I recently asked an AI a chemistry question which may have an extremely obvious answer. I never studied chemistry so I can't tell you if it was. I included as much information about the situation I found myself in as I could in the prompt. I wouldn't be surprised if the ai's response was based on the detail that's normally important but didn't apply to the situation, just like the 50 meters
If you're curious or actually knowledgeable about chemistry, here's what happened.
My apartment's dishwasher has gaps in the enamel from which rust can drip onto plates and silverware. I tried soaking but I presume to be a stainless steel knife with a drip of rust on it in citric acid. The rust turned black and the water turned a dark but translucent blue/purple.
I know nothing about chemistry. My smartest move was to not provide the color and ask what the color might have been. It never guessed blue or purple.
In fact, it first asked me if this was highschool or graduate chemistry. That's not... and it makes me think I'll only get answers to problems that are easily graded, and therefore have only one unambiguous solution
I'm a little confused by your question myself. Stainless steel rust should be that same brown color. Though it can get very dark when dried. Blue is weird but purple isn't an uncommon description, assuming everything is still dark and there's lots of sediment.
But what's the question? Are you trying to fix it? Just determine what's rusting?
Thanks, Excellent catch! Everyone is saying this is a "brain teaser." However, this reminded me of the LLM that thought it was the golden gate bridge. I hadn't been able to say it (or think it) succinctly. From Claude, "when we turn up the strength of the “Golden Gate Bridge” feature, Claude’s responses begin to focus on the Golden Gate Bridge. Its replies to most queries start to mention the Golden Gate Bridge, even if it’s not directly relevant." Here's the link for those interested. https://www.anthropic.com/news/golden-gate-claude
To be pedantic, median is cheaper than sorting. O(n) with a quicksort-like algorithm.
Also, if you're taking an average of floating point numbers, you might want to sort it first and add from smallest to largest, to better preserve precision
An aside, but I recently learned -- if one is willing to use a very modest amount of memory -- summing floating-point numbers with no loss of precision is effectively a solved problem with the XSUM algorithm.
That paper explains some useful optimisation details, but obviously since the floats are all (either infinity or) some multiple of a known tiny fraction (their smallest non-zero number), we can definitely sum them accurately.
I think you either haven't thought about this or you did your math wrong.
You need (2^e)+m+1 bits. That is more bits than would fit in the cheap machine integer type you just have lying around, but it's not that many in real terms.
Let's do a tiny one to see though first, the "half-precision" or f16 type, 5 bits of exponent, 10 bits of fraction, 1 sign bit. We need 43 bits. This will actually fit in the 64-bit signed integer type on a modern CPU.
Now lets try f64, the big daddy, 11 exponent, 52 fraction, 1 sign bit so total 2048 + 52 + 1 = 2101 bits. As I said it doesn't fit in our machine integer types but it's much smaller than a kilobyte of RAM.
Edited: I can't count, though it doesn't make a huge difference.
You also need some extra bits at the top so that it doesn't overflow (e.g., on an adversarial input filled with copies of the max finite value, followed by just as many copies of its negation, so that it sums to 0). The exact number will depend on the maximum input length, but for arrays stored in addressible memory it will add up to no more than 64 or so.
Thanks, you're right there for accumulating excess, I don't think you can actually get to 64 extra bits but sure, lets say 64 extra bits if you want a general purpose in memory algorithm, it's not cheap but we shouldn't be surprised it can be done.
My [interpretation? fanfic?] is that Julia is like a carnivore, and humanity is not it's first prey. Every creature that eats, eats to steal the disentropy of it's meal. Plants can steal order from sunlight, and certain microbes can steal order from thermal vents, but carnivores, herbivores, and decomposers steal order from the work of other organisms. The improbability of living is sustained by arranging stolen amino acids into one's own proteins, powered by the toppleing of sugar towers back into a jumbled mess.
Julia does not reassemble amino acids like earth life does. But it does absorb disentropy from it's prey. The extreme specificity of an interstellar spacecraft, it's contents and occupants, is absorbed by Julia, so that it can move, grow, and attract more prey.
Decidability is of academic interest, and might be a hint if something is feasible.
But there are (1) ways of sidestepping undecidability, e.g. A valid C++/Rust program is one for which the typechecker terminates in x steps without overflowing the stack
And (2) things which are decidable, but physically impossible to calculate, e.g the last digit of the 10^10^10 th prime
What matters is being able to reject all incorrect programs, and accept most human written valid programs
Decidability of a type system is like well-typedness of a program. It doesn't guarantee it's sensible, but not having the property is an indicator of problems.
I'm not entirely smart enough to connect all of these things together but I think there is a kind of subtlety here thats being stepped on.
1. Complete, Decidable, Well founded are all distinct things.
2. Zig (which allows types to be types) is Turing complete at compile time regardless. So the compiler isn't guaranteed to halt regardless and it doesn't practically matter.
3. The existance of a set x contains x is not enough by itself to create a paradox and prove false. All it does is violate the axiom of foundation, not create a russles paradox.
4. The axiom of foundation is a weird sort of arbitrariness in that it implies this sort of DAG nature to all sets under set membership operation.
6. The Axiom of Foundation exists to stop you from making weird cycles, but there is parallel to the axiom of choice, which directly asserts the existance of non computable sets using a non algorithmicly realizable oracle anyway....
Your other points are more relevant to the content of the article, but point 2. relates the practical consequences of undecidable type-checking, so I'll reply to that.
I don't have a problem with compile time code execution potentially not terminating, since it's clear to the programmer why that may happen. However, conventional type checking/inference is more like solving a system of constraints, and the programmer should understand what the constraints mean, but not need to know how the constraint solver (type checker) operates. If it's undecidable, that means there is a program that a programmer knows should type check, but the implementation won't be happy with; ruining the programmer's blissful ignorance of the internals.
> 2. Zig (which allows types to be types) is Turing complete at compile time regardless. So the compiler isn't guaranteed to halt regardless and it doesn't practically matter.
Being Turing complete at compile time causes the same kinds of problems as undecidable typechecking, sure. That doesn't make either of those things a good idea.
> 3. The existance of a set x contains x is not enough by itself to create a paradox and prove false. All it does is violate the axiom of foundation, not create a russles paradox.
A set that violates an axiom is immediately a paradox from which you can prove anything. See the principle of explosion.
> 4. The axiom of foundation is a weird sort of arbitrariness in that it implies this sort of DAG nature to all sets under set membership operation.
Well, sure, that's what a set is. I don't think it's weird; quite the opposite,
> 5. This isn't nessesarily some axiomatically self evident fact. Aczel's anti foundation axiom works as well and you can make arbitrary sets with weird memberships if you adopt that.
I don't think this kind of thing is established enough to say that it works well. There aren't enough people working on those non-standard axioms and theories to conclude that they're practical or meet our intuitions.
> 6. The Axiom of Foundation exists to stop you from making weird cycles, but there is parallel to the axiom of choice, which directly asserts the existance of non computable sets using a non algorithmicly realizable oracle anyway....
The Axiom of Foundation exists to make induction work, and so does the Axiom of Choice. They both express a sense that if you can start and you can always make progress, eventually you can finish. It's very hard to prove general results without them.
But like, of all the
expressive power vs analyzability
trade-offs you can make,
there's a huge leap in expressive power
when you give away decidability.
Undecidability is not a sign
that the foundation has cracks
(not well founded),
but it might be a sign
that you put the foundation on wheels
so you can drive it at highway speeds,
with all the dangers that entails.
It's not a trade everyone would make,
but the languages I prefer do.
jukebox? jeux is found by my solver, although I'm not sure Sam would include it. I'm working on a site for solving Spelling Bee, Letter Boxed, Strands, etc.
What was your algorithm? Compute a bitset for every word, for each word with 7 unique letters, check against every other word if it has a subset of those letters? Surely there's a better than O(n^2) way
Start with your chosen dictionary, remove small words and all 's'es.
Then find all words with 7 unique letters, then get the unique set of letter sets. These are your valid puzzles times 7 for each selection of a center letter.
Construct a trie of your dictionary. For each letter of each puzzle, walk the trie using only the puzzle letters. When you find a word, if it used the center letter, add to list.
~I watched a video series where someone did the techniques in isolation in reverse order and I thought it was a good idea. That way you're always learning on perfect ingredients. i.e. buy chips and learn to temper, then buy roasted nibs and learn to refine, then buy dried and learn to roast, then buy pods and learn to ferment and dry.~
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