there ought to be a special section on HN entitled "things that will make you feel thoroughly inadequate".
All elementary functions from a single binary operator
121–130 of 317 posts
Re: All elementary functions from a single binary operator
#122Earlier quoted context omitted.
> Hypergeometric functions are functions with 4 parameters. Granted, but the claim in the abstract says: >> computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations And I don't see how this is true as to hypergeometric functions in a way that isn't shared by the approach in the paper. > Finding a binary operation that can do this, like in TFA, is far more difficu…
There is an essential difference between binary functions and unary functions. With binary functions you can compose them using a very complex composition graph. With unary functions you can compose them only linearly, so in general it is impossible to make a binary function with unary functions. You can make binary functions from unary functions only by using at least one other binary function. For instance, you can…
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Re: All elementary functions from a single binary operator
#123Earlier quoted context omitted.
There is an essential difference between binary functions and unary functions. With binary functions you can compose them using a very complex composition graph. With unary functions you can compose them only linearly, so in general it is impossible to make a binary function with unary functions. You can make binary functions from unary functions only by using at least one other binary function. For instance, you can…
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I think that you may have replied before I saved my entire response, so I am not sure how much of it you had read before replying yourself.
I have replied to your last statement:
> "you can use the second parameter of a binary function to identify a unary function just as you can use the fourth parameter of a quaternary function to identify a trinary one."
As I have explained above, what you propose does not work. It works in functions with 3 or more parameters, but it does not work in binary functions, because you cannot make binary functions from unary functions (without using some auxiliary binary functions).
Re: All elementary functions from a single binary operator
#124Earlier quoted context omitted.
Don't know adding, but multiplication has diagram on the last page of the PDF. xy = eml(eml(1, eml(eml(eml(eml(1, eml(eml(1, eml(1, x)), 1)), eml(1, eml(eml(1, eml(y, 1)), 1))), 1), 1)), 1) From Table 4, I think addition is slightly more complicated?
x+y = ln(exp(x) * exp(y)) exp(a) = eml(a, 1) ln(a)=eml(1,eml(eml(1,a),1)) Plugging those in is an excercise to the reader
You use multiplication in the first line, which you have not expressed through eml yet.
Because of how exp and log turn addition into multiplication and vice versa, once you have the one, you get the other easily.
Re: All elementary functions from a single binary operator
#125So, like brainf*ck (the esoteric programming language), but for maths?
But even tighter. With eml and 1 you could encode a funtion in rpn as bits. Although you also need to encode where to put the input. The real question is what emoji to use for eml when written out.
In rpn notation you just put the input on the stack, right? The encodings seems like they could get pretty big, and encodings certainly wouldn't be unique, but you should be able to encode pretty much any constant you could think of.
Re: All elementary functions from a single binary operator
#126Re: All elementary functions from a single binary operator
#127Earlier quoted context omitted.
eml(1,eml(x,1)) = e + x and eml(eml(1,x),1) = e^e * x
Okay, I’m tired. Not quite inverse but per the title , must be a way.
I was mistaken above in the first identity, it is
eml(1,eml(x,1)) = e - x
Which then if you iterate gives x (ie is inverse of itself).
eml(1,eml(eml(1,eml(x,1)),1)) = x
Re: All elementary functions from a single binary operator
#128For completeness, there is also Peirce’s arrow aka NOR operation which is functionally complete. Fun applications iirc VMProtect copy protection system has an internal VM based on NOR. Quick google seach brings up https://github.com/pr701/nor_vm_core , which has a basic idea
That’s boolean functional completeness, which is kind of a trivial result (NAND, NOR). It mirrors this one insofar as the EDL operator is also a combination of a computation and a negation in the widest senses.
Re: All elementary functions from a single binary operator
#129This makes a good benchmark LLMs: ``` look at this paper: https://arxiv.org/pdf/2603.21852 now please produce 2x+y as a composition on EMLs ``` Opus(paid) - claimed that "2" is circular. Once I told it that ChatGPT have already done this, finished successfully. ChatGPT(free) - did it from the first try. Grok - produced estimation of the depth of the formula. Gemini - success Deepseek - Assumed some pre-existing knowl…
> Once I told it that ChatGPT have already done this, finished successfully. TIL you can taunt LLMs. I guess they exhibit more competitive spirit than I thought.
Opus seems to be wired currently to get you to spend more money. Once you tell it "Stop defrauding me, just get to the right solution" it often gets it.
Re: All elementary functions from a single binary operator
#130This is amazing! I love seeing FRACTRAN-shaped things on the homepage :) This reminds me of how 1-bit stacks are encoded in binary: A stack of zeros and ones can be encoded in a single number by keeping with bit-shifting and incrementing. Pushing a 0 onto the stack is equivalent to doubling the number. Pushing a 1 is equivalent to doubling and adding 1. Popping is equivalent to dividing by 2, where the remainder is t…
Did you just explain base-2 numbers on the HN forums as if it were novel?