiply.
It is also called a Markov Chain if you have a discrete set of possibilities at each step so you can form a matrix.
Interesting, I never saw the arithmetic probability formula for xor.
Though the original topic was truth values, not probabilities. To make it short, conjunctions in probability are multiplicative when they are independent, and disjunctions in probability are additive when they are mutually exclusive.
Re:
by EGreg on Sunday November 05, 2023 @03:08PM
(#38155107)
Well I actually came up with it on the spot. Just think of the venn diagram, and add the areas of A and B, which counts A^B twice, and then just remove it twice. :)
The abs function works because it is equal to the maximum of x and -x.
The logical "and" and logical "or" functions are precisely particular cases of the minimum and maximum functions.
The so-called "exclusive or" function is simultaneously a particular case of addition and of subtraction, i.e. addition modulo 2 and subtraction modulo 2 are the same function. Therefore applying the 4 functions minimum, maximum, addition and subtraction to 1-bit numbers produces only 3 functions, AND, OR and XOR.
It makes no sense to search other functions for expressing logical "and" and logical "or" except for minimum and maximum, because this is what they are. The search could only find alternative more complicated ways to express minimum and maximum.
The minimum and maximum functions also work in logic systems with more than 2 values of truth. Also in hardware, analog circuits for minimum and maximum are one of the 2 ways of implementing the logical "and" and "or" functions, the other way being with series and parallel connections.
The main mistake of George Boole was his unsuccessful attempt to find an equivalence between logical "and" and "or" and integer multiplication and addition, because he was not habituated to use minimum and maximum, so he did not see the correct relationships.
The functions maximum and minimum are arithmetic functions as important as addition and subtraction. The only reason for which they have not been counted among the basic arithmetic functions is because when computed by a human they required much less effort than even addition, so there was no need for a long training of how to do computations with them, like for the more complex arithmetic functions.
Most modern instruction-set architectures now include maximum and minimum instructions, together with the addition and subtraction instructions.
As someone without much mathematical ropes, this might also be related to non-integer algebra and the distinction of "smooth" (edit: or differentiable) and discontinuous functions, no?
> The functions maximum and minimum are arithmetic functions as important as addition and subtraction. The only reason for which they have not been counted among the basic arithmetic functions is because when computed by a human they required much less effort than even addition
As I have mentioned, in the logic systems with 3 or more discrete values of truth (e.g. false, unknown and true, or false, unlikely, unknown, likely and true) and also in the logic systems with a continuous range of truth values (e.g. those based on probabilities), to the base logic functions AND, OR and NOT, correspond the base logic functions minimum, maximum and inversion a.k.a. reflection functions (the last may be e.g. integer negation when the truth values are symmetric around zero, or it may be e.g. 1-x when the truth values are between 0 and 1).
A set with minimum and maximum operations has a special algebraic structure named distributive lattice, which is a structure as useful and frequently encountered as structures like the algebraic groups and rings that characterize addition-like and multiplication-like operations.
It's interesting that maximum/minimum for disjunction/conjunction doesn't apply for probabilities, except when the probabilities "overlap as much as possible" (when at least one implies the other).
And when the probabilities overlap as little as possible, then the formulas are:
P(a or b) = min(1, P(A)+P(B))
P(a and b) = max(0, P(a)+P(b)-1)
I guess there is some lesson about logic and truth here, but I can't quite see it...
I really enjoyed reading this comment, thanks. I love the reason why min/max aren’t thought of as basic arithmetic building blocks (they’re too easy), it makes so much sense now you put it that way.
The maximum function isn't elementary arithmetic or modulo. You can define it as a limit, but people are unlikely to be convinced that that's a simple, elementary function.
By the sense of the word "elementary", the maximum and minimum functions are exactly as elementary as addition and subtraction.
Most textbooks define a set of elementary functions which does not include maximum and minimum only because they use "elementary" as an abbreviation for "elementary and differentiable", and they do not bother to explain this to the students.
Re:
by layer8 on Sunday November 05, 2023 @11:01AM
(#38152269)
OR still forms a semiring with AND, and that semiring operation is still called addition. OR is also like addition in saturation arithmetics. In those senses, it’s an additive operation, and it is indeed called logical addition:
Posting update since it's too late to edit: I confused multiplication with maximum. It's been too long since I took a course in formal logic. Though for binary values, multiplication works as well.
That's cool, but it's expected since CSS is Turing complete as long as you consider an appropriate accompanying HTML file and user interactions to be part of the execution.
As I said, not a tutorial. But I did want to introduce some interesting CSS "switches" and "booleans" that may become useful in some strange scenario in the future for you!
Re:
by JaDogg on Sunday November 05, 2023 @06:41AM
(#38150268)
This is scary good. Well done. If your product have any frontend and you have the power to decide ensure at least one of your hires can write CSS at this level.
A better test for someone using CSS would be activities that CSS is typically used for. Have them make a complex layout that is responsive at multiple display sizes, or have them architect a set of reusable classes for a design library or something. You may think that this sort of cleverness is generalizable: that someone who would come up with this solution would also be good at laying out web components. But there is no relationship between this (very neat) trick and CSS skills, so it would be a terrible test to give people.
If someone asked me to write CSS at this level for an interview I'd probably question the job, because if you have to write CSS at this level for work something is horribly wrong.
Re:
by yedava on Sunday November 05, 2023 @12:40PM
(#38153414)
When you get an idea like "I'm going to X with Y, where Y is not designed to do X", it probably takes several hours or a couple of days to implement it. Are you going to let your hire take that much time? And more importantly, are you going to pay the hire for that time, regardless of whether they succeed in doing it?
> This is scary good. Well done. If your product have any frontend and you have the power to decide ensure at least one of your hires can write CSS at this level.
I'm having trouble determening as written if this is a joke or not.
If anyone looked at this and thought something along the lines of "wow, think of the possibilities", I would would ask only that they don't put more thought into it. Other thoughts could bring pain ... to the browser that has to run it, the user staring at the page as the fans spin up, and the developer who has the pleasure to work with it next.
This was clearly one of those interesting-yet-useless things that was worth a read.
Re:
by JaDogg on Monday November 06, 2023 @03:45AM
(#38160135)
This wasn't a joke; I was being serious. However, this incident highlights how much I do not know. Anyway, thank you all for providing alternative viewpoints. I am willing to look like an idiot if it means I can gain knowledge from various people.
Might I suggest a different animation. The current one swaps two elements (bars) by vertically shrinking the taller bar and growing the shorter one. Instead, the bars should stay the same height but move horizontally. I would rather see values move than slots in the array morph to the new value they should hold. The current animation feels like you are pumping water from one tank to another.
The visualization you're suggesting would be orders of magnitude harder to create, since this one visualizes what actually happens in his algorithm and the better one would require an algorithm that tracks more things.
Since the steps are hardcoded in the CSS, this would better be named "Bubble sort visualization in pure CSS". I don't know what I was expecting honestly, but seeing all the sorting steps in the stylesheet wasn't it.
Pretty cool trick nevertheless, well done to the author!
> A sorting algorithm that checks if the array is sorted until a miracle occurs. It continually checks the array until it is sorted, never changing the order of the array.
It basically waits for god intervention / single event upset. It is guaranteed to be optimal in at least one quantum universe, though.
I stopped reading there.
Let us know when the second part is out. Why write half an article?
I learned from the article and found the subject interesting, but the breathlessly excited tone and forced cuteness were offputting.
Twice the page views? To "drive engagement"?
It is actually pretty good - the code uses 10 comparisons, while the optimal sorting network for 5 elements uses 9: https://bertdobbelaere.github.io/sorting_networks.html#N5L9D...
There is a reason why conjunction (AND) is also called logical multiplication, and disjunction (OR) logical addition. There is not much difference between this and:
This is often useful to think about when working on bitset or SIMD algorithms.Really? The connection between AND and multiplication is obvious, but disjunction doesn't behave like addition. (For one thing, just like AND, it destroys information, which multiplication does do and addition doesn't.) Addition would usually be XOR.
In fact, AND and XOR are what you get when you apply multiplication and addition to the integers (mod 2).
By that argument, conjunction is also called "addition". Perhaps there's a different reason?
The choice of terminology, contrasting "logical addition" with "logical multiplication", obviously indicates that logical addition is supposed to be analogous to addition. What is the analogy?
Actually, you can add 1 and -1 to get 0, which breaks the model... Hmm...
P(A u B) = P(A) + P(B) - P(A^B)
P(A ^ B) = P(A u B) - P(A) - P(B)
And to the guy who said xor is addition — no it’s not:
P(A xor B) = P(A) + P(B) - 2 * P(A^B)
In Probability, when you have a space of outcomes, doing a Union on two disjoint events (sets of outcomes) the probabilities add. Whereas doing an Intersection on two non-disjoint events those probabilities multiply. If two events have no outcomes in common, for instance, the probability of them both occurring is zero.
When you “condition” probability on an event X happening, you restrict the space of all outcomes to that set X, and intersect every event with it.
When you condiition B[n] = A[n] given that (A[n-1] AND … A[1]) having already happened, you have a sales funnel and each step is independent of the previous, so the probabilities multiply.
It is also called a Markov Chain if you have a discrete set of possibilities at each step so you can form a matrix.
Though the original topic was truth values, not probabilities. To make it short, conjunctions in probability are multiplicative when they are independent, and disjunctions in probability are additive when they are mutually exclusive.
A chain of conjunctions doesn't necessarily guarantee the truth value of the first proposition being preserved, so we have the much less interesting:
Yeah, I just noticed:
a and b = a + b - (a or b)
Or with probability
P(a and b) = P(a) + P(b) - P(a or b)
Meaning, using only elementary arithmetic and modulo?
Seems like there should be either an obvious answer or an interesting one :)
Then the answer seems to be no, right?
Edit: The abs() function seems to be enough, awesome:
https://math.stackexchange.com/a/1641271
The logical "and" and logical "or" functions are precisely particular cases of the minimum and maximum functions.
The so-called "exclusive or" function is simultaneously a particular case of addition and of subtraction, i.e. addition modulo 2 and subtraction modulo 2 are the same function. Therefore applying the 4 functions minimum, maximum, addition and subtraction to 1-bit numbers produces only 3 functions, AND, OR and XOR.
It makes no sense to search other functions for expressing logical "and" and logical "or" except for minimum and maximum, because this is what they are. The search could only find alternative more complicated ways to express minimum and maximum.
The minimum and maximum functions also work in logic systems with more than 2 values of truth. Also in hardware, analog circuits for minimum and maximum are one of the 2 ways of implementing the logical "and" and "or" functions, the other way being with series and parallel connections.
The main mistake of George Boole was his unsuccessful attempt to find an equivalence between logical "and" and "or" and integer multiplication and addition, because he was not habituated to use minimum and maximum, so he did not see the correct relationships.
The functions maximum and minimum are arithmetic functions as important as addition and subtraction. The only reason for which they have not been counted among the basic arithmetic functions is because when computed by a human they required much less effort than even addition, so there was no need for a long training of how to do computations with them, like for the more complex arithmetic functions.
Most modern instruction-set architectures now include maximum and minimum instructions, together with the addition and subtraction instructions.
As someone without much mathematical ropes, this might also be related to non-integer algebra and the distinction of "smooth" (edit: or differentiable) and discontinuous functions, no?
> The functions maximum and minimum are arithmetic functions as important as addition and subtraction. The only reason for which they have not been counted among the basic arithmetic functions is because when computed by a human they required much less effort than even addition
A set with minimum and maximum operations has a special algebraic structure named distributive lattice, which is a structure as useful and frequently encountered as structures like the algebraic groups and rings that characterize addition-like and multiplication-like operations.
And when the probabilities overlap as little as possible, then the formulas are:
P(a or b) = min(1, P(A)+P(B))
P(a and b) = max(0, P(a)+P(b)-1)
I guess there is some lesson about logic and truth here, but I can't quite see it...
Most textbooks define a set of elementary functions which does not include maximum and minimum only because they use "elementary" as an abbreviation for "elementary and differentiable", and they do not bother to explain this to the students.
a or b = a + b - (a * b)
The last term is equivalent to a conjunction. Which also makes sense when you think of probability:
P(a or b) = P(a) + P(b) - P(a and b)
I've seen this formula I think, and for mod 2 addition, there is no difference I guess.
https://en.wikipedia.org/wiki/Logical_disjunction
https://mathworld.wolfram.com/deMorgansDualityLaw.html
*Obviously, abusing CSS isn’t everyone’s idea of fun.
I'm having trouble determening as written if this is a joke or not.
If anyone looked at this and thought something along the lines of "wow, think of the possibilities", I would would ask only that they don't put more thought into it. Other thoughts could bring pain ... to the browser that has to run it, the user staring at the page as the fans spin up, and the developer who has the pleasure to work with it next.
This was clearly one of those interesting-yet-useless things that was worth a read.
https://news.ycombinator.com/item?id=38151018
Might I suggest a different animation. The current one swaps two elements (bars) by vertically shrinking the taller bar and growing the shorter one. Instead, the bars should stay the same height but move horizontally. I would rather see values move than slots in the array morph to the new value they should hold. The current animation feels like you are pumping water from one tank to another.
Pretty cool trick nevertheless, well done to the author!
> A sorting algorithm that checks if the array is sorted until a miracle occurs. It continually checks the array until it is sorted, never changing the order of the array.
It basically waits for god intervention / single event upset. It is guaranteed to be optimal in at least one quantum universe, though.
Welcome to our phone screen at ${faang}. For the warmup question, let's solve this ${lc_hard} together in css.